CURRICULUM VITÆ. Mathematical interests : Number Theory, Logic (Model Theory), Algebraic Geometry, Complex and p-adic Analysis.

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1 Xavier VIDAUX Associate Professor Universidad de Concepción Facultad de Ciencias Físicas y Matemáticas Departamento de Matemáticas Casilla 160 C Concepción Chile CURRICULUM VITÆ Telephone (office) (secretary) Fax xvidaux@udec.cl Web Page Date of birth 28th of July, 1972 Place of birth Perpignan, France Nationality French Marital status Single Languages French (native), English, Spanish, Modern Greek Mathematical interests : Number Theory, Logic (Model Theory), Algebraic Geometry, Complex and p-adic Analysis. Doctorate of the University of Angers (France) : Équivalence Élémentaire de Corps Elliptiques et Dixième Problème de Hilbert pour les Fonctions Méromorphes p-adiques Globales (Elementary Equivalence of Elliptic Fields and Hilbert s Tenth Problem for p-adic Global Meromorphic Functions) passed with the grade of félicitations du jury (with high honors) on the 28th of September, Jury : A. Macintyre (president, referee), T. Pheidas (supervisor), J.-L. Duret (supervisor), J. V. Geel, A. Escassut, L. Lipshitz (referee) UNIVERSITY STUDIES 1999/2001 3rd and 4th year of doctorate studies in Mathematical Logic, supervised by J.-L. Duret (Angers, France) and T. Pheidas (Heraklion, Greece). Work on Hilbert s Tenth Problem. 1997/1999 1st and 2nd year of doctorate, supervised by J.-L. Duret. Work on elementary classes of elliptic fields. 1995/1996 DEA (Master) of Algebraic Geometry at the University of Angers. RESEARCH FELLOWSHIPS 2009/2012 Chilean research fellowship FONDECYT /2008 Chilean research fellowship FONDECYT /2005 European Marie Curie Individual Fellowship, as a Postdoc in Oxford, UK. 2000/2001 Fellowship from the Greek public organism IKY.

2 PUBLICATIONS some preprints can be downloaded from : 1. Polynomial parametrizations of length 4 Büchi sequences, submitted (2010). Abstract : Büchi s problem asks whether there exists a positive integer M such that any sequence (x n ) of at least M integers, whose second difference of squares is the constant sequence (2), satisifies x 2 n = (x + n) 2 for some x Z. A positive answer to Büchi s problem would imply that there is no algorithm to decide whether or not an arbitrary system of quadratic diagonal forms over Z can represent an arbitrary given vector of integers. We give explicitly an infinite family of polynomial parametrizations of non-trivial length 4 Büchi sequences of integers. In turn, these parametrizations give an explicit infinite family of curves (which we suspect to be hyperelliptic) with the following property: any integral point on one of these curves would give a length 5 non-trivial Büchi sequence of integers (it is not known whether any such sequence exists). AMS Subject Clasification : 11D09 2. The analogue of Büchi s problem for function fields, with A. Shlapentokh, submitted (2010). Abstract : Büchi s n Squares Problem asks for an integer M such that any sequence (x 0,..., x M 1 ), whose second difference of squares is the constant sequence (2) (i.e. x 2 n 2x 2 n 1 + x2 n 2 = 2 for all n), satisfies x2 n = (x + n) 2 for some integer x. Hensley s problem for r-th powers (where r is an integer 2) is a generalization of Büchi s problem asking for an integer M such that, given integers ν and a, the quantity (ν + n) r a cannot be an r-th power for M or more values of the integer n, unless a = 0. The analogues of these problems for rings of functions consider only sequences with at least one non-constant term. Let K be a function field of a curve of genus g. We prove that Hensley s problem for r-th powers has a positive answer for any r if K has characteristic zero, improving results by Pasten and Vojta. In positive characteristic p we obtain a weaker result, but which is enough to prove that Büchi s problem has a positive answer if p 312g+169 (improving results by Pheidas and the second author). AMS Subject Clasification : 03B25, 11D41, 11U05 3. A characterization of Büchi s integer sequences of length 3, with P. Saez, submitted (2010). Abstract : We give a new characterization of Büchi sequences (sequences whose sequence of squares has constant second difference (2)) of length 3 over the integers. Known characterizations of integer Buchi sequences of length 3 are actually characterizations over Q, plus some divisibility criterions that keep integer sequences. AMS Subject Clasification : 11D09 4. A survey on Büchi s problem : new presentations and open problems, with H. Pasten and T. Pheidas, to appear in the Proceedings of the Hausdorff Institute for Mathematics (2009). Abstract : In any commutative ring A with unit, Büchi sequences are those sequences whose second difference of squares is the constant sequence (2). Sequences of elements x n satisfying x 2 n = (x + n) 2 for some fixed x are Büchi sequences that we call trivial. Since we want to study sequences whose elements do not belong to certain subrings (e.g. for fields of rational functions F (z) over a field F we are interested in sequences that are not over F ) the concept of trivial sequences may vary. Büchi s Problem for a ring A asks whether there exists a positive integer M such that any Büchi sequence of length M or more is trivial.

