arXiv subjects

Nathanaël Mariaule

Publications and source records attributed to Nathanaël Mariaule.

5 recordsLinked to original sources

arXiv2019

Expansions of the $p$-adic numbers that interprets the ring of integers

Let $\widetilde{\mathbb{Q}_p}$ be the field of $p$-adic numbers in the language of rings. In this paper we consider the theory of $\widetilde{\mathbb{Q}_p}$ expanded by two predicates interpreted by multiplicative subgroups $α^\mathbb{Z}$ and $β^\mathbb{Z}$ where $α, β\in\mathbb{N}$ are multiplicatively independent. We show that the theory of this structure interprets Peano arithmetic if $α$ and $β$ have positive $p$-adic valuation. If either $α$ or $β$ has zero valuation we show that the theory of $(\widetilde{\mathbb{Q}_p}, α^\mathbb{Z}, β^\mathbb{Z})$ does not interpret Peano arithmetic. In that case we also prove that the theory is decidable iff the theory of $(\widetilde{\mathbb{Q}_p}, α^\mathbb{Z}\cdot β^\mathbb{Z})$ is decidable.

math.LO↗

arXiv2018

Expansion of Presburger arithmetic with the exchange property

Let $G$ be a model of Presburger arithmetic. Let $\mathcal{L}$ be an expansion of the language of Presburger $\mathcal{L}_{Pres}$. In this paper we prove that the $\mathcal{L}$-theory of $G$ is $\mathcal{L}_{Pres}$-minimal iff it has the exchange property and any bounded definable set has a maximum.

math.LO↗

arXiv2018

Model theory of the field of $p$-adic numbers expanded by a multiplicative subgroup

Let $G$ be a multiplicative subgroup of $\mathbb{Q}_p$. In this paper, we describe the theory of the pair $(\mathbb{Q}_p, G)$ under the condition that $G$ satisfies Mann property and is small as subset of a first-order structure. First, we give an axiomatisation of the first-order theory of this structure. This includes an axiomatisation of the theory of the group $G$ as valued group (with the valuation induced on $G$ by the $p$-adic valuation). If the subgroups $G^{[n]}$ of $G$ have finite index for all $n$, we describe the definable sets in this theory and prove that it is NIP. Finally, we extend some of our results to the subanalytic setting.

math.LO↗

arXiv2017

Effective model-completeness for p-adic analytic structures

In their paper 'p-adic and real subanalytic sets, J. Denef and L. van den Dries prove that the theory of the ring of p-adic integers admits the elimination of quantifiers in the language of p-adic restricted analytic functions expanded by a division symbol. In this paper, we are interested in restriction of this language: Let F be any family of restricted analytic functions, we construct an expansion of F so that the theory of the ring of p-adic integers is model-complete in the corresponding language. Next, we give conditions on F so that the model-completeness is effective. Finally, we apply our results in the context of p-adic exponential rings.

math.LO↗

arXiv2014

p-adic exponential ring, p-adic Schanuel's conjecture and decidability

Let exp(x) be the function determined by the classical power series of the exponentiation. Then E_p(x):=exp(px) is well-defined on Zp, the ring of p-adic integer (for p not equal to 2, we set E_2(x)=exp(4x)). Furthermore, E_p determines a structure of exponential ring on Zp. In this paper, we prove that if a p-adic version of Schanuel's conjecture is true then the theory of (Zp, +, ., 0, 1, E_p) is decidable.

math.LO↗