arXiv subjects

Sasha Kononova

Publications and source records attributed to Sasha Kononova.

4 recordsLinked to original sources

arXiv2026

Distinguishing elliptic curves modulo $p$ and identifying images of product representations

Given two elliptic curves defined over $\mathbb{Q}$ and a rational prime $p$, we study the product of their residual Galois representations. Using Goursat's lemma, we explicitly enumerate and completely characterize all possible images of such product representations. We also define associated invariants to these image groups, which we call \textit{witness ratios}, and we explain their computational utility and their relationship to the well-known Sturm bound for testing congruences between modular forms.

math.NT↗

arXiv2026

On the size and complexity of scrambles

The scramble number of a graph, a natural generalization of bramble number, is an invariant recently developed to study chip-firing games and graph gonality. We introduce the carton number of a graph, defined to be the minimum size of a maximum order scramble, to study the computational complexity of scramble number. We show that there exist graphs with carton number exponential in the size of the graph, proving that scrambles are not valid NP certificates. We characterize families of graphs whose scramble number and gonality can be constant-factor approximated in polynomial time and show that the disjoint version of scramble number is fixed parameter tractable. Lastly, we find that vertex congestion is an upper bound on screewidth and thus scramble number, leading to a new proof of the best known bound on the treewidth of line graphs and a bound on the scramble number of planar graphs with bounded degree.

math.CO↗

arXiv2025

Chip-Firing Games on Banana Trees

We study chip-firing games on multigraphs whose underlying simple graphs are trees, paths, and stars, denoted as banana trees, paths, and stars respectively. We present a polynomial time algorithm to compute the divisorial gonality of banana paths, and give combinatorial formulas for the related invariants of scramble number and screewidth for any banana tree. Furthermore, we leverage banana paths to show that gonality can increase or decrease by an arbitrary amount upon deletion of a single edge, even when the resulting graph is connected. Lastly, we study banana trees and Brill-Noether theory to prove that the gonality conjecture holds for all banana trees.

math.CO↗

arXiv2024

The gonality of chess graphs

Chess graphs encode the moves that a particular chess piece can make on an $m\times n$ chessboard. We study through these graphs through the lens of chip-firing games and graph gonality. We provide upper and lower bounds for the gonality of king's, bishop's, and knight's graphs, as well as for the toroidal versions of these graphs. We also prove that among all chess graphs, there exists an upper bound on gonality solely in terms of $\min\{m,n\}$, except for queen's, toroidal queen's, rook's, and toroidal bishop's graphs.

math.CO↗