3 We survey the current status of knowledge for Büchi s problem and its analogues for higher-order differences and higher powers. We propose several new and old open problems. We present a few new results and various sketches of proofs of old results (in particular : Vojta s conditional proof for the case of integers and a quite detailed proof for the case of polynomial rings in characteristic zero), and present a new and short proof of the positive answer to Büchi s problem over finite fields with p elements (originally proved by Hensley). We discuss applications to Logic (which were the initial aim for solving these problems). 5. Erratum - The analogue of Büchi s problem for rational functions, with T. Pheidas (University of Crete-Heraklion, Greece), to appear in the Journal of the London Mathematical Society (2009). 6. The analogue of Büchi s problem for cubes in polynomial rings, with T. Pheidas (University of Crete-Heraklion, Greece), Pacific Journal of Mathematics 238-2, pp (2008). Abstract : Let F be a field of zero characteristic. We give the following answer to a generalization of a problem of Büchi over F [t]: A sequence of 92 or more cubes in F [t], not all constant, with third difference constant and equal to 6, is of the form (x + n) 3, n = 0,..., 91, for some x F [t] (cubes of successive elements). We use this, in conjunction to the negative answer to the analogue of Hilbert s Tenth Problem for F [t] in order to show that the solvability of systems of degree-one equations, where some of the variables are assumed to be cubes and (or) non-constant, is an unsolvable problem over F [t]. AMS Subject Clasification : 03C60, 12L05, 11U05, 11C08 7. The analogue of Büchi s problem for rational functions, with T. Pheidas (University of Crete-Heraklion, Grecia), Journal of the London Mathematical Society 74-3, pp (2006). Abstract : Büchi s problem asked whether there exists an integer M such that the surface defined by a system of equations of the form : x 2 n + x 2 n 2 = 2x 2 n n = 2,..., M 1 has no integer points other than those that satisfy : ±x n = ±x 0 + n (the ± signs are independent). If answered positively, it would imply that there is no algorithm which decides, given an arbitrary system Q = (q 1,..., q r ) of integral quadratic forms and an arbitrary r-tuple B = (b 1,..., b r ) of integers whether Q represents B - see Extensions of Büchi s Problem : Questions of Decidability for Addition and k th Powers, avec T. Pheidas (Université de Crète-Héraklion, Grèce), Fundamenta Mathematicae 185, pp (2005). Thus It would imply the following strengthening of the negative answer to Hilbert s Tenth Problem : the positive existential theory of the rational integers in the language of addition and a predicate for the property x is a square would be undecidable. Despite some progress, including a conditional positive answer (depending on conjectures of Lang), Büchi s problem remains open. In this article we prove (A) An analogue of Büchi s problem in rings of polynomials of characteristic either 0 or p 17 and for fields of rational functions of characteristic 0 and (B) An analogue of Büchi s problem in fields of rational functions of characteristic p 19, but only for sequences that satisfy a certain additional hypothesis. As a consequence we prove the following result in Logic : Let F be a field of characteristic either 0 or 17 and let t be a variable. Let L t be the first order language which contains symbols for 0 and 1, a symbol for addition, a

