]> No Title

Decision problems in Algebra and analogues of Hilbert's tenth problem

A tutorial presented at

American Institute of Mathematics

and

Newton Institute of Mathematical Sciences

Thanases Pheidas University of Crete and

Karim Zahidi

University of Antwerp

Contents:

1. Part $\mathrm{A}$: Decidability results

la. Presburger arithmetic

lb. Addition and divisibility

lc. Addition and exponentiation

ld. The analogue of Hilbert's tenth problem for algebraic groups

le. The results of Ax for (almost all primes'

lf. Model-completeness

$\mathrm{l}\mathrm{g}$. Complete theories

lh. Power-series and germs of analytic functions: existential decidability and Artin approximation

li. Positive-existential versus existential

2. Part $\mathrm{B}$: A qualitative analogue of HTP and the Conjectures of S. Lang

$\mathrm{2}\mathrm{a}$. A language for geometric problems

$\mathrm{2}\mathrm{b}$. A geometric problem

$\mathrm{2}\mathrm{c}$. Varieties with an infinite number of points over a ring

3. Part $\mathrm{C}$: Hilbert's tenth problem for analytic and meromorphic functions

4. Part $\mathrm{D}$: Comments on the analogue of Hilbert's tenth problem for $\mathbb{Q}$

Introduction

One of the first tasks undertaken by Model Theory was to produce elimination results, for example methods of eliminating quantifiers in formulas of certain structures. In almost all cases those methods have been effective and thus provide algorithms for

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examining the truth of possible theorems. On the other hand, Gödel's Incompleteness Theorem and many subsequent results show that in certain structures constructive elimination is impossible. The current article is $\mathrm{a}$ (very incomplete) effort to survey some results of each kind, with a focus on the decidability of existential theories, and ask some questions in the intersection of Logic and Number Theory. It has been written having in mind a mathematician without prior exposition to Model Theory. Our presentation will consist of four parts.

Part A deals with positive (decidability) results for analogues of Hilbert's tenth problem for substructures of the integers and for certain local rings.

Part $\mathrm{B}$ focuses on the 'parametric problem' and the relevance of Hilbert's tenth problem to conjectures of Lang.

Part $\mathrm{C}$ deals with the analogue of Hilbert's tenth problem for rings of Analytic and Meromorphic functions.

Part $\mathrm{D}$ is an informal discussion on the chances of proving a negative (or could it be positive?) answer to the analogue of Hilbert's tenth problem for the field of rational numbers.

A central undecidability result in our presentation will be Hilbert's 10th problem, which asked:

Give a procedure which, in a finite number of steps, can determine whether a polyno- mial equation (in several variables) with integer coefficients has or does not have integer solutions.

The answer by Matiyasevich ([39]), following work of Davis, Putnam and J. Robin- son, was negative ${\mathrm{\left(}}^{\mathrm{\left(}}\mathrm{n}\mathrm{o}$ such algorithm can exist').

Analogous questions can be asked for domains other than the ring of integers. In trying to ask questions, ""similar“ to Hillbert's tenth problem (from now on denoted H10) in rings other than the rational integers, one has to specify the kind of 'diophantine equations' one considers. For example, say that we want to ask H10 for the ring of polynomials $\mathbb{C}\mathrm{\left[}\mathit{z}\mathrm{\right]}$ in one variable, $\mathit{z}$, with complex coefficients. Let ${D}$ be a class of 'diophantine equations“ over $\mathbb{C}\mathrm{\left[}\mathit{z}\mathrm{\right]}$, that is, polynomial equations in many variables, with coefficients in $\mathbb{C}\mathrm{\left[}\mathit{z}\mathrm{\right]}$. The analogue of H10 for this class is ""does there exist an algorithm to decide, given any equation in ${D}$, whether that equation has or does not have solutions in $\mathbb{C}\mathrm{\left[}\mathit{z}\mathrm{\right]}\mathrm{?}$“. It is obvious that if one chooses ${D}$ to be the set of all 'diophantine equations' over $\mathbb{C}\mathrm{\left[}\mathit{z}\mathrm{\right]}$ then the answer to the question is NO, simply because ${D}$ is uncountable (algorithms, in the classical sense, treat only countable problems). On the other hand, one may take ${D}$ to be the set of 'diophantine equations' which have coefficients in $\mathbb{Z}\mathrm{\left[}\mathit{z}\mathrm{\right]}$ or in $\mathbb{Z}\mathrm{\left[}\mathit{i}\mathrm{\right]}\mathrm{\left[}\mathit{z}\mathrm{\right]}\mathrm{\left(}\mathit{i}\mathrm{=}\sqrt{\mathrm{-}\mathrm{1}}\mathrm{\right)}$ or in $\mathbb{Z}$. Each of these choices gives a different 'analogue of H10 for $\mathbb{C}\mathrm{\left[}\mathit{z}\mathrm{\right]}\mathrm{\text{'}}$. In this paper we will be specifying ${D}$ by specifying the language in which we work. Notice that the analogue of H10 for $\mathbb{C}\mathrm{\left[}\mathit{z}\mathrm{\right]}$ for the class ${D}$, asked for systems of diophantine equations (rather than single equations), is really the question of decidability of the positive-existential theory of $\mathbb{C}\mathrm{\left[}\mathit{z}\mathrm{\right]}$ in the language $\mathit{L}$ which contains symbols for addition, multiplication, equality, and constant-symbols for the coefficients of the equations of ${D}$ (or constant-symbols whose interpretations generate exactly the

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set of coefficients of equations of ${D}$). For example, the analogue of H10 (for systems of equations) for $\mathbb{C}\mathrm{\left[}\mathit{z}\mathrm{\right]}$ with ${D}$ the set of equations with coefficients in $\mathbb{Z}$ is equivalent to the question of decidability of the positive-existential theory of $\mathbb{C}\mathrm{\left[}\mathit{z}\mathrm{\right]}$ in the language ${\mathit{L}}_{\mathit{r}}\mathrm{=}\mathrm{\left\{}\mathrm{+}\mathrm{,}\mathrm{}\mathrm{\cdot }\mathrm{;}\mathrm{=}\mathrm{;}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right\}}$, while that with ${D}$ the set of equations with coefficients in $\mathbb{Z}\mathrm{\left[}\mathit{z}\mathrm{\right]}$ is equivalent to the question of decidability of the positive-existential theory of $\mathbb{C}\mathrm{\left[}\mathit{z}\mathrm{\right]}$ in the language ${\mathit{L}}_{\mathit{z}}\mathrm{=}\mathrm{\left\{}\mathrm{+}\mathrm{,}\mathrm{}\mathrm{\cdot }\mathrm{;}\mathrm{=}\mathrm{;}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{,}\mathrm{}\mathit{z}\mathrm{\right\}}\mathrm{.}$

We will consider structures (models) such as the field $\mathbb{C}\mathrm{\left(}\mathit{z}\mathrm{\right)}$ . Each structure comes with a language, i.e. a set of symbols for the relations, functions and distinguished elements of the structure. For example we consider $\mathbb{C}\mathrm{\left(}\mathit{z}\mathrm{\right)}$ as an ${\mathit{L}}_{\mathit{z}}$-structure, with symbol for the relations $\mathrm{=}$, the functions $\mathrm{+}$(addition) and . (multiplication) and the distinguished elements $\mathrm{0}\mathrm{,}\mathrm{1}$ and $\mathit{z}$. The first order sentences of the language of the structure are the sentences built using the symbols of the language, variables ranging over th universe of the structure, quantifiers ( $\mathrm{\exists }$ and $\mathrm{\forall }$) and logical connectives, by the usual rules. The existential (resp. positive-existential) sentences are those that start with existential quatnifiers which are followed by a quantifier-free formula (resp. by a quantifier-free formula which is a disjunction of conjunctions of relations - negations of relations are not allowed in this case). The (full) theory (resp. existential theory, positive-existential theory) of the structure is the set of sentences (resp. existential sentences, positive-existential sentences) which are true in the structure.

We say that the theory (resp. existential theory, positive-existential theory) of a structure is decidable if there exists an algorithm which determines whether any given sentence (resp. existential sentence, positive-existential sentence) is true or false in the structure-otherwise we say that the theory is undecidable.

We present a list of decidability properties of some structures of common use. The first three lines contain substructures of the ring of rational integers, $\mathrm{\left(}\mathbb{Z}\mathrm{,}\mathrm{}\mathrm{+}\mathrm{,}\mathrm{}\mathit{n}\mathrm{\to }{\mathrm{2}}^{\mathit{n}}\mathrm{,}\mathrm{}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$ is the structure of $\mathbb{Z}$ with additon and the function $\mathit{n}\mathrm{\to }{\mathrm{2}}^{\mathit{n}}$ and $\mathrm{\left(}\mathbb{Z}\mathrm{,}\mathrm{}\mathrm{+}\mathrm{,}\mathrm{}\mathrm{|}\mathrm{,}\mathrm{}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$ the struc- ture of $\mathbb{Z}$ with addition and divisibility; the fourth line is the structure of addition and divisibility in a ring ${{O}}_{\mathit{K}}$ of integers of the number field $\mathit{K}\mathrm{-}$ which is assumed not to be imaginary quadratic; the positive-existential theories of those strucrtures have been shown to be of the same hardness as the positive-existential thories of the correspond- ing ring structures, but it is an open problem whether the latter are undecidable for arbitrary $\mathit{K}$ (see the comment below). In the first four lines the columns (ex. th.' and (full th.' show the decidability properties of the existential and the full theory of the structure, respectively. The remaining structures are rings. $\mathbb{Q}$ is the field of ratiuonal numbers, $\mathbb{R}$ the feld of real numbers, $\mathbb{C}$ the field of complex numbers, ${\mathrm{F}}_{\mathit{q}}$ is the finite field with $\mathit{q}$ elements, $\mathit{B}\mathrm{\left[}\mathit{z}\mathrm{\right]}$ the ring of polynomials in the variable $\mathit{z}$ with coefficients in the ring $\mathit{B}\mathrm{,}$ $\mathit{B}\mathrm{\left(}\mathit{z}\mathrm{\right)}$ the corresponding field of rational functions in $\mathit{z}\mathrm{,}$ ${H}\mathrm{\left(}\mathit{D}\mathrm{\right)}$ the ring of analytic functions of the variable $\mathit{z}$ as that ranges in an open superset of the subset $\mathit{D}$ in the complex plane, $\mathrm{M}\mathrm{\left(}\mathit{D}\mathrm{\right)}$ is the corresponding field of meromorphic functions, $\mathit{U}$ is the open unit disk. ${\mathit{L}}_{\mathit{T}}$ is the language $\mathrm{\left\{}\mathrm{+}\mathrm{,}\mathrm{}\mathrm{\cdot }\mathrm{;}\mathrm{=}\mathrm{;}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right\}}$ which, for rings of functions is augmented by the predicate $\mathit{T}$ which is interpreted as $\mathrm{\left(}\mathit{x}$ is not a constant function'. For rings of functions of the variable $\mathit{z}$ the language ${\mathit{L}}_{\mathit{z}}$ is as above. The first column

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shows whether the positive existential theory of the ring in the language ${\mathit{L}}_{\mathit{T}}$ is decidable or not ${\mathrm{\left(}}^{\mathrm{\left(}}\mathrm{Y}\mathrm{\text{'}}$ means decidable, 'N' means undecidable, 'conj. $\mathrm{N}$' means 'conjectured to be undecidable', (? denotes an open problem), the second column corresponds to the similar properties in the language ${\mathit{L}}_{\mathit{z}}$ and the third column to that of the full theory in the language ${\mathit{L}}_{\mathit{T}}$ for the rings $\mathbb{Z}\mathrm{,}$ ${{O}}_{\mathit{K}}$ and $\mathbb{Q}$ and the language ${\mathit{L}}_{\mathit{z}}$ for the remaining rings.

