Yefei Ma | Searcharxiv

arXiv2026

Wug-snake graphs and Markov numbers of matrix semigroups

Classically, Markov numbers are recovered as perfect matching numbers of domino snake graphs. We extend this correspondence by introducing weighted universal generalised snake graphs, or wug-snake graphs. These are weighted ordered bipartite graphs whose perfect matching sequences encode linear recurrences. To every wug-snake graph we associate a continuant matrix and prove that the determinant of this matrix equals the weighted perfect matching sum. We then introduce polyomino wug-tiles, bodies of wug-snake graphs that act linearly on state vectors. Every integer matrix admits a canonical polyomino wug-tile. Our main result identifies the wug-snake determinant of a tile representing a matrix $A$ with the Markov-Davenport form of $A$. Consequently, algebraic and geometric Markov numbers of matrices and matrix semigroups can be expressed as weighted perfect matching determinants. Further we define Frobenius maps for matrix semigroups and discuss examples recovering classical Markov numbers and higher-dimensional lattice realisations.

math.CO↗

arXiv2025

Equidistribution of integers represented by standard quadratic form under arithmetic constraints

We study the equidistribution of integers of the form $n= x_1^2 + \cdots + x_d^2$ under the arithmetic constraints given by $(\mathbb{Z}/p\mathbb{Z})^d$. The first step in addressing this problem is to construct modular forms whose Fourier expansion coefficients correspond to the counting problem over the quadric in $\mathbb{Z}^d$ induced by the standard quadratic form, subject to the aforementioned arithmetic constraints. The weak modular property of these modular forms allows us to use representation theory to identify the congruence subgroup to which our modular forms correspond. We then establish a necessary and sufficient condition for functions on $(\mathbb{Z}/p\mathbb{Z})^d$ that defines a cusp form. Finally, we conclude that the equidistribution phenomenon occurs locally on $p+1$ orbits for $d \geq 4$.

math.NT↗