4 symbol for the property x is a square and symbols for multiplication by each element of the image of Z[t] in F [t]. Let R be a subring of F (t), containing the natural image of Z[t] in F (t). Assume that one of the following is true : R F [t]. The characteristic of F is either 0 or p 19. Then multiplication is positive-existentially definable over the ring R, in the language L t. Hence the positive-existential theory of R in L t is decidable if and only if the positiveexistential ring-theory of R in the language of rings, augmented by a constant-symbol for t, is decidable. 8. The analogue of Büchi s problem for polynomials, with T. Pheidas (University of Crete-Heraklion, Greece), Lecture Notes in Computer Science ISSU 3526, pp (2005). Abstract : Büchi s problem asked whether a surface of a specific type, defined over the rationals, has integer points other than some known ones. A consequence of a positive answer would be the following strengthening of the negative answer to Hilbert s tenth problem : the positive existential theory of the rational integers in the language of addition and a predicate for the property x is a square would be undecidable. Despite some progress, including a conditional positive answer (pending on conjectures of Lang), Büchi s problem remains open. In this article we prove an analogue of Büchi s problem in rings of polynomials of characteristic either 0 or p 13. As a consequence we prove the following result in Logic : Let F be a field of characteristic either 0 or 17 and let t be a variable. Let R be a subring of F [t], containing the natural image of Z[t] in F [t]. Let L t be the first order language which contains a symbol for addition in R, a symbol for the property x is a square in F [t] and symbols for multiplication by each element of the image of Z[t] in F [t]. Then multiplication is positive-existentially definable over the ring R, in the language L t. Hence the positive-existential theory of R in L t is decidable if and only if the positive-existential ring-theory of R in the language of rings, augmented by a constant-symbol for t, is decidable. 9. Extensions of Büchi s Problem : Questions of Decidability for Addition and k th Powers, with T. Pheidas (University of Crete-Heraklion, Greece), Fundamenta Mathematicae 185, pp (2005). Abstract : We generalize a question of Büchi : Let R be an integral domain, C a subring and k 2 an integer. Is there an algorithm to decide the solvability in R of any given system of polynomial equations, each of which is linear in the k th powers of the unknowns, with coefficients in C? We state a number-theoretical problem, depending on k, a positive answer to which would imply a negative answer to the question for R = C = Z. We reduce a negative answer for k = 2 and for R = F (t), a field of rational functions of zero characteristic, to the undecidability of the ring theory of F (t). We address the similar question, where we allow, along with the equations, also conditions of the form x is a constant and x takes the value 0 at t = 0, for k = 3 and for function fields R = F (t) of zero characteristic, with C = Z[t]. We prove that a negative

5 answer to this question would follow from a negative answer for a ring between Z and the extension of Z by a primitive cube root of An Analogue of Hilbert s Tenth Problem for Fields of Meromorphic Functions over non-archimedean Valued Fields, Journal of Number Theory, Vol. 101, Issue 1, pp (2003). Abstract : Let K be a complete and algebraically closed valued field. We prove that the set of rational integers is positive existentially definable in the field M of meromorphic functions on K in the language L R of rings augmented by a constant symbol for the independent variable z and by a symbol for the unary relation the function x takes the value 0 at 0. Consequently, we prove that the positive existential theory of M in the language L R is undecidable. In order to obtain these results, we obtain a complete characterization of all analytic projective maps (over K) from an elliptic curve E minus a point to E, for any elliptic curve defined over the field of constants. 11. Multiplication Complexe et Équivalence Élémentaire dans le Langage des Corps, Journal of Symbolic Logic 67 n2, pp (2002). Résumé : Soit K et K deux corps elliptiques avec multiplication complexe sur un corps algébriquement clos k de caractéristique 0, non k-isomorphes, et soit C et C deux courbes ayant pour corps de fonctions K et K respectivement. Nous démontrons que si les anneaux d endomorphismes de C et de C ne sont pas isomorphes, alors K et K ne sont pas élémentairement équivalents dans le langage des corps enrichi d une seule constante (l invariant modulaire). Ce travail fait suite à un travail de David A. Pierce qui se place dans le langage des k-algèbres. 12. Équivalence Élémentaire de Corps Elliptiques, Comptes Rendus de l Académie des Sciences de Paris, Série I 330, pp. 1-4 (2000). Résumé : Il s agit de démontrer une partie de la conjecture suivante : deux corps elliptiques sur un corps algébriquement clos k sont k-isomorphes si et seulement s ils sont élémentairement équivalents dans le langage des corps enrichi d une constante (l invariant modulaire). C est une extension des résultats de Duret sur l équivalence élémentaire des corps de fonctions.