arXiv subjects

Makoto Kawashima

Publications and source records attributed to Makoto Kawashima.

16 recordsLinked to original sources

arXiv2026

Linear independence of values of hypergeometric functions and arithmetic Gevrey series

We prove new linear independence results for the values of generalized hypergeometric functions ${}_pF_q$ at several distinct algebraic points, over suitable algebraic number fields. Our approach provides a uniform construction of Padé approximants of type II, together with a novel non-vanishing argument for generalized Wronskians of Hermite type. This method applies uniformly across all parameter regimes. Even in the case $p = q+1$, we extend known results from single-point to multi-points settings over general number fields, in both complex and $p$-adic settings. When $p < q+1$, we establish linear independence results over arbitrary number fields; and for $p > q+1$, we confirm that the values do not satisfy global linear relations in the $p$-adic setting in a framework of arithmetic Gevrey series. The results generalize and strengthen earlier works, demonstrating the flexibility of our Padé construction for families of contiguous hypergeometric functions, through a new non-vanishing proof for the determinant, that is crucial for the universality.

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arXiv2026

Hermite's approach to Abelian integrals revisited

In this article, we establish a new linear independence criterion for the values of certain {\it Lauricella hypergeometric series} $F_D$ with rational parameters, in both the complex and $p$-adic settings, over an algebraic number field. This result generalizes a theorem of C.~Hermite \cite{Hermite} on the linear independence of certain Abelian integrals. Our proof relies on explicit Padé-type approximations to solutions of a reducible Jordan-Pochhammer differential equation, which extends the Padé approximations for certain Abelian integrals in \cite{Hermite}. The main novelty of our approach lies in the proof of the non-vanishing of the determinants associated with these Padé-type approximants.

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arXiv2026

Linear independence of periods related to polylogarithms

This paper provides the first criteria for the linear independence of multiple polylogarithm values over algebraic number fields. In particular, we derive novel results regarding the linear independence of products of polylogarithms at distinct points over an algebraic number field. Our approach is based on the explicit construction of Padé-type approximants tailored for multiple polylogarithms.

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arXiv2026

Counting Theorems for Algebraic Relations

Let X be a set definable in a sharply o-minimal structure. We consider the problem of counting the number of points where X intersects algebraic varieties V over Q of dimension k < codim X, as a function of T := deg(V) + h(V), where h(V) is the log-height of V. In particular, we conjecture that after removing a suitable "algebraic part", this number grows polynomially in T -- a generalization of Wilkie's conjecture. We show that this full conjecture implies some open problems in algebraic independence theory. We also formulate a weaker conjecture stating that all intersections above are contained in a poly(T) amount of balls of radius e^{-T}. We then consider the case where X (subset of C^n) is a (compact piece of a) trajectory of a polynomial differential equation satisfying a variant of Nesterenko's D-property. Our main theorem is a proof of the weakened conjecture for such curves when k < sqrt(n) - 1.

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arXiv2025

Padé approximations for products of functions

In this article, we construct new Padé approximations for the \emph{product} of binomial functions and powers of logarithmic functions. While several explicit Padé approximants are known for powers of exponential functions, binomial functions, and logarithmic functions individually, an explicit Padé construction for the product of these functions has not yet been directly achieved. Our main result yields arithmetic applications, providing new linear independence measures for linear forms in $(1+α)^{ω_i}\log^{j_i}(1+α)$ for $1 \le i \le m$ and $0 \le j_i \le r_i - 1$, where $0 < m, r_1, \ldots, r_m \in \mathbb{Z}_{\geq 1}$, $ω_1, \ldots, ω_m \in \mathbb{Q}$, and $0 \le ω_1 < \cdots < ω_m < 1$. These results hold with algebraic coefficients in both the complex and $p$-adic cases. Additionally, we establish that Padé approximation of a single polylogarithm is, in general, perfect.

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arXiv2024

On the linear independence of $p$-adic polygamma values

In this article, we present a new linear independence criterion for values of the $p$-adic polygamma functions defined by J.~Diamond. As an application, we obtain the linear independence of some families of values of the $p$-adic Hurwitz zeta function $ζ_p(s,x)$ at distinct shifts $x$. This improves and extends a previous result due to P.~Bel [5], as well as irrationality results established by F.~Beukers [7]. Our proof is based on a novel and explicit construction of Padé-type approximants of the second kind of Diamond's $p$-adic polygamma functions. This construction is established by using a difference analogue of the Rodrigues formula for orthogonal polynomials.

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arXiv2023

Rodrigues formula and linear independence for values of hypergeometric functions with parameters vary

In this article, we prove a generalized Rodrigues formula for a wide class of holonomic Laurent series, which yields a new linear independence criterion concerning their values at algebraic points. This generalization yields a new construction of Padé approximations including those for Gauss hypergeometric functions. In particular, we obtain a linear independence criterion over a number field concerning values of Gauss hypergeometric functions, allowing the parameters of Gauss hypergeometric functions to vary.

