**Xavier VIDAUX**

__PUBLICATIONS__

*(almost only preprints appear here, not the revised versions - hence they might contain some minor mistakes)*

**14. Optimal bounds for Büchi's problem in modular arithmetic,** with P. Saez and M. Vsemirnov, ** Journal of Number Theory 149**, pp. 368-403 (2014). doi 10.1016/j.jnt.2014.10.008

AMS Subject Classification: 11B50, 11T99, 12Y05, 68Q17

AMS Subject Classification: 03B25, 11U05, 12L05

AMS Subject Classification: 11D09

AMS Subject Classification: 11D09

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+168 (improving results by Pheidas and the second author).

AMS Subject Classification: 03B25, 11D41, 11U05

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).

AMS Subject Classification: 11D09

AMS Subject Classification: 03C60; 12L05

AMS Subject Classification: 03C60, 12L05, 11U05, 11C08

x_{n}^2+x_{n-2}^2=2x_{n-1}^2+2, n=2,..., M-1,

has no integer points other than those that satisfy |x_n|=|x_0|+n. 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 T. Pheidas and X. Vidaux, Fund. Math. 185, pp. 171-194 (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 paper we prove the following:

an analogue of Büchi's problem in rings of polynomials of characteristic either 0 or p>16 and for fields of rational functions of characteristic 0; and

an analogue of Büchi's problem in fields of rational functions of characteristic p>18, 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 at least 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 symbol for the property "x is a square" and symbols for multiplication by each element of the image of

R is a subset of F[t];

the characteristic of F is either 0 or p>18.

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. (Received February 25 2005) (Revised February 13 2006)

AMS Subject Classification: 03C60; 12L05

AMS Subject Classification: 03C60; 12L05

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 =

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 =

AMS Subject Classification: 03C60; 12L05

Last time modified : 23rd of December, 2014