Notes on Hilbert's 16th: experiencing Viro's theory

Abstract

This text is intended to become in the long run Chapter 3 of our long saga dedicated to Riemann, Ahlfors and Rohlin. Yet, as its contents evolved as mostly independent (due to our inaptitude to interconnect both trends as strongly as we wished), it seemed preferable to publish it separately. More factually, our account is an attempt to get familiarized with the current consensus about Hilbert's 16th in degree 8. This is a nearly finished piece of mathematics, thanks heroic breakthroughs by Viro, Fiedler, Korchagin, Shustin, Chevallier, Orevkov, yet still leaving undecided six tantalizing bosons among a menagerie of 104 logically possible distributions of ovals (respecting B\'ezout, Gudkov periodicity, and the Fiedler-Viro imparity law sieving away 4+36 schemes). This quest inevitably involves glimpsing deep into the nebula referrable to as Viro's patchworking, and the likewise spectacular obstructional laws of Fiedler, Viro, Shustin, Orevkov. In the overall, the game is much comparable to a pigeons hunting video-game, where 144 birds are liberated in nature, with some of them strong enough to fly higher and higher in the blue sky as to rejoin the stratospheric paradise of eternal life (construction of a scheme in the algebro-geometric category). Some other, less fortunate, birds were killed (a long time ago) and crashed down miserably over terrestrial crust (prohibition). Alas, the hunt is unfinished with still six birds, apparently too feeble to rally safely the paradise, yet too acrobatic for any homosapiens being skillful enough to shoot them down.