arXiv subjects

Matthew Babbitt

Publications and source records attributed to Matthew Babbitt.

2 recordsLinked to original sources

arXiv2025

Summability of Elliptic Functions via Residues

Summability has been a central object of study in difference algebra over the past half-century. It serves as a cornerstone of algebraic methods to study linear recurrences over various fields of coefficients and with respect to various kinds of difference operators. Recently, Dreyfus, Hardouin, Roques, and Singer introduced a notion of elliptic orbital residues, which altogether serve as a partial obstruction to summability for elliptic functions with respect to the shift by a non-torsion point over an elliptic curve. We explain how to refine this into a complete obstruction, which promises to be useful in applications of difference equations over elliptic curves, such as elliptic hypergeometric functions and the combinatorics of walks in the quarter plane.

math.NT↗

arXiv2014

On k-visibility graphs

We examine several types of visibility graphs in which sightlines can pass through $k$ objects. For $k \geq 1$ we bound the maximum thickness of semi-bar $k$-visibility graphs between $\lceil \frac{2}{3} (k + 1) \rceil$ and $2k$. In addition we show that the maximum number of edges in arc and circle $k$-visibility graphs on $n$ vertices is at most $(k+1)(3n-k-2)$ for $n > 4k+4$ and ${n \choose 2}$ for $n \leq 4k+4$, while the maximum chromatic number is at most $6k+6$. In semi-arc $k$-visibility graphs on $n$ vertices, we show that the maximum number of edges is ${n \choose 2}$ for $n \leq 3k+3$ and at most $(k+1)(2n-\frac{k+2}{2})$ for $n > 3k+3$, while the maximum chromatic number is at most $4k+4$.

math.CO↗