Note that it is known that the theories of many rings ${{O}}_{\mathit{K}}$ are undecidable (e.g. for abelian $\mathit{K}$) and it has been conjectured that all of them are, but the question for arbitrary $\mathit{K}$ remains open.

 ex. th. (in ${\mathit{L}}_{\mathit{T}}$) ex. th. in ${\mathit{L}}_{\mathit{z}}$ full th. $\mathrm{\left(}\mathbb{Z}\mathrm{,}\mathrm{}\mathrm{+}\mathrm{,}\mathrm{}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$ $\mathrm{Y}$ $\mathrm{Y}$ $\mathrm{\left(}\mathrm{,}\mathrm{}\mathrm{+}\mathrm{,}\mathrm{}\mathit{n}\mathrm{\to }{\mathrm{2}}^{\mathit{n}}\mathrm{,}\mathrm{}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$ $\mathrm{Y}$ $\mathrm{Y}$ $\mathrm{\left(}\mathbb{Z}\mathrm{,}\mathrm{}\mathrm{+}\mathrm{,}\mathrm{}\mathrm{|}\mathrm{,}\mathrm{}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$ $\mathrm{Y}$ $\mathrm{N}$ $\mathrm{\left(}{{O}}_{\mathit{K}}\mathrm{,}\mathrm{}\mathrm{+}\mathrm{,}\mathrm{}\mathrm{|}\mathrm{,}\mathrm{}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right)}$ conj. $\mathrm{N}$ $\mathrm{N}$ $\mathbb{Z}$ $\mathrm{N}$ $\mathrm{N}$ ${{O}}_{\mathit{K}}$ conj. $\mathrm{N}$ $\mathrm{N}$ $\mathbb{Q}$ $\mathrm{?}$ $\mathrm{N}$ ${\mathrm{F}}_{\mathit{q}}\mathrm{\left[}\mathit{z}\mathrm{\right]}\mathrm{,}\mathrm{}\mathbb{R}\mathrm{\left[}\mathit{z}\mathrm{\right]}\mathrm{,}\mathrm{}\mathbb{C}\mathrm{\left[}\mathit{z}\mathrm{\right]}$ $\mathrm{N}$ $\mathrm{N}$ $\mathrm{N}$ ${\mathrm{F}}_{\mathit{q}}\mathrm{\left(}\mathit{z}\mathrm{\right)}$ $\mathrm{?}$ $\mathrm{N}$ $\mathrm{N}$ $\mathbb{R}\mathrm{\left(}\mathit{z}\mathrm{\right)}$ $\mathrm{?}$ $\mathrm{N}$ $\mathrm{N}$ $\mathbb{C}\mathrm{\left(}\mathit{z}\mathrm{\right)}$ $\mathrm{?}$ $\mathrm{?}$ $\mathrm{?}$ ${H}\mathrm{\left(}\mathrm{\left\{}\mathit{a}\mathrm{\right\}}\mathrm{\right)}$ $\mathrm{Y}$ $\mathrm{Y}$ $\mathrm{Y}$ ${H}\mathrm{\left(}{U}\mathrm{\right)}$ $\mathrm{Y}$ $\mathrm{?}$ $\mathrm{N}$ ${H}\mathrm{\left(}\mathbb{C}\mathrm{\right)}$ $\mathrm{?}$ $\mathrm{?}$ $\mathrm{N}$ $\mathrm{M}\mathrm{\left(}{U}\mathrm{\right)}$ $\mathrm{Y}$ $\mathrm{?}$ $\mathrm{?}$ $\mathrm{M}\mathrm{\left(}\mathbb{C}\mathrm{\right)}$ $\mathrm{?}$ $\mathrm{?}$ $\mathrm{?}$

For a fast introduction to applications of Model Theory to Algebra the reader may consult [7] and [61]. The solution of H10 can be found in [13], and is explained very nicely to the non-expert in [12]. Surveys of questions similar to the present paper's are [43], [47], [48] and [55]. Surveys of elimination ('decidability') techniques and results can be found in [56] (and many later more specialized articles, from the Algebraist's point of view).

We are indebted to F. Campana, J. Demeyer and T. Scanlon for various comments towards improvements of this paper.

1 Part $\mathrm{A}$: Decidability results

For a quick introduction to a basic elimination technique, whose origin lies in the early days of Model Theory, solve the following Exercise.

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Say that we work in a language $\mathit{L}$. A theory $\mathit{T}$ (i.e. a subset of the set of formulas) of $\mathit{L}$ admits elimination of quantifiers if any $\mathit{L}$-formula $\mathit{\phi }\mathrm{\left(}\mathit{x}\mathrm{\right)}$ is equivalent in $\mathit{T}$ to an existential formula.

Exercise 1.1 (a) Let $\mathit{T}$ be an $\mathit{L}$-theory. Assume that any formula of the form $\mathrm{\exists }\mathit{y}\mathit{\psi }\mathrm{\left(}\mathit{x}\mathrm{,}\mathrm{}\mathit{y}\mathrm{\right)}$ , where $\mathit{y}$ is one variable, $\mathit{x}$ is a tuple of variables and $\mathit{\psi }$ is quantifier-free, is equivalent in $\mathit{T}$ to a quantifier-free formula. Prove that $\mathit{T}$ admits elimination of quantifiers.

(b) Consider the field $\mathbb{C}$ of complex numbers as an ${\mathit{L}}_{\mathit{r}}$-structure. Consider a system of polynomial equations

$\mathit{S}\mathrm{\left(}\mathit{x}\mathrm{,}\mathrm{}\mathit{y}\mathrm{\right)}\mathrm{:}{\wedge }_{\mathit{i}}{\mathit{f}}_{\mathit{i}}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\wedge }{\wedge }_{\mathit{j}}{\mathit{g}}_{\mathit{j}}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\ne }\mathrm{0}$

where $\mathit{x}$ is one variable, $\mathit{y}\mathrm{=}\mathrm{\left(}{\mathit{y}}_{\mathrm{1}}\mathrm{,}\mathrm{}\mathrm{\text{...}}\mathrm{,}\mathrm{}{\mathit{y}}_{\mathit{m}}\mathrm{\right)}$ and each ${\mathit{y}}_{\mathit{k}}$ is a variable, the indices $\mathit{i}$ and $\mathit{j}$ range in some finites sets I and $\mathit{J}$ respectively, and each ${\mathit{f}}_{\mathit{i}}$ and ${\mathit{g}}_{\mathit{i}}$ is a polynomial in $\mathbb{Z}\mathrm{\left[}\mathit{x}\mathrm{,}\mathrm{}{\mathit{y}}_{\mathrm{1}}\mathrm{,}\mathrm{}\mathrm{\text{...}}\mathrm{,}\mathrm{}{\mathit{y}}_{\mathit{m}}\mathrm{\right]}$. Prove that the formula $\mathrm{\exists }\mathit{x}\mathit{S}\mathrm{\left(}\mathit{x}\mathrm{,}\mathrm{}\mathit{y}\mathrm{\right)}$ is equivalent to a quantifier-free formula.

[Hint: Use the theory of resultants (see any graduate book in Algebra) to do this in the case that $\mathit{J}\mathrm{=}\mathrm{\varnothing }$, then do it in the case that both I and $\mathit{J}$ are singletons.]

Conclude that the theory of $\mathbb{C}$ admits elimination of quantifiers. Observe that the procedure of finding a quantifier-free formula, equivalent to a given formula, is effective. Conclude that the theory of $\mathbb{C}$ is decidable, that is, there is an algorithm to determine whether any given sentence of ${\mathit{L}}_{\mathit{r}}$ is true or false in $\mathbb{C}\mathrm{.}$

1.1 Presburger arithmetic

A basic, old result is the decidability of Presburger arithmetic, i.e. the theory of the ordered additive group of integers. Presburger showed that $\mathrm{Z}$ admits elimination of quantifiers in a language which extends the language ${\mathit{L}}_{\mathit{P}}\mathrm{=}\mathrm{\left\{}\mathrm{+}\mathrm{,}\mathrm{}\mathrm{\ge }\mathrm{;}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right\}}$ with predicates for $\mathit{a}\mathrm{\equiv }\mathit{b}\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{}\mathit{m}$ for every integer $\mathit{m}$ (actually Presburger considered the theory of natural numbers, not the integers, and in a language slighly different from ${\mathit{L}}_{\mathit{P}}$: instead $\mathrm{o}\mathrm{f}\mathrm{+}\mathrm{i}\mathrm{t}$ contained a function-symbol for the 'successor function' $\mathit{S}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\mathit{x}\mathrm{+}\mathrm{1}$; but the expressive powers of these two languages are obviously the same).

Exercise 1.2 Show that the theory of $\mathbb{Z}$ in the language ${\mathit{L}}_{\mathit{P}}$ is $\mathit{\zeta }\mathit{m}\mathit{o}\mathit{d}\mathit{e}\mathit{l}$-complete' in the following way:

(a) For each $\mathit{k}\mathrm{\in }\mathbb{N}$ with $\mathit{k}\mathrm{\ge }\mathrm{2}$ define the one-place predicate ${\mathit{M}}_{\mathit{k}}$ to mean $\mathit{\zeta }{\mathit{M}}_{\mathit{k}}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{↔}$ $\mathit{x}$ is a multiple of $\mathit{k}\mathrm{\text{'}}$. Extend the language ${\mathit{L}}_{\mathit{P}}$ by the predicates ${\mathit{M}}_{\mathit{k}}$ to obtain the lan- guage ${\mathit{L}}_{\mathit{P}}^{\mathit{M}}\mathrm{.}$

(b) Observe that each quantifier-free formula of ${\mathit{L}}_{\mathit{P}}^{\mathit{M}}$ is equivalent to a finite disjunc- tion of formulas of the form

$\underset{\mathit{i}}{\wedge }\mathrm{\left(}{\mathit{f}}_{\mathit{i}}\mathrm{\left(}\overline{\mathit{x}}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathrm{\right)}\mathrm{\wedge }\underset{\mathit{j}}{\wedge }\mathrm{\left(}{\mathit{M}}_{{\mathit{k}}_{\mathit{j}}}\mathrm{\left(}{\mathit{g}}_{\mathit{j}}\mathrm{\left(}\overline{\mathit{x}}\mathrm{\right)}\mathrm{\right)}\mathrm{\right)}\mathrm{\wedge }\underset{\mathit{i}}{\wedge }\mathrm{\left(}{\mathit{h}}_{\mathit{i}}\mathrm{\left(}\overline{\mathit{x}}\mathrm{\right)}\mathrm{\ge }\mathrm{0}\mathrm{\right)}$

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where each ${\mathit{f}}_{\mathit{i}}$ and ${\mathit{g}}_{\mathit{j}}$ is a polynomial of degree 1 in the variables of the tuple $\overline{\mathit{x}}\mathrm{.}$

(c) Prove that each existential ${\mathit{L}}_{\mathit{P}}^{\mathit{M}}$-formula, i.e. of the form $\mathrm{\exists }\overline{\mathit{x}}\mathit{\phi }\mathrm{\left(}\overline{\mathit{x}}\mathrm{,}\mathrm{}\mathrm{y}\mathrm{\right)}$ where $\mathit{\phi }\mathrm{\left(}\overline{\mathit{x}}\mathrm{,}\mathrm{}\mathrm{y}\mathrm{\right)}$ is quantifier free, is equivalent to a quantifier-free ${\mathit{L}}_{\mathit{P}}^{\mathit{M}}$-formula. You can do this by eliminating the existential quantifiers, one at a time. For example, to eliminate the quantifier $\mathrm{\exists }\mathit{x}$ from

$\mathrm{\exists }\mathit{x}\mathrm{\left[}\mathit{a}\mathrm{-}\mathit{x}\mathrm{\ge }\mathrm{0}\mathrm{\wedge }\mathit{x}\mathrm{\ge }\mathit{b}\mathrm{\wedge }{\mathit{M}}_{\mathrm{2}}\mathrm{\left(}\mathit{x}\mathrm{+}\mathit{c}\mathrm{\right)}\mathrm{\right]}$

(a, b and c can be polynomials in some variables and x does not occur in any of them) observe that it is equivalent to the disjunction of the formulas that correspond to the cases (A) $\mathit{a}\mathrm{\ge }\mathit{b}\mathrm{+}\mathrm{1}\mathrm{,}$ $\mathrm{\left(}\mathit{B}\mathrm{\right)}\mathit{a}\mathrm{=}\mathit{b}$ and $\mathit{b}$ is even and $\mathit{c}$ is even, and (C) $\mathit{a}\mathrm{=}\mathit{b}$ and $\mathit{a}$ is odd and $\mathit{c}$ is odd.

