Extremal invariant polynomials not satisfying the Riemann hypothesis

TL;DR

This paper demonstrates that certain extremal invariant polynomials, related to weight enumerators of linear codes, do not satisfy the Riemann hypothesis, challenging assumptions about their properties.

Contribution

It provides the first explicit examples of extremal invariant polynomials that violate the Riemann hypothesis, expanding understanding of their behavior.

Findings

Existence of extremal invariant polynomials not satisfying the Riemann hypothesis

Counterexamples to the assumption that extremal weight enumerators always satisfy the hypothesis

Invariance under the MacWilliams transform does not guarantee the Riemann hypothesis

Abstract

Zeta functions for linear codes were defined by Iwan Duursma in 1999. They were generalized to the case of some invariant polynomials by the preset author. One of the most important problems is whether extremal weight enumerators satisfy the Riemann hypothesis. In this article, we show there exist extremal polynomials of the weight enumerator type which are invariant under the MacWilliams transform and do not satisfy the Riemann hypothesis.