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arXiv2023

Linear Forms in Polylogarithms

Let $r, \,m$ be positive integers. Let $x$ be a rational number with $0 \le x <1$. Consider $Φ_s(x,z) =\displaystyle\sum_{k=0}^{\infty}\frac{z^{k+1}}{{(k+x+1)}^s}$ the $s$-th Lerch function with $s=1, 2, \cdots, r$. When $x=0$, this is a polylogarithmic function. Let $α_1, \cdots, α_m$ be pairwise distinct algebraic numbers of arbitrary degree over the rational number field, with $0<|α_j|<1 \,\,\,(1\leq j \leq m)$. In this article, we show a criterion for the linear independence, over an algebraic number field containing $\mathbb{Q}(α_1, \cdots, α_m)$, of all the $rm+1$ numbers : $Φ_1(x,α_1)$, $Φ_2(x,α_1), $ $\cdots , Φ_r(x,α_1)$, $Φ_1(x,α_2)$, $Φ_2(x,α_2), $ $\cdots , Φ_r(x,α_2), \cdots, \cdots, Φ_1(x,α_m)$, $Φ_2(x,α_m)$, $\cdots , Φ_r(x,α_m)$ and $1$. This is the first result that gives a sufficient condition for the linear independence of values of the Lerch functions at several distinct algebraic points, not necessarily lying in the rational number field nor in quadratic imaginary fields. We give a complete proof with refinements and quantitative statements of the main theorem announced in [10], together with a proof in detail on the non-vanishing Wronskian of Hermite type.

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arXiv2023

Linear independence criteria for generalized polylogarithms with distinct shifts

For a given rational number $x$ and an integer $s\geq 1$, let us consider a generalized polylogarithmic function, often called the Lerch function, defined by $$Φ_{s}(x,z)= \sum_{k=0}^{\infty}\frac{z^{k+1}}{(k+x+1)^s}\enspace.$$ We prove the linear independence over any number field $K$ of the numbers $1$ and $Φ_{s_j}(x_j,α_i)$ with any choice of distinct shifts $x_1,\ldots, x_d$ with $0\le x_1<\ldots<x_d<1$, as well as any choice of depths $1\leq s_1\leq r_1,\ldots, 1\leq s_d\leq r_d$, at distinct algebraic numbers $α_1,\ldots,α_m\in K$ subject to a metric condition. As is usual in the theory, the points $α_i$ need to be chosen sufficiently close to zero with respect to a given fixed place $v_0$ of $K$, Archimedean or finite. This is the first linear independence result with distinct shifts $x_1, \ldots, x_d$ that allows values at different points for generalized polylogarithmic functions. Previous criteria were only for the functions with one fixed shift or at one point. Further, we establish another linear independence criterion for values of the generalized polylogarithmic function with cyclic coefficients. Let $q\geq 1$ be an integer and $\boldsymbol{a}=(a_1,\ldots, a_q)\in K^q$ be a $q$-tuple whose coordinates supposed to be cyclic with the period $q$. Consider the generalized polylogarithmc function with coefficients $$Φ_{\boldsymbol{a},s}(x,z)= \sum_{k=0}^{\infty}\frac{a_{k+1\bmod(q)}\cdot z^{k+1}}{(k+x+1)^s}\enspace.$$ Under suitable condition, we show that the values of these functions are linearly independent over $K$. Our key tool is a new non-vanishing property for a generalized Wronskian of Hermite type associated to our explicit constructions of Padé approximants for this family of generalized polylogarithmic function.

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arXiv2023

Can polylogarithms at algebraic points be linearly independent?

Let $r,m$ be positive integers. Let $0\le x <1$ be a rational number. Let $Φ_s(x,z)$ be the $s$-th Lerch function $\sum_{k=0}^{\infty}\tfrac{z^{k+1}}{(k+x+1)^s}$ with $s=1,2,\ldots ,r$. When $x=0$, this is the polylogarithmic function. Let $α_1,\ldots ,α_m$ be pairwise distinct algebraic numbers with $0<|α_j|<1$ $(1 \le j \le m)$. In this article, we state a linear independence criterion over algebraic number fields of all the $rm+1$ numbers $:$ $Φ_1(x,α_1),Φ_2(x,α_1),\ldots, Φ_r(x,α_1),Φ_1(x,α_2),Φ_2(x,α_2),\ldots, Φ_r(x,α_2),\ldots,Φ_1(x,α_m),Φ_2(x,α_m),\ldots, Φ_r(x,α_m)$ and $1$. This is the first result that gives a sufficient condition for the linear independence of values of the $r$ Lerch functions $Φ_1(x,z),Φ_2(x,z),\ldots, Φ_r(x,z)$ at $m$ distinct algebraic points without any assumption for $r$ and $m$, even for the case $x=0$, the polylogarithms. We give an outline of our proof and explain basic idea.

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arXiv2022

$S$-unit equation in two variables and Padé approximations

In this article, we use Padé approximations constructed for binomial functions, to give a new upper bound for the number of the solutions of the $S$-unit equation. Combining explicit formulae of these Padé approximants with a simple argument relying on Mahler measure and on the local height, we refine the bound due to J.-H. Evertse.

