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Hideki Matsumura

Publications and source records attributed to Hideki Matsumura.

9 recordsLinked to original sources
arXiv2026

Combinatorial designs and the Prouhet--Tarry--Escott problem

This is the first paper that provides a systematic treatment of the $r$-dimensional PTE problem in additive number theory, abbreviated by PTE$_r$, through its connection with combinatorial design theory, the branch of combinatorial mathematics that deals with finite set systems or arrangements with the ^^ balancedness' conditions. We first propose a combinatorial reconsideration of the definition of nontrivial solution introduced by Alpers and Tijdeman (2007), and then prove a fundamental lower bound for the size of such solutions. We exhibit high-dimensional minimal solutions with respect to the fundamental bound, which inherently have the structure of distinctive block designs or orthogonal arrays (OAs). Next, we develop a powerful method for constructing PTE$_r$ solutions via various classes of combinatorial designs such as block designs and OAs. Furthermore, we explore two dimension-lifting methods for constructing PTE$_r$ solutions: one is a combinatorial composition that produces PTE$_r$ solutions by embedding lower-dimensional solutions into OAs with $r$ columns, and the other is a recursive technique in which a PET$_r$ solution is constructed by taking the Cartesian product of two lower-dimensional solutions. It is emphasized that our results generalize many previous works, including a measure-theoretic construction by Lorentz (1949) and its geometric analog by Alpers and Tijdeman (2007), a key lemma in Jacroux's work (1995) on the construction of sets of integers with equal power sums, and the famous Borwein solution and its two-dimensional extension by Matsumura and Sawa (2025). In addition, we prove a characterization theorem for ideal solutions of the PTE$_1$ and discuss the connection with a curious phenomenon, called half-integer design, that is rarely reported in the combinatorial design theory or spherical design theory.

math.CO↗
arXiv2025

Ellipsoidal designs and the Prouhet--Tarry--Escott problem

The notion of ellipsoidal design was first introduced by Pandey (2022) as a full generalization of spherical designs on the unit circle $S^1$. In this paper, we elucidate the advantages of examining the connections between ellipsoidal design and the two-dimensional Prouhet--Tarry--Escott problem, say ${\mathrm PTE}_2$, originally introduced by Alpers and Tijdeman (2007) as a natural generalization of the classical one-dimensional PTE problem (${\mathrm PTE}_1$). We first provide a combinatorial criterion for the construction of solutions of ${\mathrm PTE}_2$ from a pair of ellipsoidal designs. We also give an arithmetic proof of the Stroud-type bound for ellipsoidal designs, and then establish a classification theorem for designs with equality. Such a classification result is closely related to an open question on the existence of rational spherical $4$-designs on $S^1$, discussed in Cui, Xia and Xiang (2019). As far as the authors know, a solution found by Alpers and Tijdeman is the first and the only known parametric ideal solution of degree $5$ for ${\mathrm PTE}_2$. Moreover, as one of our main theorems, we prove that the Alpers--Tijdeman solution is equivalent to a certain two-dimensional extension of the famous Borwein solution for ${\mathrm PTE}_1$. As a by-product of this theorem, we discover a family of ellipsoidal $5$-designs among the Alpers--Tijdeman solution.

math.CO↗
arXiv2023

A generalization of formulas for the discriminants of quasi-orthogonal polynomials with applications to hypergeometric polynomials

Let $K$ be a field. In this article, we derive a formula for the discriminant of a sequence $\{r_{A,n}+c r_{A,n-1}\}$ of polynomials. Here, $c \in K$ and $\{r_{A,n} \}$ is a sequence of polynomials satisfying a certain recurrence relation that is considered by Ulas or Turaj. There are several works calculating the discriminants of given polynomials. For example, Kaneko--Niiho and Mahlburg--Ono independently proved the formula for the discriminants of certain hypergeometric polynomials that are related to $j$-invariants of supersingular elliptic curves. Sawa--Uchida proved the formula for the discriminants of quasi-Jacobi polynomials. In this article, we present a uniform way to prove a vast generalization of the above formulas. In the proof, we use the formulas for the resultants Res$(r_{A,n},r_{A,n-1})$ by Ulas and Turaj that are generalizations of Schur's classical formula for the resultants.