(d) Fix a language and a theory. Assume that each existential formula is equivalent (in the theory) to some quantifier-free formula. Then show that each formula of the language have the same property ($\mathit{i}\mathrm{.}\mathit{e}$. is equivalent to a quantifier-free formula).

One can ask how one can enrich the Presburger language ${\mathit{L}}_{\mathit{P}}$ and still retain decid- ability for the existential theory of $\mathrm{Z}$ in this enriched language. Of course, as soon as multiplication is definable from the functions and predicates in the language, one gets undecidability. The bibliography on this subject is quite extensive. For some informa- tion see [8] and the bibliography of [47]. Below we will see some results of this kind, as well.

1.2 Addition and divisibility

The results we present are from [34], [35] and [36].

A natural extension of ${\mathit{L}}_{\mathit{P}}$ consists of adding a binary predicate for the divisibility relation $\mathrm{|}$ $\mathrm{\left(}$that $\mathrm{i}\mathrm{s}\mathrm{,}\mathrm{}\mathit{a}\mathrm{|}\mathit{b}\mathrm{}\mathrm{i}\mathrm{f}\mathrm{}\mathrm{a}\mathrm{n}\mathrm{d}$ only $\mathrm{i}\mathrm{f}\mathrm{}\mathrm{\exists }\mathit{c}\mathrm{:}\mathit{b}\mathrm{=}\mathit{a}\mathit{c}\mathrm{\right)}$ . J. Robinson showed that multiplication can be defined from addition and divisibility by a first-order formula and hence, the first- order theory of $\mathrm{Z}$ in this language is undecidable. In contrast, Lipshitz, in [34] (and, independently, Bel'tyukov) showed that the existential theory of $\mathrm{Z}$ in the language $\mathit{L}\mathrm{=}{\mathit{L}}_{\mathit{P}}\mathrm{\cup }\mathrm{\left\{}\mathrm{|}\mathrm{\right\}}$ is decidable. The same is true for any ring of integers of an imaginary quadratic extension of the rationals (in this case the predicate $\mathrm{>}$ has to be excluded from the language). This result is optimal in several ways: multiplication is positive- existential in the ${\mathit{L}}_{\mathrm{1}}$-theory of any ring of algebraic integers in a number field other than the rationals and imaginary quadratic ([35] and [36]); hence the ${\mathit{L}}_{{\mathrm{1}}^{\mathrm{-}}}\mathrm{e}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{1}$-theory of those rings has the same decidability property as the ring theory which has been conjectured -by Denef and Lipshitz, but not yet proved -to be undecidable. These results were later generalized to polynomial rings over fields: the existential theory of addition and divisibility in a polynomial ring $\mathit{k}\mathrm{\left[}\mathit{t}\mathrm{\right]}$ over a field $\mathit{k}$ (in the language $\mathit{L}\mathrm{=}{\mathit{L}}_{\mathit{P}}\mathrm{\cup }\mathrm{\left\{}\mathit{t}\mathrm{,}\mathrm{}\mathrm{|}\mathrm{\right\}}\mathrm{\right)}$ , is decidable if and only if the existential theory of $\mathit{k}$ is decidable; the positive existential theory of $\mathit{A}\mathrm{\left[}{\mathit{t}}_{\mathrm{1}}\mathrm{,}\mathrm{}{\mathit{t}}_{\mathrm{2}}\mathrm{\right]}$ in the language $\mathit{L}\mathrm{=}{\mathit{L}}_{\mathit{P}}\mathrm{\cup }\mathrm{\left\{}{\mathit{t}}_{\mathrm{1}}\mathrm{,}\mathrm{}{\mathit{t}}_{\mathrm{2}}\mathrm{,}\mathrm{}\mathrm{|}\mathrm{\right\}}$ is undecidable for any commutative domain $\mathit{A}\mathrm{.}$

We give a short account of the proof of [34]:

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Let ${\mathit{L}}_{\mathrm{1}}\mathrm{=}{\mathit{L}}_{\mathit{P}}\mathrm{\cup }\mathrm{\left\{}\mathrm{|}\mathrm{\right\}}$ where $\mathrm{|}$ will be interpreted by $\mathrm{\left(}\mathit{a}\mathrm{|}\mathit{b}\mathrm{↔}\mathrm{\exists }\mathit{c}\mathrm{\left[}\mathit{b}\mathrm{=}\mathit{a}\mathit{c}\mathrm{\right]}\mathrm{\text{'}}$

We say that a class of quantifier-free formulas 'satisfies the local-to-global principle' if it is satisfiable when and only when it is satisfiable in every completion of $\mathbb{Z}$ with respect to a prime (including the 'prime at infinity'). Observe that the quantifier-free formulas of ${\mathit{L}}_{\mathrm{1}}$ do not satisfy the local-to-global principle, e.g. $\mathit{x}\mathrm{+}\mathrm{2}\mathrm{|}\mathrm{2}\mathit{x}\mathrm{+}\mathrm{3}\mathrm{\wedge }\mathit{x}\mathrm{\ge }\mathrm{0}$ is satisfiable in each completion (in each ${\mathbb{Q}}_{\mathit{p}}$ and in $\mathbb{R}$) but not in $\mathbb{Z}\mathrm{.}$

(a) Show that every existential sentence of ${\mathit{L}}_{\mathrm{1}}$ is equivalent to a disjunction of for- mulas of the form $\mathrm{\exists }\mathit{x}\mathit{\phi }\mathrm{\left(}\mathit{x}\mathrm{\right)}$ where

$\mathit{\phi }\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{:}\underset{\mathit{i}}{\wedge }\mathrm{\left(}{\mathit{f}}_{\mathit{i}}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{|}{\mathit{g}}_{\mathit{i}}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\right)}\underset{\mathit{j}}{\wedge }\mathrm{\left(}{\mathit{h}}_{\mathit{j}}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\ge }\mathrm{0}\mathrm{\right)}$ (1)

where each ${\mathit{f}}_{\mathit{i}}\mathrm{,}$ ${\mathit{g}}_{\mathit{i}}$ and ${\mathit{h}}_{\mathit{j}}$ is a polynomial of degree at most 1 in the variables of the tuple of variables $\mathit{x}\mathrm{=}\mathrm{\left(}{\mathit{x}}_{\mathrm{1}}\mathrm{,}\mathrm{}\mathrm{\text{...}}\mathrm{,}\mathrm{}{\mathit{x}}_{\mathit{m}}\mathrm{\right)}$ . To do this one has to

(a1) Eliminate non-divisibilities: observe that $\mathrm{¬}\mathit{a}\mathrm{|}\mathit{b}$ is equivalent to 'a greatest com- mon divisor of $\mathit{a}$ and $\mathit{b}$ is different from $\mathit{a}$ and $\mathrm{-}\mathit{a}$' which is equivalent to

$\mathrm{\exists }\mathit{s}\mathrm{,}\mathrm{}\mathit{x}\mathrm{,}\mathrm{}\mathit{y}\mathrm{\left[}\mathit{s}\mathrm{|}\mathit{a}\mathrm{\wedge }\mathit{s}\mathrm{|}\mathit{b}\mathrm{\wedge }\mathit{a}\mathrm{|}\mathit{x}\mathrm{\wedge }\mathit{b}\mathrm{|}\mathit{y}\mathrm{\wedge }\mathit{s}\mathrm{=}\mathit{x}\mathrm{-}\mathit{y}\mathrm{\wedge }\mathrm{\left(}\mathrm{\ne }\mathit{s}\mathrm{+}\mathit{a}\mathrm{=}\mathrm{0}\mathrm{\right)}\mathrm{\wedge }\mathrm{\left(}\mathrm{\ne }\mathit{s}\mathrm{-}\mathit{a}\mathrm{=}\mathrm{0}\mathrm{\right)}\mathrm{\right]}$

(a2) Eliminate equations, e.g. if the equation $\mathrm{2}\mathit{x}\mathrm{-}\mathit{y}\mathrm{=}\mathrm{0}$ occurs then one can substitute all occurences of the variable $\mathit{x}$ by $\frac{\mathit{y}}{\mathrm{2}}$, clear denominators by multiplying all terms by 2, and add the divisibility $\mathrm{2}\mathrm{|}\mathit{y}\mathrm{.}$

In the rest work with notation as in (1).

(b) Impose all possible relative orderings on the variables of $\mathit{x}$, e.g. ${\mathit{x}}_{\mathrm{1}}\mathrm{\ge }\mathrm{\text{...}}\mathrm{\ge }{\mathit{x}}_{\mathit{m}}\mathrm{,}$ and all possible orderings on the terms ${\mathit{f}}_{\mathit{i}}\mathrm{,}$ ${\mathit{g}}_{\mathit{i}}$ and ${\mathit{h}}_{\mathit{i}}$. Each one of these cases gives a sentence (the disjunction of all these is equivalent to the initial sentence). So assume that 1 corresponds to such a sentence.

Draw conclusions that eliminate some variables, for example, if a divisibility $\mathit{x}\mathrm{+}$ $\mathit{f}\mathrm{\left(}\mathit{y}\mathrm{\right)}\mathrm{|}\mathit{x}\mathrm{+}\mathit{g}\mathrm{\left(}\mathit{y}\mathrm{\right)}$ occurs and $\mathit{x}\mathrm{,}$ $\mathit{x}\mathrm{+}\mathit{f}\mathrm{\left(}\mathit{y}\mathrm{\right)}\mathrm{,}$ $\mathit{x}\mathrm{+}\mathit{g}\mathrm{\left(}\mathit{y}\mathrm{\right)}\mathrm{\ge }\mathrm{0}$ and the variable $\mathit{x}$ is $\mathrm{\ge }$ to all the variables of the tuple $\mathit{y}$, then draw the conclusion that either $\frac{\mathit{x}\mathrm{+}\mathit{g}\mathrm{\left(}\mathit{y}\mathrm{\right)}}{\mathit{x}\mathrm{+}\mathit{f}\mathrm{\left(}\mathit{y}\mathrm{\right)}}$ is an integer $\mathrm{\le }\mathit{M}\mathrm{+}\mathrm{1}$ where $\mathit{M}$ is the maximum absolute value of the coefficients of $\mathit{f}\mathrm{-}\mathit{g}$ (Exercise: Why?); Consider the cases and eliminate the variable $\mathit{x}$ (conclude with fewer variables).

(c) Impose certai $\mathrm{n}$implied' divisisbilities, foe example, from ${\mathit{f}}_{\mathrm{1}}\mathrm{|}{\mathit{f}}_{\mathrm{2}}$ and ${\mathit{f}}_{\mathrm{2}}\mathrm{|}{\mathit{f}}_{\mathrm{3}}$ conclude that ${\mathit{f}}_{\mathrm{1}}\mathrm{|}{\mathit{f}}_{\mathrm{3}}$ (and add it to the given list of divisibilities).

(d) Iterate (b) and (c). Show that a finite number of iterations concludes with systems which are 'diagonal', in the sense that in each divisibility $\mathit{f}\mathrm{|}\mathit{g}$ there is a variable that occurs in $\mathit{g}$, which is 'bigger' than all variables of $\mathit{f}$; in addition these systems are closed under the operations mentioned in (c).

(e) Show that diagonal systems satisfy the local-to-global principle. In fact show that to each diagonal system $\mathit{\phi }\mathrm{\left(}\mathit{x}\mathrm{\right)}$ one can effectively associate a natural number $\mathit{N}$ such that $\mathit{\phi }\mathrm{\left(}\mathit{x}\mathrm{\right)}$ is satisfiable in $\mathbb{Z}$ if and only if it is satisfiable in each ${\mathbb{Q}}_{\mathit{p}}$ with $\mathit{p}\mathrm{\le }\mathit{N}\mathrm{.}$ Then use the decidability of the theory of each ${\mathbb{Q}}_{\mathit{p}}$ to conclude.

