Self-inversive polynomials, curves, and codes

TL;DR

This paper explores the mathematical relationships between self-inversive polynomials, algebraic curves, and self-dual codes, providing new characterizations of superelliptic curves and proposing a conjecture on code properties.

Contribution

It establishes a link between superelliptic curves with certain automorphism groups and self-inversive or self-reciprocal polynomials, and introduces a conjecture on extremal self-dual codes.

Findings

Superelliptic curves can be expressed using self-inversive or self-reciprocal polynomials.

A conjecture is proposed regarding the coefficients of zeta polynomials of extremal self-dual codes.

Connections between algebraic curves, polynomials, and coding theory are elucidated.

Abstract

We study connections between self-inversive and self-reciprocal polynomials, reduction theory of binary forms, minimal models of curves, and formally self-dual codes. We prove that if $\mathcal X$ is a superelliptic curve defined over $\mathbb C$ and its reduced automorphism group is nontrivial or not isomorphic to a cyclic group, then we can write its equation as $y^n = f(x)$ or $y^n = x f(x)$, where $f(x)$ is a self-inversive or self-reciprocal polynomial. Moreover, we state a conjecture on the coefficients of the zeta polynomial of extremal formally self-dual codes.