A little scholium on Hilbert-Rohn via the total reality of $M$-curves: Riemann's flirt with Miss Ragsdale

TL;DR

This paper provides an elementary proof of Hilbert's nesting conjecture for $M$-sextics using total reality methods, discusses its implications for prohibiting certain schemes, and explores potential applications to the Ragsdale conjecture.

Contribution

It introduces a simple, robust total reality approach to Hilbert's 16th problem for degree 6 and suggests its possible extension to higher degrees and related conjectures.

Findings

Proof of Hilbert's nesting for $M$-sextics using total reality

Prohibition of Rohn's scheme 10/1 via the method

Potential application to Ragsdale conjecture and weaker bounds

Abstract

This note presents an elementary proof of Hilbert's 1891 Ansatz of nesting for $M$-sextics, along the line of Riemann's Nachlass 1857 and a simple Harnack-style argument (1876). Our proof seems to have escaped the attention of Hilbert (and all subsequent workers) [but alas turned out to contain a severe gap, cf. Introduction for more!]. It uses a bit Poincar\'e's index formula (1881/85). The method applies as well to prohibit Rohn's scheme 10/1, and therefore all obstructions of Hilbert's 16th in degree $m=6$ can be explained via the method of total reality. (The same ubiquity of the method is conjectured in all degrees, and then suspected to offer new insights.) More factually, a very simple and robust phenomenon of total reality on $M$-curves of even order is described (the odd-order case being already settled in Gabard 2013), and it is speculated that this could be used as an attack upon the (still open) Ragsdale conjecture for $M$-curves (positing that $| \chi|\le k^2$). Of course a giant gap still remains to be bridged in case the latter conjecture is true at all. Alas, the writer has little experimental evidence for the truth of the conjecture, and the game can be a hazardous one. However we suspect that the method of total reality should at least be capable of recovering the weaker Petrovskii bound, or strengthened variants due to Arnold 1971. This text has therefore merely didactic character and offers no revolutionary results, but tries to reactivate a very ancient method (due basically to Riemann 1857) whose swing seems to have been somewhat underestimated, at least outside of the conformal-mapping community.