arXiv subjects

Ryo Yoshizumi

Publications and source records attributed to Ryo Yoshizumi.

4 recordsLinked to original sources

arXiv2026

UTVPI-representable integer point sets: discrete convexity, polymorphisms, and pairwise closure

We study subsets of the integer lattice represented by single-variable-per-inequality (SVPI), difference-constraint (DC), unit two-variable-per-inequality (UTVPI), and two-variable-per-inequality (TVPI) systems. We relate five viewpoints: inequality representation, discrete convexity, polymorphisms, reconstruction from two-coordinate projections, and fixed points of closure operators. Our central result completely characterizes UTVPI-representability. For every set $S\subseteq\mathbb Z^n$ with $n>1$, \[ \begin{aligned} &S\text{ is UTVPI-representable}\\ &\;\Longleftrightarrow\; S\text{ is closed under the directed midpoint and median operations}\\ &\;\Longleftrightarrow\; S\text{ is integrally convex and $2$-decomposable}. \end{aligned} \] The median condition may instead be replaced by closedness under some majority operation, and the same class is the fixed-point class of a pairwise directed-midpoint closure operator. Thus, all five viewpoints yield equivalent characterizations of UTVPI-representability. In particular, $2$-decomposability is exactly the global condition needed to lift the known two-dimensional equivalence between integral convexity and UTVPI-representability to arbitrary dimension. This theorem is embedded in a broader pairwise-closure theory. For a family $F$ of operations, we define a closure operator by closing every two-coordinate projection under $F$ and joining the resulting sets. Its fixed points are precisely the sets that are both $2$-decomposable and $F$-closed, and we establish a local-to-global criterion for such characterizations. A closed-convex-hull analogue characterizes TVPI-representability. We also characterize SVPI-representability by natural multioperations, prove limitations of operation-based characterizations for several related classes, and determine the complete inclusion hierarchies in the general, Boolean, and two-dimensional settings.

cs.DM↗

arXiv2026

Geometric construction of modular polynomials with level structures

The classical modular polynomial for $j$-invariants describes the relation between two elliptic curves connected by isogenies. This polynomial has been applied to various algorithms in computational number theory, such as point counting on elliptic curves. In addition, computing the modular polynomial itself is also an important problem, and various algorithms to compute it have been proposed. On the other hand, modular polynomials for other invariants of higher level structures have also been studied. For example, the modular polynomials for the Legendre $λ$-invariant and the Weber functions are well-known. In this paper, we give another approach to construct modular polynomials of higher level purely algebraically. In particular, we show the existence of modular polynomials for invariants directly related to models of elliptic curves, such as the coefficients of Montgomery and Hessian curves. We also show that these modular polynomials have integer coefficients and are symmetric and irreducible in certain cases, and give an algorithm to compute them, which is based on the deformation method by Kunzweiler and Robert.

math.NT↗

arXiv2025

Listing superspecial curves of genus three using Richelot isogeny graphs

In algebraic geometry, superspecial curves are important research objects. While the number of superspecial genus-3 curves in characteristic $p$ is known, the number of hyperelliptic ones among them has not been determined even for small $p$. In this paper, in order to compute the latter number, we give an explicit algorithm for computing the Richelot isogeny graph of superspecial principally polarized abelian varieties of dimension 3 using theta functions. In particular, one can determine whether a given vertex in the graph corresponds to the Jacobian of a genus-3 curve or not, and restore the defining equation of such a genus-3 curve from its theta constants. Our algorithm enables efficient enumeration of superspecial genus-3 curves, as all operations can be performed in $\mathbb{F}_{p^2}$. By implementing the algorithm in Magma, we successfully counted the number of hyperelliptic curves among them for all primes $11 \leq p < 100$.

math.AG↗

arXiv2024

Characterizing the integer points in 2-decomposable polyhedra by closedness under operations

Characterizing the solution sets in a problem by closedness under operations is recognized as one of the key aspects of algorithm development, especially in constraint satisfaction. An example from the Boolean satisfiability problem is that the solution set of a Horn conjunctive normal form (CNF) is closed under the minimum operation, and this property implies that minimizing a nonnegative linear function over a Horn CNF can be done in polynomial time. In this paper, we focus on the set of integer points (vectors) in a polyhedron, and study the relation between these sets and closedness under operations from the viewpoint of 2-decomposability. By adding further conditions to the 2-decomposable polyhedra, we show that important classes of sets of integer vectors in polyhedra are characterized by 2-decomposability and closedness under certain operations, and in some classes, by closedness under operations alone. The most prominent result we show is that the set of integer vectors in a unit-two-variable-per-inequality polyhedron can be characterized by closedness under the median and directed discrete midpoint operations, each of these operations was independently considered in constraint satisfaction and discrete convex analysis.

cs.DS↗