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Exercise 1.3 The following give some of the main ideas in [35].

Consider the ring of integers ${O}$ of a real quadratic number field $\mathit{K}$ (e.g. $\mathit{K}\mathrm{=}$ $\mathbb{Q}\mathrm{\left(}\sqrt{\mathrm{5}}\mathrm{\right)}\mathrm{\right)}$ . Consider known the following fact: There is a unit ${\mathit{\epsilon }}_{\mathrm{0}}$ which is $\mathit{\zeta }$fundamental ' i.e. all units of ${O}$ are of the $\mathit{f}\mathit{o}\mathit{r}\mathit{m}\mathrm{±}{\mathit{\epsilon }}_{\mathrm{0}}^{\mathit{n}}$ with $\mathit{n}\mathrm{\in }\mathbb{Z}\mathrm{.}$

(a) Show that if $\mathit{n}\mathrm{|}\mathit{m}$ in $\mathbb{Z}$ then ${\mathit{\epsilon }}_{\mathrm{0}}^{\mathit{n}}\mathrm{-}\mathrm{1}\mathrm{|}{\mathit{\epsilon }}_{\mathrm{0}}^{\mathit{m}}\mathrm{-}\mathrm{1}$ in ${O}\mathrm{.}$

It can be shown that a sort of converse is also true: There is a $\mathit{d}\mathrm{\in }\mathbb{Z}$ such that setting $\mathit{\epsilon }\mathrm{=}{\mathit{\epsilon }}_{\mathrm{0}}^{\mathit{d}}$ we have that $\mathit{n}\mathrm{|}\mathit{m}$ in $\mathbb{Z}$ if and only if ${\mathit{\epsilon }}^{\mathit{n}}\mathrm{-}\mathrm{1}\mathrm{|}{\mathit{\epsilon }}^{\mathit{m}}\mathrm{-}\mathrm{1}$ in ${O}$. For the rest assume that one has in the language a name (constant-symbol) for a unit $\mathit{\epsilon }$ with these properties and assume that the set $\mathrm{\left\{}{\mathit{\epsilon }}^{\mathit{n}}\mathrm{:}\mathit{n}\mathrm{\in }\mathbb{Z}\mathrm{\right\}}$ is positive existential in the language ${\mathit{L}}_{\mathrm{1}}\mathrm{\cup }\mathrm{\left\{}\mathit{\epsilon }\mathrm{\right\}}$ (all the mentioned facts are true).

(b) Assume that $\mathit{m}\mathrm{,}$ $\mathit{n}\mathrm{\ne }\mathrm{0}$ and $\mathit{m}\mathrm{\ne }\mathit{n}$. Show that ${\mathit{\epsilon }}^{\mathit{m}}\mathrm{=}{\mathit{\epsilon }}^{\mathrm{2}\mathit{n}}$ if and only if

${\mathit{\epsilon }}^{\mathit{n}}\mathrm{-}\mathrm{1}\mathrm{|}{\mathit{\epsilon }}^{\mathit{m}}\mathrm{-}{\mathit{\epsilon }}^{\mathit{n}}\mathrm{\wedge }{\mathit{\epsilon }}^{\mathit{m}}\mathrm{-}{\mathit{\epsilon }}^{\mathit{n}}\mathrm{|}{\mathit{\epsilon }}^{\mathit{n}}\mathrm{-}\mathrm{1}$ .

(c) Show: If $\mathit{x}\mathrm{\in }{O}\mathrm{\setminus }\mathrm{\left\{}\mathrm{O}\mathrm{\right\}}$ then there is an $\mathit{n}\mathrm{\ne }\mathrm{0}$ such that $\mathit{x}\mathrm{|}{\mathit{\epsilon }}^{\mathit{n}}\mathrm{-}\mathrm{1}$ (hint: if $\mathit{x}$ is not a unit then the ring ${O}\mathrm{/}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ is finite, hence the natural image of $\mathit{\epsilon }$ has finite multiplicative order).

(d) Let $\mathit{\phi }\mathrm{\left(}\mathit{x}\mathrm{\right)}$ denote the formula

$\mathit{x}\mathrm{\ne }\mathrm{0}\mathrm{\wedge }\mathit{x}\mathrm{\ne }\mathrm{1}\mathrm{\wedge }\mathit{x}\mathrm{\ne }\mathrm{-}\mathrm{1}\mathrm{\wedge }\mathit{x}\mathrm{|}{\mathit{\epsilon }}^{\mathit{n}}\mathrm{-}\mathrm{1}\mathrm{\wedge }\mathit{x}\mathrm{-}\mathrm{1}\mathrm{|}{\mathit{\epsilon }}^{\mathit{n}}\mathrm{-}\mathrm{1}\mathrm{\wedge }\mathit{x}\mathrm{+}\mathrm{1}\mathrm{|}{\mathit{\epsilon }}^{\mathit{n}}\mathrm{-}\mathrm{1}$

Assume that if $\mathit{n}\mathrm{\ne }\mathrm{0}$ and $\mathit{\phi }\mathrm{\left(}\mathit{x}\mathrm{\right)}$ is true then $\mathrm{2}\mathrm{|}{\mathit{x}}^{\mathrm{2}}\mathrm{|}\mathrm{<}{\mathit{\epsilon }}^{\mathit{n}}$($\mathrm{|}\mathit{w}\mathrm{|}$ is the absolute value of $\mathit{w}$). Show: For $\mathit{x}\mathrm{,}$ $\mathit{y}\mathrm{\in }{O}\mathrm{\setminus }\mathrm{\left\{}\mathrm{O}\mathrm{,}\mathrm{}\mathrm{1}\mathrm{,}\mathrm{}\mathrm{-}\mathrm{1}\mathrm{\right\}}$ the following is true: $\mathit{y}\mathrm{=}{\mathit{x}}^{\mathrm{2}}$ if and only if for some $\mathit{n}\mathrm{\in }\mathbb{Z}\mathrm{\setminus }\mathrm{\left\{}\mathrm{O}\mathrm{\right\}}$ we have

$\mathit{\phi }\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\wedge }\mathit{\phi }\mathrm{\left(}\mathit{y}\mathrm{\right)}\mathrm{\wedge }{\mathit{\epsilon }}^{\mathit{n}}\mathrm{-}\mathit{x}\mathrm{|}{\mathit{\epsilon }}^{\mathrm{2}\mathit{n}}\mathrm{-}\mathit{y}$

(e) Consider known the following: The relation $\mathit{x}\mathrm{\ne }\mathrm{0}$ is positive-existential in ${\mathit{L}}_{\mathrm{1}}\mathrm{\cup }\mathrm{\left\{}\mathit{\epsilon }\mathrm{\right\}}\mathrm{.}$

Use $\mathrm{\left(}\mathit{a}\mathrm{\right)}\mathrm{-}\mathrm{\left(}\mathit{d}\mathrm{\right)}$ to produce a formula $\mathit{\psi }\mathrm{\left(}\mathit{x}\mathrm{,}\mathrm{}\mathit{y}\mathrm{\right)}$ of ${\mathit{L}}_{\mathrm{1}}\mathrm{\cup }\mathrm{\left\{}\mathit{\epsilon }\mathrm{\right\}}$ such that for any $\mathit{x}\mathrm{,}$ $\mathit{y}\mathrm{\in }{O}$ the following is true in ${O}$:

$\mathit{\psi }\mathrm{\left(}\mathit{x}\mathrm{,}\mathrm{}\mathit{y}\mathrm{\right)}\mathrm{↔}\mathit{y}\mathrm{=}{\mathit{x}}^{\mathrm{2}}$

thus squaring over ${O}$ is positive-existential in ${\mathit{L}}_{\mathrm{1}}\mathrm{\cup }\mathrm{\left\{}\mathit{\epsilon }\mathrm{\right\}}$. Prove that this implies that multiplication over ${O}$ is positive-existential in ${\mathit{L}}_{\mathrm{1}}\mathrm{\cup }\mathrm{\left\{}\mathit{\epsilon }\mathrm{\right\}}$. Conclude that the positive- existential theory of ${\mathit{L}}_{\mathrm{1}}\mathrm{\cup }\mathrm{\left\{}\mathit{\epsilon }\mathrm{\right\}}$ is undecidable.

1.3 Addition and exponentiation

The following are results from [54].

The first-order theory of $\mathrm{N}$ in the language $\mathit{L}\mathrm{=}{\mathit{L}}_{\mathit{P}}\mathrm{\cup }\mathrm{\left\{}{\mathrm{exp}}_{\mathrm{2}}\mathrm{\right\}}$, where ${\mathrm{exp}}_{\mathrm{2}}$ the function which sends a natural number $\mathit{n}$ to ${\mathrm{2}}^{\mathit{n}}$, is decidable.

Towards proving this, one has to study the behavior of 'exponential polynomials', e.g. of the form ${\mathrm{2}}^{{\mathrm{2}}^{\mathrm{3}\mathit{x}\mathrm{-}\mathrm{1}}}\mathrm{-}{\mathrm{2}}^{\mathrm{5}\mathit{x}\mathrm{+}\mathrm{5}}\mathrm{+}\mathrm{1}$. We will not go into the details.

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Exercise 1.4 Show that there is an algorithm which, given any diophantine equation $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\mathrm{0}$($\mathit{x}$ is a tuple of $\mathit{m}$ variables and $\mathit{f}$ a polynomial in $\mathit{x}$ over $\mathbb{Z}$), decides whether that has or does not have solutions in the set $\mathrm{\left\{}{\mathrm{2}}^{\mathit{n}}\mathrm{:}\mathit{n}\mathrm{\in }\mathbb{N}{\mathrm{\right\}}}^{\mathit{m}}\mathrm{.}$

Remark: By results of Lang the statement of the Exercise is true for any finitely generated multiplicative (i.e. closed under multiplication) set, e.g. the set $\mathrm{\left\{}{\mathrm{2}}^{\mathit{n}}{\mathrm{3}}^{\mathit{m}}$ : $\mathit{n}\mathrm{,}$ $\mathit{m}\mathrm{\in }\mathbb{N}\mathrm{\right\}}\mathrm{.}$

1.4 The analogue of Hilbert's tenth problem for algebraic groups

The following (among several) result can be found in [28].

Theorem 1.5 There is an algorithm which, given any algebraic group $\mathit{G}$ defined over $\mathbb{Z}$ and any positive integer $\mathit{k}$, decides whether $\mathit{G}$ contains a subgroup of index $\mathit{k}\mathrm{.}$

Two examples of algebraic groups defined over $\mathbb{Z}$ are:

1. $\mathbb{Z}\mathrm{,}$ $\mathbb{Z}\mathrm{×}\mathbb{Z}$ e.t.c. with the group structure of component-wise addition.

2. The solution set of the equation ${\mathit{X}}^{\mathrm{2}}\mathrm{-}\mathit{d}{\mathit{Y}}^{\mathrm{2}}\mathrm{=}\mathrm{1}$ where $\mathit{d}$ is a square-free integer; The group $\mathrm{l}\mathrm{a}\mathrm{w}\mathrm{\oplus }\mathrm{i}\mathrm{s}$ defined by

$\mathrm{\left(}{\mathit{x}}_{\mathrm{1}}\mathrm{,}\mathrm{}{\mathit{y}}_{\mathrm{1}}\mathrm{\right)}\mathrm{\oplus }\mathrm{\left(}{\mathit{x}}_{\mathrm{2}}\mathrm{,}\mathrm{}{\mathit{y}}_{\mathrm{2}}\mathrm{\right)}\mathrm{=}\mathrm{\left(}{\mathit{x}}_{\mathrm{1}}{\mathit{x}}_{\mathrm{2}}\mathrm{+}\mathit{d}{\mathit{y}}_{\mathrm{1}}{\mathit{y}}_{\mathrm{2}}\mathrm{,}\mathrm{}{\mathit{x}}_{\mathrm{1}}{\mathit{y}}_{\mathrm{2}}\mathrm{+}\mathit{x}\mathit{y}\mathrm{\right)}$ .

