## 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.

### Example¶

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.