M-curves of degree 9 or 11 with one unique non-empty oval

TL;DR

This paper investigates the topology of certain real algebraic curves called M-curves of degrees 9 and 11, establishing lower bounds on the number of specific ovals using complex orientation techniques.

Contribution

It provides new bounds on the number of non-empty ovals in M-curves of degrees 9 and 11 with a particular real scheme, advancing understanding of their topological configurations.

Findings

For degree 9, at least 2 non-empty ovals are required.

For degree 11, at least 3 non-empty ovals are necessary.

Uses complex orientations to derive these bounds.

Abstract

In this note, we consider M-curves of odd degree with real scheme of the form $< \mathcal{J} \amalg \alpha \amalg 1 < \beta > >$. With help of complex orientations, we prove that for $m = 9$, $\alpha \geq 2$, and for $m = 11$, $\alpha \geq 3$.