This justifies the claim that 'the more structure one has, the more likely decidability is'

1.5 The results of Ax for 'almost all primes'

In [1] Ax gave an algorithm which, for any given diophantine equation, tests whether the equation has a solution modulo every prime number. In [2] he extended this to an algorithm that decides the truth of a sentence of ${\mathit{L}}_{\mathit{R}}$ in all finite fields. The proofs make use of Weil's result on the congruence zeta function and of Čebotarev's density theorem.

For further results in this direction see [25] and [60].

1.6 Model-completeness

A set (subset of a power of $\mathbb{Z}$ or any effectively constructible countable set) is recursive if memebership in it can be tested by an algoritm; it is recursively enumerable if its elements can be listed (eventually all) by an algorithm.

A theory of a language (finite or countable-and-recursive) $\mathit{L}$ is a subset of the set of sentences of $\mathit{L}$. The theory of a structure (model) ${A}$ in a language $\mathit{L}$ is model-complete if every formula of $\mathit{L}$ is equivalent to an existential formula of $\mathit{L}$ over ${A}\mathrm{.}$

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The following is a classical decidability argument in Model Theory:

Assume:

(a) ${A}$ is a countable and recursive structure in the countable language $\mathit{L}\mathrm{.}$

(b) The theory of ${A}$ in $\mathit{L}$ is effectively model-complete, that is, there is an algorithm which to any sentence of $\mathit{L}$ associates an equivalent (over ${A}$) existential sentence.

Conclude: The theory of ${A}$ in $\mathit{L}$ is decidable.

Proof: Let $\mathit{\sigma }$ be a sentence of $\mathit{L}$. Find an existential sentence, say $\mathrm{\exists }\mathit{x}\mathit{\phi }\mathrm{\left(}\mathit{x}\mathrm{\right)}$ , equivalent to $\mathit{\sigma }$, and an existential sentence, say $\mathrm{\exists }\mathit{y}\mathit{\psi }\mathrm{\left(}\mathit{y}\mathrm{\right)}$ , equivalent to $\mathrm{¬}\mathit{\sigma }$ where $\mathit{x}$ is a tuple of $\mathit{m}$ variables, $\mathit{y}$ is a tuple of $\mathit{n}$ variables and the formulas $\mathit{\phi }$ and $\mathit{\psi }$ are quantifier-free. Enumerate the tuples of $\mathit{m}$ elements of ${A}$ and the tuples of $\mathit{n}$ elements of ${A}$. By day plug in $\mathit{\phi }\mathrm{\left(}\mathit{x}\mathrm{\right)}$ tuples of $\mathit{m}$ elements and check whether they make it true. By night do similarly with $\mathit{\psi }\mathrm{.}$

One of $\mathit{\sigma }$ or $\mathrm{¬}\mathit{\sigma }$ is true, hence either we will satisfy $\mathit{\phi }$ on a day (in which case $\mathit{\sigma }$ is true) or we will satisfy $\mathit{\psi }$ on a night (in which case $\mathrm{¬}\mathit{\sigma }$ is true). $\mathrm{⟨}\mathrm{⟩}$

Exercise 1.6 (a) Prove that if in a theory $\mathit{T}$ every universal formula is equivalent to an existential one then $\mathit{T}$ is model-complete.

Consider the language $\mathit{L}$ which extends the language of rings by a symbol $\mathrm{\ge }\mathit{f}\mathit{o}\mathit{r}$ the ordinary ordering relation. A polynomial (or rational function) $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}$ in the $\mathit{m}$ variables $\mathit{x}$, with coefficients in $\mathbb{R}$, is positive-definite if for all $\mathit{a}\mathrm{\in }{\mathbb{R}}^{\mathit{m}}$ we have $\mathit{f}\mathrm{\left(}\mathit{a}\mathrm{\right)}\mathrm{\ge }\mathrm{0}\mathrm{.}$

(b) Show that every positive-definite polynomial in one variable $\mathit{x}$ is the sum of two squares of polynomials of $\mathit{x}$ with coefficients in $\mathbb{R}$. (This admits a generalization to $\mathit{m}$ variables: A positive-definite rational function in $\mathit{m}$ variables is equal to the sum of ${\mathrm{2}}^{\mathit{m}}$-many squares of rational functions; this is a positive answer to Hilbert's $\mathrm{1}\mathrm{7}$-th problem.)

(c) Use (b) to prove that if $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\in }\mathbb{Z}\mathrm{\left[}\mathit{x}\mathrm{\right]}$ and $\mathit{x}$ is one variable then the universal formula $\mathrm{\forall }\mathit{x}\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{\ge }\mathrm{0}$ is equivalent to an existential formula.

1.7 Complete theories

Suppose that $\mathit{T}$ is a theory of the recursive language $\mathit{L}$. We say that $\mathit{T}$ is complete if for any sentence $\mathit{\sigma }$ of $\mathit{L}$ either $\mathit{\sigma }$ or $\mathrm{¬}\mathit{\sigma }$ is a consequence of $\mathit{T}$. Then one sees easily that

Theorem 1.7 Assume that $\mathit{T}$ is a recursively enumerable and complete theory of the recursive language L. Then there is an algorithm which tests any given sentence $\mathit{\sigma }$ of $\mathit{L}$ for being or not being a consequence of $\mathit{T}\mathrm{.}$

Exercise 1.8 Give an outline of a proof of the Theorem.

For example, the theory of algebraically closed fields of a fixed characteristic (the axioms for a field, together with axioms which fix the characteristic, together with axioms which state, for each $\mathit{n}$, ""every polynomial of degree $\mathit{n}$ has a zero“), which is known to be complete, has a recursive set of consequences.

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Exercise 1.9 Describe in detail the latter theory.

The latter Theorem transforms the question of whether the set of consequences of a given theory is decidable (i.e. recursive) to the question of whether the theory is complete (whenever that is true). This is a very common approach in Model Theory to decidability questions.

It is obvious that the latter Theorem may be not true if $\mathit{T}$ is not complete. Examples of this kind exist in the bibliography.

For relevant bibliography see [7].

1.8 Power-series and germs of analytic functions: existential decidability and Artin approximation

The ring-theories of ${\mathbb{Q}}_{\mathit{p}}$ ( $\mathit{p}$ is a prime number) are decidable (results of Nerode, Ax and Kochen [3], Ershov [24], also cf. [38], [9], and [17]).

If $\mathit{F}$ is a field of characteristic $\mathrm{0}$ then the ring-theory of the field of formal fractional power series $\mathit{F}\mathrm{\left(}\mathrm{\left(}\mathit{T}\mathrm{\right)}\mathrm{\right)}$ in one variable $\mathit{T}$ (and that of $\mathit{F}\mathrm{\left[}\mathrm{\left[}\mathit{T}\mathrm{\right]}\mathrm{\right]}$) is decidable ([32] and [65]). But if $\mathit{T}$ is two or more variables then we have undecidability ([14]).

For more relevant results see [5] and [20].

We wish to address the question of the decidability of the existential theory of a ring ${{H}}_{\mathit{z}}\mathrm{\left(}\mathit{D}\mathrm{\right)}$ of functions of a variable $\mathit{z}$, analytic on a set $\mathit{D}$ (i.e. analytic on some open super-set of $\mathit{D}$). We will address first the question for $\mathit{D}\mathrm{=}$( $\mathrm{a}$ singleton), say $\mathit{D}\mathrm{=}\mathrm{\left\{}\mathrm{0}\mathrm{\right\}}\mathrm{\right)}$ . Already in [32] it was shown that the ring-theory of ${{H}}_{\mathit{z}}\mathrm{\left(}\mathrm{0}\mathrm{\right)}$ is decidable. But the techniques of that paper do not transfer to bigger $\mathit{D}$, so we will look into other approaches. One approach is via 'Approximation Properties'

Definition 1.10 (i) Let $\mathit{R}$ be a local ring and $\stackrel{\mathrm{ˆ}}{\mathit{R}}$ be its completion. We say that $\mathit{R}$ has the Approximation Property if every system of polynomial equations over $\mathit{R}$, which has a solution in $\stackrel{\mathrm{ˆ}}{\mathit{R}}$, has a solution in $\mathit{R}\mathrm{.}$

(ii) $\mathbb{C}\mathrm{\left\{}{\mathit{z}}_{\mathrm{1}}\mathrm{,}\mathrm{}{\mathit{z}}_{\mathit{q}}\mathrm{\right\}}$ is the ring of formal power series, in the variables ${\mathit{z}}_{\mathrm{1}}\mathrm{,}$ ${\mathit{z}}_{\mathit{q}}$ over $\mathrm{C}\mathrm{,}$ which converge in some neighborhood of the origin.

(iii) Let $\mathit{k}$ be a field. Then $\mathit{k}\mathrm{⟨}{\mathit{z}}_{\mathrm{1}}\mathrm{,}$ ${\mathit{z}}_{\mathit{q}}\mathrm{⟩}$ denotes the ring of formal power series in ${\mathit{z}}_{\mathrm{1}}\mathrm{,}$ ${\mathit{z}}_{\mathit{q}}$ over $\mathit{k}$ which are algebraic over $\mathit{k}\mathrm{\left[}{\mathit{z}}_{\mathrm{1}}\mathrm{,}\mathrm{}{\mathit{z}}_{\mathit{q}}\mathrm{\right]}\mathrm{.}$

Artin proved:

Theorem 1.11 Let $\mathit{k}$ be any field. The following two rings have the Approximation Property:

(i) $\mathit{k}\mathrm{⟨}{\mathit{z}}_{\mathrm{1}}\mathrm{,}$ ${\mathit{z}}_{\mathit{q}}\mathrm{⟩}$ and

(ii) $\mathrm{C}\mathrm{\left\{}{\mathit{z}}_{\mathrm{1}}\mathrm{,}\mathrm{}\mathrm{\text{...}}{\mathit{z}}_{\mathit{q}}\mathrm{\right\}}\mathrm{.}$

Theorem 1.12 Let $\mathit{k}$ be an arbitrary field and $\mathit{z}$ be a variable. Then, a system of polynomial equations, with coefficients in $\mathit{k}\mathrm{\left[}\mathrm{\left[}\mathit{z}\mathrm{\right]}\mathrm{\right]}$, has solutions in $\mathit{k}\mathrm{\left[}\mathrm{\left[}\mathit{z}\mathrm{\right]}\mathrm{\right]}$ if and only if, for any $\mathit{n}\mathrm{\in }{\mathrm{N}}^{\mathrm{+}}$, it has solutions modulo $\mathrm{\left(}\mathit{z}{\mathrm{\right)}}^{\mathit{n}}\mathrm{.}$

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By Theorem 1.11, a system of equations, with coefficients in $\mathbb{C}\mathrm{\left[}\mathit{z}\mathrm{\right]}$, has solutions in ${{H}}_{\mathit{z}}\mathrm{\left(}\mathrm{0}\mathrm{\right)}$ if and only if it has solutions in $\mathbb{C}\mathrm{⟨}\mathit{z}\mathrm{⟩}\mathrm{.}$

A much stronger version of Theorem 1.12, suitable for algorithmic computation of solutions of equations, is true: for each system of equations, there is a constant $\mathit{N}\mathrm{,}$ depending on the system (actually: algorithmically computable and depending only on the number of variables and the degrees of the involved polynomials), such that the system has solutions in $\mathit{k}\mathrm{\left[}\mathrm{\left[}\mathit{z}\mathrm{\right]}\mathrm{\right]}$ if and only if it has solutions modulo $\mathrm{\left(}\mathit{z}{\mathrm{\right)}}^{\mathit{N}}$. Results of this type occur in the literature under the name 'Strong Approximation theorems'

