Limits of combinatorial patchworking

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 $d\cdot\Delta_2$ for each nonempty real scheme of degree $d\leq 7$, we provide an explicit method for constructing polynomials realizing these real schemes.

This resolves a question of Itenberg and Viro (1996).