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

Abstract

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

8 pages, 5 figures, one proof simplified