Strong Approximation implies decidability of the positive existential theory of $\mathit{k}\mathrm{\left[}\mathrm{\left[}\mathit{z}\mathrm{\right]}\mathrm{\right]}$ (in order to see whether a system of equations has solutions, compute the constant $\mathit{N}$ of the previous paragraph and check whether the system has solutions modulo $\mathrm{\left(}\mathit{z}{\mathrm{\right)}}^{\mathit{N}}$). But, since our main interest is to investigate the possibility of adapting our arguments to ${{H}}_{\mathit{z}}\mathrm{\left(}\mathit{D}\mathrm{\right)}$ for $\mathit{D}$ not a singleton, and because there is no known analogue of Strong Approximation for these rings, we will now present a decidability result, based only on Theorems 1.11 and 1.12. We will show how one can use these Theorems in order to detect whether a given system of polynomial equations, with coefficients in $\stackrel{\mathrm{˜}}{\mathbb{Q}}\mathrm{\left[}\mathit{z}\mathrm{\right]}$ $\mathrm{\left(}\mathit{z}\mathrm{=}\mathrm{\left(}{\mathit{z}}_{\mathrm{1}}\mathrm{,}\mathrm{}{\mathit{z}}_{\mathit{n}}\mathrm{\right)}$ and $\stackrel{\mathrm{˜}}{\mathbb{Q}}$ denotes the algebraic closure of $\mathbb{Q}$), has solutions over ${{H}}_{\mathit{z}}\mathrm{\left(}\mathrm{\left\{}\mathrm{0}{\mathrm{\right\}}}^{\mathit{n}}\mathrm{\right)}$ . Let such a system be given; by Theorem 1.11 it suffices to check whether the system has solutions over $\mathbb{C}\mathrm{⟨}\mathit{z}\mathrm{⟩}$. First, observe that if the system has solutions over $\mathbb{C}\mathrm{⟨}\mathit{z}\mathrm{⟩}$ then it has solutions whose constant coefficients are algebraic numbers, that is, the system has solutions over $\stackrel{\mathrm{˜}}{\mathbb{Q}}\mathrm{⟨}\mathit{z}\mathrm{⟩}$. So, here is an algorithm that decides whether the system has solutions over ${{H}}_{\mathit{z}}\mathrm{\left(}\mathrm{\left\{}\mathrm{0}{\mathrm{\right\}}}^{\mathit{n}}\mathrm{\right)}$ : We run in parallel the following two processes. The first process lists the tuples of $\stackrel{\mathrm{˜}}{\mathbb{Q}}\mathrm{⟨}\mathit{z}\mathrm{⟩}$ and determines whether each one of them is a solution. The second process determines whether the system has solutions modulo ${\mathit{z}}^{\mathit{m}}$ for $\mathit{m}\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{2}$ (observe that the reduction of the system modulo any ${\mathit{z}}^{\mathit{m}}$ reduces to solving a new system over $\stackrel{\mathrm{˜}}{\mathbb{Q}}$, which can be done by an algorithm, since the existential theory of any algebraically closed field, such as $\stackrel{\mathrm{˜}}{\mathbb{Q}}$, is decidable). If the system has solutions over $\stackrel{\mathrm{˜}}{\mathbb{Q}}\mathrm{⟨}\mathit{z}\mathrm{⟩}$ then the first process will find them, eventually. If the system has no solutions, then the second process will eventually find an $\mathit{m}$ for which the system has no solution modulo ${\mathit{z}}^{\mathit{m}}$. We note that a similar algorithm works for systems of algebraic differential equations (cf. [21]).

It is evident that, in the domains where one has analogues of Theorems 1.11 and 1.12, one may expect decidability results, similar to the above. Unfortunately, analogues of Theorem 1.12 in rings ${{H}}_{\mathit{z}}\mathrm{\left(}\mathit{D}\mathrm{\right)}$ , for $\mathit{D}$ not a singleton, are not known. Theorem 1.11 has been extended to the very general case of a regular ring (by Spivakovski). Surprisingly perhaps, there is a similar result for compact domains $\mathit{D}$ by van den Dries:

Theorem 1.13 Assume that $\mathit{D}$ is a compact subset of $\mathbb{C}$ and denote by ${\mathbb{C}}_{\mathit{D}}\mathrm{⟨}\mathit{z}\mathrm{⟩}$ the ring of functions on $\mathit{D}$, in the variable $\mathit{z}$, which are algebraic over $\mathbb{C}\mathrm{\left(}\mathit{z}\mathrm{\right)}$ and analytic on an open superset of D. Let $\mathit{x}\mathrm{=}$ $\mathrm{\left(}{\mathit{x}}_{\mathrm{1}}\mathrm{,}\mathrm{}{\mathit{x}}_{\mathit{m}}\mathrm{\right)}\mathrm{,}$ ${\mathit{f}}_{\mathit{i}}\mathrm{\in }{\mathbb{C}}_{\mathit{D}}\mathrm{⟨}\mathit{z}\mathrm{⟩}\mathrm{\left[}\mathit{x}\mathrm{\right]}$. Let $\mathit{\epsilon }\mathrm{>}\mathrm{0}\mathrm{.}$

Assume that the system of equations ${\wedge }_{\mathit{i}}{\mathit{f}}_{\mathit{i}}\mathrm{\left[}{\mathit{x}}_{\mathrm{1}}\mathrm{,}\mathrm{}{\mathit{x}}_{\mathit{m}}\mathrm{\right]}$ has a solution $\mathit{\alpha }\mathrm{=}$ $\mathrm{\left(}{\mathit{\alpha }}_{\mathrm{1}}\mathrm{,}\mathrm{}{\mathit{\alpha }}_{\mathit{m}}\mathrm{\right)}$ in ${{H}}_{\mathit{z}}\mathrm{\left(}\mathit{D}\mathrm{\right)}$ . Then it also has a solution $\mathit{\beta }\mathrm{=}\mathrm{\left(}{\mathit{\beta }}_{\mathrm{1}}\mathrm{,}\mathrm{}{\mathit{\beta }}_{\mathit{m}}\mathrm{\right)}$ with ${\mathit{\beta }}_{\mathit{i}}\mathrm{\in }{\mathbb{C}}_{\mathit{D}}\mathrm{⟨}\mathit{z}\mathrm{⟩}$ and such that

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$\mathrm{|}\mathit{\beta }\mathrm{-}\mathit{\alpha }{\mathrm{|}}_{\mathrm{\infty }}\mathrm{<}\mathit{\epsilon }$ where $\mathrm{|}$ ${\mathrm{|}}_{\mathrm{\infty }}$ denotes the supremum norm on $\mathit{D}\mathrm{.}$

It is easy to see that Theorem 1.11 reduces the question of decidability of the existential theory of ${{H}}_{\mathit{z}}\mathrm{\left(}\mathit{D}\mathrm{\right)}$ , for $\mathit{D}$ compact, to the similar question for the existential theory of the ring of algebraic functions which are analytic on $\mathit{D}$. But the problem remains open:

Question 1.14 Is the existential theory of ${{H}}_{\mathit{z}}\mathrm{\left(}\mathit{D}\mathrm{\right)}$ decidable for $\mathit{D}\mathrm{=}{\mathbb{C}}^{\mathit{j}}$?for $\mathit{D}\mathrm{=}\mathrm{\left(}\mathit{t}\mathit{h}\mathit{e}$ open unit disc)? for $\mathit{D}\mathrm{=}$($\mathit{t}\mathit{h}\mathit{e}$ closed unit disc)?

In a later section we give more information on this problem.

1.9 Positive-existential versus existential

In some cases we do have a decision method for the positive-existential theory of a ring but we do not know the similar result for the existential theory. Such is the case, for example, for the power series ring ${\mathrm{F}}_{\mathit{p}}\mathrm{\left[}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{\right]}$ where $\mathit{t}$ is one variable.

In [22] it is shown that if there is Resolution of Singularities in positive characteristic $\mathit{p}$ then the positive-existential theory of ${\mathrm{F}}_{\mathit{p}}\mathrm{\left[}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{\right]}$ is decidable.

Exercise 1.15 Prove that for any prime $\mathit{p}\mathrm{\ge }\mathrm{3}$ the set ${\mathrm{F}}_{\mathit{p}}\mathrm{\left[}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{\right]}$ is positive-existentially definable in the field ${\mathrm{F}}_{\mathit{p}}\mathrm{\left(}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\right)}$ in the following way:

For any $\mathit{x}\mathrm{\in }{\mathrm{F}}_{\mathit{p}}\mathrm{\left(}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\right)}\mathrm{,}$ $\mathit{x}\mathrm{\in }{\mathrm{F}}_{\mathit{p}}\mathrm{\left[}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{\right]}$ if and only if

$\mathrm{\exists }\mathit{y}\mathrm{\left[}\mathrm{1}\mathrm{+}\mathit{t}{\mathit{x}}^{\mathrm{2}}\mathrm{=}{\mathit{y}}^{\mathrm{2}}\mathrm{\right]}$ .

Conclude that the existential theory of ${\mathrm{F}}_{\mathit{p}}\mathrm{\left[}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{\right]}$ is decidable if and only if the positive- existential theory of ${\mathrm{F}}_{\mathit{p}}\mathrm{\left(}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\right)}$ is decidable.

A rough presentation of some ideas in the proof is the following:

Assume that $\mathit{V}$ and $\mathit{Y}$ are two affine varieties over ${\mathrm{F}}_{\mathit{p}}\mathrm{\left[}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{\right]}$, each of which is given by a finite set of polynomial equations. We want to decide whether the quasi-variety $\mathit{V}\mathrm{\setminus }\mathit{Y}$ has points rational over ${\mathrm{F}}_{\mathit{p}}\mathrm{\left[}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{\right]}\mathrm{.}$

Step 1: Reduce the problem to the one whether $\mathit{V}$ has non-singular points, rational over ${\mathrm{F}}_{\mathit{p}}\mathrm{\left[}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{\right]}$. This is done via the Hensel-Rychlic Lemma (a sort of formal analogue of the Inverse Function Theorem for real functions).

Step 2: A Resolution of Singularities $\mathrm{\left(}\mathrm{R}\mathrm{o}\mathrm{S}\mathrm{\right)}$ of the variety $\mathit{V}$ over the field ${\mathrm{F}}_{\mathit{p}}\mathrm{\left(}\mathrm{\left(}\mathit{t}\mathrm{\right)}\mathrm{\right)}$ is $\mathrm{a}$ (not necessarily irreducible) non-singular variety $\mathit{W}$, together with a surjective morphism $\mathit{f}$ : $\mathit{W}\mathrm{\to }\mathit{V}$. A basic observation is:

Observation: If every variety over a countable recursive field $\mathit{K}$ has $\mathrm{R}\mathrm{o}\mathrm{S}$ then that is effective $\mathrm{\left(}$that $\mathrm{i}\mathrm{s}\mathrm{,}\mathrm{}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{}\mathrm{c}\mathrm{a}\mathrm{n}$ find $\mathrm{a}\mathrm{}\mathrm{R}\mathrm{o}\mathrm{S}\mathrm{\right)}$ .

This is done roughly as follows: List all possible pairs $\mathrm{\left(}\mathit{W}\mathrm{,}\mathrm{}\mathit{f}\mathrm{\right)}$ and for each one of them, in the order of enumeration, check whether $\mathit{f}$ maps onto $\mathit{V}$ (this can be done effectively but we will not go into the details here). Since we have assumed that some $\mathrm{R}\mathrm{o}\mathrm{S}$ exists, at some point we will find one.