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arXiv2023

Quadrature formulas for Bessel polynomials

A quadrature formula is a formula computing a definite integration by evaluation at finite points. The existence of certain quadrature formulas for orthogonal polynomials is related to interesting problems such as Waring's problem in number theory and spherical designs in algebraic combinatorics. Sawa and Uchida proved the existence and the non-existence of certain rational quadrature formulas for the weight functions of certain classical orthogonal polynomials. Classical orthogonal polynomials belong to the Askey-scheme, which is a hierarchy of hypergeometric orthogonal polynomials. Thus, it is natural to extend the work of Sawa and Uchida to other polynomials in the Askey-scheme. In this article, we extend the work of Sawa and Uchida to the weight function of the Bessel polynomials. In the proofs, we use the Riesz--Shohat theorem and Newton polygons. It is also of number theoretic interest that proofs of some results are reduced to determining the sets of rational points on elliptic curves.

math.NT↗
arXiv2022

Elliptic analogue of irregular prime numbers for the $p^{n}$-division fields of the curves $y^{2} = x^{3}-(s^{4}+t^{2})x$

A prime number $p$ is said to be irregular if it divides the class number of the $p$-th cyclotomic field $\mathbb{Q}(ζ_{p}) = \mathbb{Q}(\mathbb{G}_m[p])$. In this paper, we study its elliptic analogue for the division fields of an elliptic curve. More precisely, for a prime number $p \geq 5$ and a positive integer $n$, we study the $p$-divisibility of the class number of the $p^{n}$-division field $\mathbb{Q}(E[p^{n}])$ of an elliptic curve $E$ of the form $y^{2} = x^{3}-(s^{4}+t^{2})x$. In particular, we construct a certain infinite subfamily consisting of curves with novel properties that they are of Mordell-Weil rank 1 and the class numbers of their $p^{n}$-division fields are divisible by $p^{2n}$. Moreover, we can prove that these division fields are not isomorphic to each other. In our construction, we use recent results obtained by the first author.

math.NT↗
arXiv2020

Infinitely many hyperelliptic curves with exactly two rational points: Part II

In the previous paper, Hirakawa and the author determined the set of rational points of a certain infinite family of hyperelliptic curves $C^{(p;i,j)}$ parametrized by a prime number $p$ and integers $i$, $j$. In the proof, we used the standard $2$-descent argument and a Lutz-Nagell theorem that was proven by Grant. In this paper, we extend the above work. By using the descent theorem, the proof for $j=2$ is reduced to elliptic curves of rank $0$ that are independent of $p$. On the other hand, for odd $j$, we consider another hyperelliptic curve $C'^{(p;i,j)}$ whose Jacobian variety is isogenous to that of $C^{(p;i,j)}$, and prove that the Mordell-Weil rank of the Jacobian variety of $C'^{(p;i,j)}$ is $0$ by $2$-descent. Then, we determine the set of rational points of $C^{(p;i,j)}$ by using the Lutz-Nagell type theorem.

math.NT↗
arXiv2019

Infinitely many hyperelliptic curves with exactly two rational points

In this paper, we construct some families of infinitely many hyperelliptic curves of genus $2$ with exactly two rational points. In the proof, we first show that the Mordell-Weil ranks of these hyperelliptic curves are $0$ and then determine the sets of rational points by using the Lutz-Nagell type theorem for hyperelliptic curves which was proven by Grant.

math.NT↗
arXiv2018

A unique pair of triangles

A rational triangle is a triangle with sides of rational lengths. In this short note, we prove that there exists a unique pair of a rational right triangle and a rational isosceles triangle which have the same perimeter and the same area. In the proof, we determine the set of rational points on a certain hyperelliptic curve by a standard but sophisticated argument which is based on the 2-descent on its Jacobian variety and Coleman's theory of $p$-adic abelian integrals.

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