Comments by Michael Joswig:

Given a so-called Wronski-system, i.e. \(d\) polynomials in \(d\) variables such that

  • all polynomials have the same Newton polytope

  • the Newton polytope has a regular triangulation whose dual graph is bipartite, which implies that the vertices are \(d+1\) colorable and the facet are 2 colorable (say black and white), and the coefficients are constant on the \(d+1\) vertex colors.


Two polynomials of the form \(a(1+xy)+b(x+y^2)+c(y+x^2)\), whose Newton polytope is a simplex of side length 2.

Then Soprunova and Sotile (2006) showed that the number of real solutions to that system is at least the signature of that coloring

\[\left(= | \# {\rm black\ facets} - \# {\rm white\ facets} | \right).\]

This is done by constructing a special map whose degree is the signature, which yields that many zeroes.

The goal is to find a algorithm to compute exactly those solutions, given a Wronski-system, for the case \(d=2\). This can be regarded as a real homotoy algorithm.