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Existence of $\mathrm{R}\mathrm{o}\mathrm{S}$ is known (for any variety) over any field of zero characteristic (due to Hironaka), but unknown in positive characteristic. In what follows we will assume existence of $\mathrm{R}\mathrm{o}\mathrm{S}$ in characteristic $\mathit{p}\mathrm{.}$

So assume that $\mathit{V}$ has a $\mathrm{R}\mathrm{o}\mathrm{S}\mathrm{\left(}\mathit{W}\mathrm{,}\mathrm{}\mathit{f}\mathrm{\right)}$ . The singular locus (set of singular points) of $\mathit{V}$ is mapped inversely by $\mathit{f}$ to some components (in the sense of the Zariski topology) of $\mathit{W}$. One subtracts those components from $\mathit{W}$ (it can be shown that this can be done effectively) to obtain a variety ${\mathit{W}}_{\mathrm{0}}$. Then the question whether $\mathit{V}$ has non-singular points over ${\mathrm{F}}_{\mathit{p}}\mathrm{\left[}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{\right]}$ is equivalent to the question whether a certain closed (in the sense of the natural topology of ${\mathrm{F}}_{\mathit{p}}\mathrm{\left[}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{\right]}$) of ${\mathit{W}}_{\mathrm{0}}$ has points over ${\mathrm{F}}_{\mathit{p}}\mathrm{\left[}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{\right]}$. It is easy to see that the latter question is equivalent to asking whether a certain affine variety has points which are rational over ${\mathrm{F}}_{\mathit{p}}\mathrm{\left[}\mathrm{\left[}\mathit{t}\mathrm{\right]}\mathrm{\right]}$; The point here is that the phrases ( ${\mathit{W}}_{\mathrm{0}}$ has points' and $\mathrm{\left(}{\mathit{W}}_{\mathrm{0}}$ has non-singular points' are equivalent $\mathrm{\left(}$since ${\mathit{W}}_{\mathrm{0}}$ is $\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{-}$singular)$\mathrm{-}\mathrm{t}\mathrm{h}\mathrm{e}$ same is true for closed subsets of ${\mathit{W}}_{\mathrm{0}}\mathrm{.}$

2 Part $\mathrm{B}$: A qualitative analogue of HTP and the Conjectures of S. Lang

We want to address the following

Question 2.1 Is there an algorithm which, given any variety $\mathit{V}\mathit{o}\mathit{v}\mathit{e}\mathit{r}\mathbb{Q}$, decides whether $\mathit{V}$ has infinitely many points over some number field $\mathit{K}$?

For example, due to the proof by Faltings of Mordell's Conjecture, the question has a positive answer for curves:

Any curve of geometric genus $\mathrm{\ge }\mathrm{2}$ has only a finite number of points over any given number field $\mathit{K}\mathrm{.}$

And it is known that curves of genus $\mathrm{0}$ or 1 have infinitely many points over some number field $\mathit{K}$. Hence the algorithm is: Compute the genus of the curve $\mathit{V}$ (the genus is a geometric invariant and known to be computable). If it is $\mathrm{\le }\mathrm{1}$ reply YES, otherwise NO.

In [33] (and more extensively in [63] and from a different point of view in [6]) S. Lang announced a number of conjectures, which, if true, would imply that varieties that have infinitely many points over some number field are characterized by certain geometric properties. As an example,

Conjecture 2.2 (Lang) Any hyperbolic variety has only a finite number of points over any number field.

A variety $\mathit{V}$ is hyperbolic if every complex analytic map $\mathit{f}$ : $\mathbb{C}\mathrm{\to }\mathit{V}$ is constant (this is one of several equivalent definitions). For example, the (irreducible) curves (varieties of dimension 1) that are hyperbolic are precisely those with geometric genus $\mathrm{\ge }\mathrm{2}\mathrm{.}$

What are the implications of this for Question 2.1? If true, Conjecture 2.2 reduces Question 2.1 to deciding whether a given variety has certain geometric properties. Some

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of those properties (e.g. genus) are known to be decidable; but some other properties are not known (to be decidable or undecidable). As an example of the second kind (to the best of the authors' knowledge), we ask

Question 2.3 Is there an algorithm which, given a variety $\mathit{V}$ over $\mathbb{Q}$, decides whether that is hyperbolic?

In the following sub-section we will address this question.

Even if the answer to it turns out to be positive, there still remain certain geometric properties which have to be decidable if the answer to Question 2.1 is positive.

As we will see in the next sub-section, there is an undecidability result for some type of 'geometric problem'; but it is probably too early to conjecture that Question 2.1 has a negative answer.

Now we give a more precise account of the conjectures of Lang and Vojta. The main notions involved are the following properties of an algebraic variety $\mathit{V}$, defined over $\mathbb{Q}$ (or a number field):

1. $\mathit{V}$ has only finitely many points over any number field.

2. $\mathit{V}$ is hyperbolic (one of several equivalent definitions being that there is no non- constant holomorphic map into $\mathit{V}$).

3. $\mathit{V}$ is Kobayashi hyperbolic. This means that $\mathit{V}$ is hyperbolic and is, in addition, equipped with a metric with some special properties ([33]).

4. $\mathit{V}$ and all its subvarieties are of general type (a notion defined in terms of algebraic geometry).

The union of the conjectures connecting these properties is that all four properties are equivalent. The only part which has been proved is that property 3 (hyperbolic varieties) is equivalent to property 4 (Kobayashi hyperbolic varieties); this is due to Bondi. All the other equivalences are conjectures.

(We are indebted to Professor Campana for explaining to us some of the details for these notions).

2.1 A language for geometric problems

When we study the decidability question for a ring of functions of one (or more) variable $\mathit{z}$, we usually augment the language of rings by a name (constant-symbol) for $\mathit{z}$. This has as a result that the varieties that we study have coefficients in $\mathbb{Z}\mathrm{\left[}\mathit{z}\mathrm{\right]}$ (or a bigger ring) and are not invariant under even the most elementary geometric transformations, for example $\mathit{z}\mathrm{↦}\mathit{z}\mathrm{+}\mathit{\beta }$ with $\mathit{\beta }$ in the base field (field of constant functions). Since we want a language that does not have this defect, we define the languge

${\mathit{L}}_{\mathit{T}}\mathrm{=}\mathrm{\left\{}\mathrm{+}\mathrm{,}\mathrm{}\mathrm{\cdot }\mathrm{;}\mathrm{=}\mathrm{,}\mathrm{}\mathit{T}\mathrm{;}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right\}}$

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which augments the language of rings by the one-place predicate $\mathit{T}$, which will be inerpreted by

$\mathit{T}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{↔}$ (the function $\mathit{x}$ is not a constant)

Notice that, given a variety $\mathit{V}$ through a finite set of defining polynomials, the sentence 'The variety $\mathit{V}$ is hyperbolic' can be expressed in ${\mathit{L}}_{\mathit{T}}$ over ${{H}}_{\mathit{z}}\mathrm{\left(}\mathbb{C}\mathrm{\right)}$ .

Question 2.4 Is the theory (resp. existential theory, positive-existential theory) of ${\mathit{L}}_{\mathit{T}}$ over ${{H}}_{\mathit{z}}\mathrm{\left(}\mathrm{\right)}$ decidable? over $\mathbb{C}\mathrm{\left(}\mathit{z}{\mathrm{\right)}}^{\mathit{j}}$? over $\mathbb{C}\mathrm{\left[}\mathit{z}{\mathrm{\right]}}^{\mathit{j}}$?

The only known relevant results are given by the following two theorems:

Theorem 2.5 ([46]) The positive-existential theory in ${\mathit{L}}_{\mathit{T}}$ of a polynomial ring $\mathit{F}\mathrm{\left[}\mathit{z}\mathrm{\right]}$ over a field $\mathit{F}$ is undecidable.

Theorem 2.6 (52]) The existential theory in ${\mathit{L}}_{\mathit{T}}$ of the ring ${{H}}_{\mathit{z}}\mathrm{\left(}\mathit{U}\mathrm{\right)}$ is decidable, where $\mathit{U}$ is either the open or the closed unit disc.

The analogous question for fields of rational functions and for ${{H}}_{\mathit{z}}\mathrm{\left(}\mathbb{C}\mathrm{\right)}$ are open.

An outline of proof of Rubel's Theorem 2.6 is:

Let $\mathit{V}$ be an affine variety defined over $\mathbb{Q}$ (or $\stackrel{\mathrm{˜}}{\mathbb{Q}}$), defined by a finite set of equations ${\mathit{f}}_{\mathit{i}}\mathrm{\left(}\mathit{X}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathrm{\left(}\mathit{X}\mathrm{=}\mathrm{\left(}{\mathit{X}}_{\mathrm{1}}\mathrm{,}\mathrm{}\mathrm{\text{...}}\mathrm{,}\mathrm{}{\mathit{X}}_{\mathit{m}}\mathrm{\right)}$ is a tuple of variables and ${\mathit{f}}_{\mathit{i}}\mathrm{\in }\mathbb{Q}\mathrm{\left[}\mathit{X}\mathrm{\right]}$).

We want to determine whether $\mathit{V}$ has points over ${{H}}_{\mathit{z}}\mathrm{\left(}\mathit{U}\mathrm{\right)}$ . We consider the system of differential equations and in-equations

$\mathit{S}\mathrm{:}\mathrm{\left(}{\wedge }_{\mathit{i}}{\mathit{f}}_{\mathit{i}}\mathrm{\left(}\mathit{X}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathrm{\right)}\mathrm{\wedge }\mathrm{\left(}{\wedge }_{\mathit{j}}\frac{\mathit{d}{\mathit{X}}_{\mathit{j}}}{\mathit{d}\mathit{z}}\mathrm{\ne }\mathrm{0}\mathrm{\right)}$

One examines $\mathit{S}$ for solutions in a differentially closed field which extends ${{H}}_{\mathit{z}}\mathrm{\left(}\mathit{U}\mathrm{\right)}$ . This is effective by results of Seidenberg. If the answer is negative then we are done: the answer over ${{H}}_{\mathit{z}}\mathrm{\left(}\mathit{U}\mathrm{\right)}$ is also negative. If the answer is positive, then, by a theorem of Ritt, $\mathit{S}$ has a solution $\overline{\mathit{X}}\mathrm{\left(}\mathit{z}\mathrm{\right)}$ such that each ${\overline{\mathit{X}}}_{\mathit{i}}$ is analytic in a neighborhood of $\mathit{z}\mathrm{=}\mathrm{0}\mathrm{.}$ Substitute $\mathit{z}$ by $\mathit{c}\mathit{z}$, for a suitable constant $\mathit{c}$ to obtain the solution $\overline{\mathit{X}}\mathrm{\left(}\mathit{c}\mathit{z}\mathrm{\right)}$ (here we need the fact that the ${\mathit{f}}_{\mathit{i}}$ are polynomials over $\mathbb{Q}$, hence their coefficients do not involve z) which is analytic on $\mathit{U}\mathrm{.}$

2.2 A geometric problem

Let $\mathit{K}$ be a number field. We consider the following problem:

Question 2.7 Is there an algorithm to decide whether an arbitrary variety over $\mathit{K}$ contains a rational curve?

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Note that for varieties of dimension one, the answer to this question is positive. Indeed if $\mathit{V}$ is a variety of dimension one, then $\mathit{V}$ will contain a rational curve if and only if one of its irreducible components is a curve of genus $\mathrm{0}$ and this curve contains a $\mathit{K}$-rational point. Since, given a variety over $\mathit{K}$ we can explicitly determine all its irreducible components, and given the fact that the genus of a curve is computable, one easily gets the algorithm asked for in the question.

For higher dimensional varieties the problem is open. However the following conjecture gives some information for surfaces:

Conjecture (S. Lang) Let $\mathit{V}$ be a variety ofgeneral type defined over a number field K. Then there exists a proper closed subvariety $\mathit{W}$, defined over $\mathit{K}$ such that for any number field $\mathit{L}$ containing $\mathit{K}\mathrm{,}$ $\mathit{V}\mathrm{\left(}\mathit{L}\mathrm{\right)}\mathrm{\setminus }\mathit{W}\mathrm{\left(}\mathit{L}\mathrm{\right)}$ is finite.

Let $\mathit{V}$ be a surface of general type, defined over $\mathit{K}$. If $\mathit{V}$ contains a rational curve $\mathit{C}\mathrm{,}$ then $\mathit{V}\mathrm{\left(}\mathit{K}\mathrm{\right)}$ is infinite.

Conversely, suppose that $\mathit{V}$ contains infinitely many points, then by Lang's conjecture there exists a subvariety, i.e. a one-dimensional variety, such that at least one of its irreducible components is a curve which contains infinitely many points (i.e. this curve is either rational or an elliptic curve of positive rank).

Summarizing we have: determining whether a surface contains a curve of genus at most 1 is as difficult as determining whether the surface has infinitely many points or not. In view of the results in the previous section, one is lead to believe that determining whether a variety defined over $\mathit{K}$ contains a $\mathit{K}$-rational curve is as difficult as HTP (K) .

