On the existence of maximizing curves of odd degrees

TL;DR

This paper establishes a non-existence criterion for maximizing curves of odd degrees and introduces a new class of plane curves called M-curves, generalizing maximizing curves.

Contribution

It provides the first non-existence criterion for odd degree maximizing curves and defines M-curves, broadening the understanding of these geometric objects.

Findings

Non-existence criterion for maximizing curves of odd degrees

Introduction of M-curves as a generalization of maximizing curves

Enhanced understanding of plane curve classifications

Abstract

In this paper we provide the non-existence criterion for the so-called maximizing curves of odd degrees. Furthermore, in the light of our criterion, we define a new class of plane curves that generalizes the notion of maximizing curves which we call as $M$-curves.