arXiv subjects

Daniel Vargas Montoya

Publications and source records attributed to Daniel Vargas Montoya.

3 recordsLinked to original sources

arXiv2023

Algebraicity modulo p of generalized hypergeometric series $_nF_{n-1}$

Let $f(z)={}_nF_{n-1}(\mathbfα,\mathbfβ)$ be the hypergeometric series with parameters $\mathbfα = (α_1,\ldots,α_n)$ and $\mathbfβ = (β_1,\ldots,β_{n-1},1)$ in $(\mathbb{Q}\cap(0,1])^n$, let $d_{\mathbfα,\mathbfβ}$ be the least common multiple of the denominators of $α_1,\ldots,α_n$, $β_1,\ldots,β_{n-1}$ written in lowest form and let $p$ be a prime number such that $p$ does not divide $d_{\mathbfα,\mathbfβ}$ and $f(z)\in\mathbb{Z}_{(p)}[[z]]$. Recently in \cite{vmsff}, it was shown that if for all $i,j\in\{1,\ldots,n\}$, $α_i-β_j\notin\mathbb{Z}$ then the reduction of $f(z)$ modulo $p$ is algebraic over $\mathbb{F}_p(z)$. A standard way to measure the complexity of an algebraic power series is to estimate its degree and its height. In this work, we prove that if $p>2d_{\mathbfα,\mathbfβ}$ then there is a nonzero polynomial $P_p(Y)\in\mathbb{F}_p(z)[Y]$ having degree at most $p^{2^nφ(d_{\mathbfα,\mathbfβ})}$ and height at most $5^n(n+1)!p^{2^{n}φ({d_{\mathbfα,\mathbfβ})}}$ such that $P_p(f(z)\bmod p)=0$, where $φ$ is the Euler's totient function. Furthermore, our method of proof provides us a way to make an explicit construction of the polynomial $P_p(Y)$. We illustrate this construction by applying it to some explicit hypergeometric series.

math.NT↗

arXiv2023

Maximal Unipotent Monodromy, congruences "à la Lucas" and Algebraic independence

Let $f(z)$ be in $1+z\mathbb{Q}[[z]]$ and $\mathcal{S}$ be an infinite set of prime numbers such that, for all $p\in\mathcal{S}$, we can reduce $f(z)$ modulo $p$. We let $f(z)_{\mid p}$ denote the reduction of $f(z)$ modulo $p$. Generally, when $f(z)$ is D-finite, $f(z)_{\mid p}$ is algebraic over $\mathbb{F}_p(z)$. It turns out that if $f(z)$ is a solution of a polynomial of the form $X-A_p(z)X^{p^l}$, we can use this type of equations to obtain results of transcendence and algebraic independence over $\mathbb{Q}(z)$. In the present paper, we look for conditions on the differential operators annihilating $f(z)$ to guarantee the existence of these particular equations. Suppose that $f(z)$ is solution of a differential operator $\mathcal{H}\in\mathbb{Q}(z)[d/dz]$ having a strong Frobenius structure for all $p\in\mathcal{S}$ and we also suppose that $f(z)$ annihilates a Fuchsian differential operator $\mathcal{D}\in\mathbb{Q}(z)[d/dz]$ such that zero is a regular singular point of $\mathcal{D}$ and the exponents of $\mathcal{D}$ at zero are equal to zero. Our main result states that, for almost every prime $p\in\mathcal{S}$, $f(z)_{\mid p}$ is solution of a polynomial of the form $X-A_p(z)X^{p^l}$, where $A_p(z)$ is a rational function with coefficients in $\mathbb{F}_p$ of height less than or equal to $Cp^{2l}$ with $C$ a positive constant that does not depend on $p$. We also study the algebraic independence of these power series over $\mathbb{Q}(z)$.

math.NT↗

arXiv2021

Algébricité modulo p, séries hypergéométriques et structures de Frobenius forte

This work is devoted to study of algebraicty modulo p of Siegel's G-functions. Our goal is to emphasize the relevance of the notion of strong Frobenius structure, clasically studied in the theory of the p-adic diffenrential equations, for the study of a Adamczewski-Delaygue's conjecture concerning of the degree of algebraicity modulo p of G-functions. For this, we first make a Christol's result explicit by showing that if $f$ is a G-function that is solution of a differential operator $L$ in $ \mathbb{Q}(z)[d/dz]$ of order $n$ endowed of a strong Frobenius structure of period $h$ for the prime number $p$ and that $f(z)$ belongs to $\mathbb{Z}_{(p)}[[z]]$, then the reduction of $f$ modulo $ p $ is algebraic over $\mathbb F_p(z)$ and its algebraicity degree is bounded by $p^{n^2h}$. By generalizing an approach introduced by Salinier, we show that if $L$ is a Fuchsian operator with coefficients in $\mathbb{Q}(z)$, whose monodromy group is rigid and whose exponents are rational numbers, then $L$ has for almost every prime number $p$ a strong Frobenius structure of period $h$, where $h$ is explicitly bounded and does not depend on $p$. A slightly different version of this result has been recently demonstrated by Crew following a different approach based on the $p$ -adic cohomology. We use these two results to solve the mentioned conjecture in the case of generalized hypergeometric series.

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