Remark: (a) It is tempting to use induction on the dimension to use the similar ar- gument for arbitrary varieties. Unfortunately this does not work: the subvariety $\mathit{W}$ of $\mathit{V}$ which Lang's conjecture asserts to exist if $\mathit{V}$ is of general type, need itself not be of general type. $\mathrm{⟨}\mathrm{⟩}$

2.3 Varieties with an infinite number of points over a ring

Here we address the decision problem of whether a given variety has an infinite number of points over a fixed ring.

Consider a commutative domain $\mathit{R}$ whose fraction field is not algebraically closed. Let ${\mathit{R}}_{\mathrm{0}}$ be a infinite recursive subring of $\mathit{R}$ with fraction field ${\mathit{k}}_{\mathrm{0}}$. Suppose further that the algebraic closure of ${\mathit{k}}_{\mathrm{0}}$ is not contained in $\mathit{R}$. Denote the cardinality of $\mathit{R}$ by $\mathrm{|}\mathit{R}\mathrm{|}\mathrm{.}$ Let $\mathit{S}$ be an arbitrary proper subset of $\mathrm{N}\mathrm{\cup }\mathrm{\left\{}\mathrm{|}\mathit{R}\mathrm{|}\mathrm{\right\}}\mathrm{.}$

Theorem 2.8 Consider the following two decision problems:

(a) Is there an algorithm to decide for an arbitrary polynomial with coefficients in ${\mathit{k}}_{\mathrm{0}}$ whether ${\mathit{N}}_{\mathit{R}}\mathrm{\left(}\mathit{f}\mathrm{\right)}\mathrm{\in }\mathit{S}$

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(b) Is there an algorithm to decide for an arbitrary polynomial with coefficients in ${\mathit{k}}_{\mathrm{0}}$ whether ${\mathit{N}}_{\mathit{R}}\mathrm{\left(}\mathit{f}\mathrm{\right)}\mathrm{=}\mathrm{0}$

A negative answer to (b) implies a negative answer to $\mathrm{\left(}\mathit{a}\mathrm{\right)}$ .

This implies that deciding whether a polynomial has infinitely or finitely many solutions in $\mathit{R}$ is as difficult as deciding whether its has a solution or not. This result was proved by M. Davis for $\mathit{R}\mathrm{=}\mathrm{Z}$ and was generalized by the second author.

3 Part $\mathrm{C}$: Hilbert's tenth problem for analytic and meromorphic functions

We will look more closely into Question 1.14. The main existing results are:

Theorem 3.1 (R. Robinson) Assume that $\mathit{D}$ contains a real open interval. Then the first order theory of ${{H}}_{\mathit{D}}\mathrm{\left(}\mathit{z}\mathrm{\right)}$ in ${\mathit{L}}_{\mathit{z}\mathrm{,}\mathit{C}}$ is undecidable.

Proof: First we state:

Lemma 3.2 (Huuskonen) Assume that $\mathit{D}\mathrm{\subset }\mathrm{C}$ has nonempty interior. Then the set of constants ($\mathit{i}\mathrm{.}\mathit{e}$. of constant functions) in ${{H}}_{\mathit{z}}\mathrm{\left(}\mathit{D}\mathrm{\right)}$ is first order definable in the language $\mathrm{\left\{}\mathrm{+}\mathrm{,}\mathrm{}\mathrm{\cdot }\mathrm{;}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{,}\mathrm{}\mathit{z}\mathrm{\right\}}\mathrm{.}$

For simplicity work in the case $\mathit{q}\mathrm{=}\mathrm{1}$ so that $\mathit{z}\mathrm{=}{\mathit{z}}_{\mathrm{1}}$ and assume that the line segment $\mathrm{\left[}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$ is contained in $\mathit{D}$; the general case is left to the reader. The following formula is equivalent to $\mathit{\alpha }\mathrm{\in }\mathrm{N}\mathrm{-}\mathrm{\left\{}\mathrm{O}\mathrm{\right\}}$:

$\mathit{C}\mathrm{\left(}\mathit{\alpha }\mathrm{\right)}\mathrm{\wedge }\mathrm{\exists }\mathit{x}\mathrm{\left[}\mathit{x}\mathrm{\left(}\mathrm{O}\mathrm{\right)}\mathrm{=}\mathrm{1}\mathrm{\wedge }\mathit{x}\mathrm{\left(}\mathrm{1}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathrm{\right]}$

$\mathrm{\wedge }\mathrm{\forall }\mathit{y}\mathrm{\left[}\mathrm{\left(}\mathit{y}\mathrm{\in }\mathbb{C}\mathrm{\wedge }\mathit{y}\mathrm{\ne }\mathrm{0}\mathrm{\wedge }\mathit{y}\mathrm{+}\mathrm{1}\mathrm{\ne }\mathrm{0}\mathrm{\right)}\mathrm{\to }\mathrm{\left(}\mathit{x}\mathrm{\left(}\frac{\mathrm{1}}{\mathit{y}}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathrm{\to }\mathrm{\left(}\mathit{x}\mathrm{\left(}\frac{\mathrm{1}}{\mathit{y}\mathrm{+}\mathrm{1}}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathrm{\right)}\mathrm{\bigvee }\mathit{y}\mathrm{=}\mathit{\alpha }\mathrm{\right)}\mathrm{\right]}\mathrm{\right]}$ Of course the expressions of the form $\frac{\mathrm{1}}{\mathit{z}}$ do not belong to ${\mathit{L}}_{\mathit{z}}$ but they can be replaced by new variables $\mathit{w}$ preceeded by $\mathrm{\exists }\mathit{w}$: $\mathit{z}\mathrm{\cdot }\mathit{w}\mathrm{=}\mathrm{1}\mathrm{.}$

$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{←}$direction holds because if $\mathit{\alpha }\mathrm{\notin }\mathrm{N}$, then the formula implies that each point $\frac{\mathrm{1}}{\mathit{n}}$, with $\mathit{n}\mathrm{\in }\mathrm{N}\mathrm{-}\mathrm{\left\{}\mathrm{O}\mathrm{\right\}}$, is a zero of the analytic function $\mathit{x}$, while $\mathit{x}\mathrm{\left(}\mathrm{O}\mathrm{\right)}\mathrm{=}\mathrm{1}$, which is impossible by the continuity of $\mathit{x}$; for $\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{\to }$direction, observe that, for $\mathit{n}\mathrm{\in }\mathrm{N}\mathrm{-}\mathrm{\left\{}\mathrm{O}\mathrm{\right\}}\mathrm{,}$ the formula is realized by taking

$\mathit{x}\mathrm{=}\mathrm{\left(}\mathrm{-}\mathrm{1}{\mathrm{\right)}}^{\mathit{n}\mathrm{-}\mathrm{1}}\mathrm{\left(}\mathit{n}\mathrm{-}\mathrm{1}\mathrm{\right)}\mathrm{!}\mathrm{\left(}\mathit{z}\mathrm{-}\mathrm{1}\mathrm{\right)}\mathrm{\left(}\mathit{z}\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\mathrm{\right)}\mathrm{\text{...}}\mathrm{\left(}\mathit{z}\mathrm{-}\frac{\mathrm{1}}{\mathit{n}\mathrm{-}\mathrm{1}}\mathrm{\right)}$

$\mathrm{⟨}\mathrm{⟩}$

It is obvious that the proof of this Theorem shows the similar result for any ring of functions which are continuous on $\mathrm{\left[}\mathrm{0}\mathrm{,}\mathrm{1}\mathrm{\right]}$, containing the identity function.

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Theorem 3.3 ([37]) The positive-existential theory of the ring ${{H}}_{\mathit{p}\mathrm{,}\mathit{z}}\mathrm{\left(}{\mathbb{C}}_{\mathit{p}}\mathrm{\right)}$ of functions of the variable $\mathit{z}$ which are analytic on the $\mathit{p}$-adic complex plane ${\mathbb{C}}_{\mathit{p}}$, in the language which extends the language of rings by a constant-symbol for $\mathit{z}$, is undecidable.

Theorem 3.4 (Vidaux, [62]) The positive-existential theory of the field ${\mathrm{M}}_{\mathit{p}\mathrm{,}\mathit{z}}\mathrm{\left(}{\mathbb{C}}_{\mathit{p}}\mathrm{\right)}$ of functions of the variable $\mathit{z}$ which are meromorphic on the $\mathit{p}$-adic complex plane ${\mathbb{C}}_{\mathit{p}}$, in the language which extends the language of rings by a constant-symbol for $\mathit{z}$ and by a predicate symbol for the property $\mathit{\zeta }\mathit{t}\mathit{h}\mathit{e}$ function $\mathit{x}$ has $\mathit{z}\mathrm{=}\mathrm{0}$ as a zero', is undecidable.

The analogous questions over the field of complex numbers $\mathbb{C}$ are open.

4 Part $\mathrm{D}$: Comments on the analogue of Hilbert's

4 Part $\mathrm{D}$: Comments on tenth problem for $\mathbb{Q}$

the analogue of Hilbert's

As mentioned earlier, a major open problem in the area of decidability of existential theories of rings is the analogue of Hilbert's tenth problem for the field $\mathbb{Q}$ of rational numbers.

Question 4.1 Does the analogue of Hilbert's tenth problem for $\mathbb{Q}$ have a negative an- swer?

A naive approach might seek a possible positive answer to the following:

Question 4.2 Is the set $\mathbb{Z}$ existentially definable in the field ${\mathbb{Q}}^{\mathit{j}}$?

which, if true, would imply obviously a YES answer to Question 4.1. But the following observation seems to make such an expectation unlikely:

All known examples of algebraic varieties over $\mathbb{Q}$ have the property that the real topo- logical closure of the Zariski closure of their rational (over $\mathbb{Q}$) points has finitely many connected components.

In consequence Mazur asked whether this is true for all algebraic varieties ([40]). He also stated a more general similar statement (an analogue where the real topology is substituted by the $\mathit{p}$-adic topologies). These questions remain open. An implication of a possible positive answer to Mazur's Question would be that Question 4.2 has a negative answer: Finitely many components project onto finitely many components, hence existential sets of $\mathbb{Q}$ (being projections of varieties) would have only finitely many components, hence $\mathbb{Z}$ can not be one of them. Actually the implications are much deeper (cf. [10]). Some of us doubt the truth of Mazur's Question (mainly because the analogue of the $\mathit{p}$-adic version fails in global fields of positive characteristic). But still, most (if not all) of us, expect the answer to Question 4.2 to be negative.

In [45] a programme is presented for proving Question 4.1. It is based on interpreting the rational integers as the points (over $\mathbb{Q}$) of an elliptic curve of rank 1 over $\mathbb{Q}\mathrm{.}$ Addition among points is given by the addition law on the curve. It therefore suffices to

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express existentially (in terms of the coordinates of the points) 'multiplication' among points. Since this seems inaccessible for the moment, one may, as a first step, try to define divisibility among points. Modulo conjectures (by Cornelissen and Everest) this may be existential. But even this does not suffice due to the fact that, as we saw, the existential theory of addition and divisibility over $\mathbb{Z}$ is decidable. Hence more (existentially definable) structure is needed. A proposal of [45] to this effect seems unlikely (by arguments of Cornelissen). But Cornelissen proposed a possible remedy: Look, not at the points of an elliptic curve, but to the points (over $\mathbb{Q}$) of an abelian variety with a group of points which, given a natural additional structure of multiplication, is isomorphic to a real quadratic extension of $\mathbb{Z}$ (say $\mathbb{Z}\mathrm{\left[}\sqrt{\mathrm{5}}\mathrm{\right]}$). Such varieties do exist! If one can define existentially divisibility among points of such a variety, then a YES answer to Question 4.1 will result from the undecidability of the existential theory of addition and divisibility over $\mathbb{Z}\mathrm{\left[}\sqrt{\mathrm{5}}\mathrm{\right]}\mathrm{!}$ See [11] for additional information on this type of approach.

Let us mention here that most, if not all, of the above approaches to resolve Question 4.1 would also disprove Mazur's Question.

Relevant material may be found also in [31], [31]), [41] and the survey [47].

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