Tootfinder

Binomial edge ideals of Cameron-Walker graphs
Takayuki Hibi, Sara Saeedi Madani
https://arxiv.org/abs/2509.01150 https://arxiv.org/pdf/2509.01150

Binomial edge ideals of Cameron-Walker graphs
Let $G$ be a Cameron--Walker graph on $n$ vertices and $J_G$ the binomial edge ideal of $G$. Let $S$ denote the polynomial ring in $2n$ variables over a field. It is shown that the following conditions are equivalent: (i) $S/J_G$ is Cohen--Macaulay; (ii) $J_G$ is unmixed; (iii) $\dim (S/J_G) = n+1$; (iv) (a) $n = 3$ and $G$ is a path of length $2$ or (b) $n = 5$ and $G$ is a path of length $4$ or (c) $n=5$ and $G$ is obtained by attaching a path of length $2$ to a triangle. Moreover, the depth …

Galois theory by calculator
Thomas W. Mattman, Robertson-Figaniak, Zoe Steele
https://arxiv.org/abs/2508.18595 https://arxiv.org/pdf/2508.18595

Galois theory by calculator
We present an algorithm to determine the Galois group of an irreducible monic polynomial $f(x) \in \mathbb{Z}[x]$ of degree at most five. Following work of Conrad, Dummit, and Stauduhar this comes down to answering two questions: Is a given integer a square? and Does a given polynomial have an integral root? Since these are both easily addressed with a calculator, our algorithm amounts to Galois theory by calculator. For example, we have an implementation at Desmos.com. In an appendix we presen…