with D. Chompitaki, H. Pasten, T. Pheidas, X. Vidaux

Abstract: One of the main open problems regarding decidability of the existential theory of rings is the analogue of Hilbert's Tenth Problem (HTP) for the ring of entire holomorphic functions in one variable. In the direction of a negative solution, we prove unsolvability of HTP for rings of exponential polynomials. This provides the first known case of HTP for a ring of entire holomorphic functions in one variable strictly containing the polynomials. The technique of proof consists of an interaction between Arithmetic, Analysis, Logic, and Functional Transcendence.

with H. Pasten

Abstract: For any family of elliptic curves over the rational numbers with fixed j-invariant, we prove that the existence of a long sequence of rational points whose x-coordinates form a non-trivial arithmetic progression implies that the Mordell-Weil rank is large, and similarly for y-coordinates. We give applications related to uniform boundedness of ranks, conjectures by Bremner and Mohanty, and arithmetic statistics on elliptic curves. Our approach involves Nevanlinna theory as well as Rémond's quantitative extension of results of Faltings.

with H. Pasten

Abstract: For a positive proportion of primes p and q, we prove that ℤ is Diophantine in the ring of integers of ℚ(∛p,√-q). This provides a new and explicit infinite family of number fields K such that Hilbert's tenth problem for O_K is unsolvable. Our methods use Iwasawa theory and congruences of Heegner points in order to obtain suitable rank stability properties for elliptic curves.

with G. Urzua

Abstract: We construct explicit families of quasi-hyperbolic and hyperbolic surfaces. This is based on earlier work of Vojta, and the recent expansion and generalization of it by the first author. In this paper we further extend it to the singular case, obtaining results for the surface of cuboids, the generalized surfaces of cuboids, and other families of Diophantine surfaces of general type. In particular, we produce explicit families of smooth complete intersection surfaces of multidegrees (m₁,...,m_n) in ℙⁿ⁺² which are hyperbolic, for any n ≥ 8 and any degrees m_i ≥ 2. We also show similar results for complete intersection surfaces in ℙⁿ⁺² for n = 4,5,6,7. These families give evidence for [Dem18, Conjecture 0.18] in the case of surfaces.

Abstract: By explicitly finding the complete set of curves of genus 0 or 1 in some surfaces of general type, we prove that under the Bombieri-Lang conjecture for surfaces, there exists an absolute bound M > 0 such that there are only finitely many sequences of length M formed by k-th rational powers with second differences equal to 2. Moreover, we prove the unconditional analogue of this result for function fields, with M depending only on the genus of the function field. We also find new examples of Brody-hyperbolic surfaces arising from the previous arithmetic problem. Finally, under the Bombieri-Lang conjecture and the ABC-conjecture for four terms, we prove analogous results for sequences of integer powers with possibly different exponents, in which case some exceptional sequences occur.

with H. Pasten

Abstract: We prove a negative solution to the analogue of Hilbert's tenth problem for rings of one variable non-Archimedean entire functions in any characteristic. In the positive characteristic case we prove more: the ring of rational integers is uniformly positive existentially interpretable in the class of {0,1,t,+,‧,=}-structures consisting of positive characteristic rings of entire functions on the variable t. From this we deduce uniform undecidability results for the positive existential theory of such structures. As a key intermediate step, we prove a rationality result for the solutions of certain Pell equation (which a priori could be transcendental entire functions).

Abstract: Given a field F and polynomials a and b in F[t], we prove that in general the sets {aλ+b : λ∈F} and {λ2+aλ+b : λ∈F} contain only finitely many powers and find bounds that are uniform from various points of view. We derive from this analysis various first-order definability and undecidability results. For example, we prove that there is no algorithm to decide whether or not an arbitrary system of linear equations over the integers, together with conditions of the form 'x is a power' and of the form 'x is non-constant' on some of the variables, has a solution in F[t].