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arXiv2022

The digit exchanges in the rotational beta expansions of algebraic numbers

In this article, we investigate the $β$-expansions of real algebraic numbers. In particular, we give new lower bounds for the number of digit exchanges in the case where $β$ is a Pisot or Salem number. Moreover, we define a new class of algebraic numbers, quasi-Pisot numbers and quasi-Salem numbers, which gives a generalization of Pisot numbers and Salem numbers. Our method is applicable also to the digit expansions of complex algebraic numbers, which gives new estimation. In particular, we investigate the digits of rotational beta expansion by Akiyama and Caalim $[3]$ and zeta-expansion by Surer $[20]$, where the base is a quasi-Pisot or quasi-Salem number.

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arXiv2022

Generalized hypergeometric $G$-functions take linear independent values

In this article, we show a new general linear independence criterion related to values of $G$-functions, including the linear independence of values at algebraic points of contiguous hypergeometric functions, which is not known before. Let $K$ be any algebraic number field and $v$ be a place of $K$. Let $r\in\mathbb{Z}$ with $r\ge2$. Consider $a_1,\ldots,a_{r}, b_1,\ldots,b_{r-1}\in \mathbb{Q}\setminus\{0\}$ not being negative integers. Assume neither $a_k$ nor $a_k+1-b_j$ be strictly positive integers $(1\le k \le r, 1\le j \le r-1)$. Let $α_1,\ldots,α_m\in K\setminus\{0\}$ with $α_1,\ldots,α_m$ pairwise distinct. By choosing sufficiently large $β\in \mathbb{Z}$ depending on $K$ and $v$ such that the points $α_1/β,\ldots,α_m/β$ are closed enough to the origin, we prove that the $rm+1$ numbers~$:$ \begin{align*} &{}_{r}F_{r-1} \biggl(\begin{matrix} a_1,\ldots, a_r\\ b_1, \ldots, b_{r-1} \end{matrix} \biggm| \dfrac{α_i}β\biggr)\enspace, \ \ {}_{r}F_{r-1} \biggl(\begin{matrix} a_1+1,\ldots,\ldots,\ldots,a_r+1\\ b_1+1, \ldots, b_{r-s}+1,b_{r-s+1},\ldots,b_{r-1} \end{matrix} \biggm| \dfrac{α_i}β\biggr)\enspace\\ &(1\le i \le m, 1\le s \le r-1)\end{align*} and $1$ are linearly independent over $K$. The essential ingredient is our term-wise formal construction of type II of Padé approximants together with new non-vanishing argument for the generalized Wronskian.

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arXiv2022

Padé approximation for a class of hypergeometric functions and parametric geometry of numbers

In this article we obtain new irrationality measures for values of functions which belong to a certain class of hypergeometric functions including shifted logarithmic functions, binomial functions and shifted exponential functions. We explicitly construct Padé approximations by using a formal method and show that the associated sequences satisfy a Poincaré-type recurrence. To study precisely the asymptotic behavior of those sequences, we establish an \emph{effective} version of the Poincaré-Perron theorem. As a consequence we obtain, among others, effective irrationality measures for values of binomial functions at rational numbers, which might have useful arithmetic applications. A general theorem on simultaneous rational approximations that we need is proven by using new arguments relying on parametric geometry of numbers.

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arXiv2019

Linear independence of values of logarithms revisited

Let $m\ge 2$ be an integer, $K$ an algebraic number field and $α\in K\setminus \{0,-1\}$ with sufficiently small absolute value. In this article, we provide a new lower bound for linear form in $1,{\rm{log}}(1+α),\ldots,{\rm{log}}^{m-1}(1+α)$ with algebraic integer coefficients in both complex and $p$-adic cases (see Theorem $2.1$ and Theorem $2.4$). Especially, in the complex case, our result is a refinement of the result of Nesterenko-Waldschmidt on the lower bound of linear form in certain values of power of logarithms. The main integrant is based on Hermite-Mahler's Padé approximation of exponential and logarithm functions.

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arXiv2018

Irrationality of special values of formal Laurent series represented by the formal Mellin transform of $G$-functions

Let $p$ be a prime number and $\mathbb{C}_p$ the completion of algebraic closure of $\mathbb{Q}_p$. Let $K$ be an algebraic number field. We fix an embedding $ι_p:\overline{\mathbb{Q}}\hookrightarrow \mathbb{C}_p$ and denote $K_p$ the completion of $K$ with respect to the embedding $ι_p$. Let $g(z)\in K[[z]]$ and denote by $\mathcal{M}(g)(z)\in \tfrac{1}{z}K\left[\left[\tfrac{1}{z}\right]\right]$ the formal Mellin transform of $g(z)$. In this article, we prove that if $\mathcal{M}(g)(z)$ has a good Padé approximation, the special values $\mathcal{M}(g)(α)$ are convergent in $K_p$ and irrational for infinitely many $α\in \mathbb{Q}\cap \left(\mathbb{Q}_p\setminus \mathbb{Z}_p\right)$ satisfying certain conditions. This result can be regarded as a partial generalization of the method of Beukers in his proof the irrationality of special values of $p$-adic Hurwitz zeta functions.

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