Limits of combinatorial patchworking

Abstract

It is shown that there are real plane algebraic curves of degree eight that cannot be realized as T-curves, i.e., via combinatorial patchworking. In fact, this holds for several real schemes (i.e., ambient isotopy types) with the maximal number of real components, called $M$-curves. On the other hand, each nonempty real scheme of lower degree, maximal or not, arises as a T-curve. By constructing one patchwork of the dilated triangle $d\cdot \hat{I}{”}_2$ for each nonempty real scheme of degree $d\le 7$, we provide an explicit method for constructing polynomials realizing these real schemes. This resolves a question of Itenberg and Viro (1996).

major revision, new results, new title, 43 pages, 12 figures, 6 tables