Tootfinder

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…

Scalar and Mean Curvature Comparison on Compact Cylinder
Jie Xu
https://arxiv.org/abs/2507.07005 https://arxiv.org/pdf/2507.07005

Scalar and Mean Curvature Comparison on Compact Cylinder
Let $ X $ be a closed, oriented Riemannian manifold. Denote by $ (M = X \times I, \partial M = X \times \lbrace 0 \rbrace \cup X \times \lbrace 1 \rbrace, g) $ a compact cylinder with smooth boundary, $ \dim M \geqslant 3 $. In this article, we address the following question: If $ g $ is a Riemannian metric having (i) positive scalar curvature (PSC metric) on $ M $ and nonnegative mean curvature on $ \partial M $; and (ii) the $ g $-angle between normal vector field $ ν_{g} $ along $ \partial …