On algorithms to obtain linear determinantal representations of smooth plane curves of higher degree

TL;DR

This paper introduces two algorithms for computing linear determinantal representations of smooth plane curves of any degree, with explicit examples for specific quartics over the rationals, advancing computational methods in algebraic geometry.

Contribution

It presents new algorithms for obtaining linear determinantal representations of smooth plane curves of arbitrary degree, including explicit classifications for certain quartics over ield.

Findings

Algorithms successfully compute representations for all degrees.

Explicit representatives provided for Klein and Fermat quartics over ield.

Advances computational techniques in algebraic geometry.

Abstract

We give two algorithms to compute linear determinantal representations of smooth plane curves of any degree over any field. As particular examples, we explicitly give representatives of all equivalence classes of linear determinantal representations of two special quartics over the field $\mathbb{Q}$ of rational numbers, the Klein quartic and the Fermat quartic. This paper is a summary of third author's talk at the JSIAM JANT workshop on algorithmic number theory in March 2018. Details will appear elsewhere.