arXiv subjects

John C. Harris

Publications and source records attributed to John C. Harris.

2 recordsLinked to original sources

arXiv2012

Resolutions of 2 and 3 dimensional rings of invariants for cyclic groups

Let $G$ be the cyclic group of order $n$ and suppose ${\bf F}$ is a field containing a primitive $n^\text{th}$ root of unity. We consider the ring of invariants ${\bf F}[W]^G$ of a three dimensional representation $W$ of $G$ where $G \subset \text{SL}(W)$. We describe minimal generators and relations for this ring and prove that the lead terms of the relations are quadratic. These minimal generators for the relations form a Gröbner basis with a surprisingly simple combinatorial structure. We describe the graded Betti numbers for a minimal free resolution of $F[W]^G$. The case where $W$ is any two dimensional representation of $G$ is also handled.

math.AC↗

arXiv2004

Non-Negative Integer Linear Congruences

We consider the problem of describing all non-negative integer solutions to a linear congruence in many variables. This question may be reduced to solving the congruence $x_1 + 2x_2 + 3x_3 + ... + (n-1)x_{n-1} \equiv 0 \pmod n$ where values of the unknowns, $x_i$, are sought among the non-negative integers. We consider the monoid of solutions of this equation and prove a conjecture of Elashvili concerning the structure of these solutions. This yields a simple algorithm for generating most (conjecturally all) of the high degree indecomposable solutions of the equation.

math.NT↗