Representation of squares by monic second degree polynomials in the field of 𝑝-adic meromorphic functions

Abstract

We prove a result on the representation of squares by monic second degree polynomials in the field of p p -adic meromorphic functions in order to solve positively Büchi’s n n squares problem in this field. Using this result, we prove the non-existence of an algorithm to decide whether a system of diagonal quadratic forms over Z [ z ] \mathbb {Z}[z] represents or not in the ring of p p -adic entire functions (in the variable z z ) a given vector of polynomials in Z [ z ] \mathbb {Z}[z] , and a similar result for p p -adic meromorphic functions when the systems allow vanishing conditions on the unknowns. This improves the known negative answers for the analogue of Hilbert’s Tenth Problem for these structures. We also improve some results by Vojta concerning the case of complex meromorphic functions, the case of function fields and finally the case of number fields, and we show an intimate relation of the latter with Bombieri’s Conjecture for surfaces over number fields.