Topological degree of fewnomial maps

Abstract

We study the topological degree on the positive orthant of a map $F:\mathbb{R}^n_{>0}\to\mathbb{R}^n$ whose components are real polynomials, possibly with real exponents, with the same support $\mathcal{A}$ of $N=n+k+1$ monomials. This degree is well defined under a half-space condition on the columns of the coefficient matrix indexed by the faces of the Newton polytope, which holds for generic exponents and coefficients. We give three formulas for it, all in terms of signs of maximal minors. The first one is a Viro formula: the degree is a signed count of the simplices of a regular triangulation of $\mathcal{A}$ which are positively decorated in the sense of Bihan, Santos and Spaenlehauer. The second one is a sum over the facets of the Newton polytope, and the third one a sum over the vertices of the polyhedron of the Gale dual system. As a consequence, the degree only depends on the oriented matroids of the exponent and coefficient matrices restricted to the vertices of the Newton polytope. We also obtain a Descartes rule of signs for the degree and, in the generic case, the upper bound $|°F|<\frac12\min(M(n,N),M(k,N))$, where $M(d,N)$ is the largest number of facets of a $d$-polytope with $N$ vertices. This bound is sharp when $n=2$ or $k=2$. Using the H-duality of Bihan, Dickenstein and Forsgård, we show that the set of possible degrees is symmetric in $n$ and $k$. Since a nonzero degree forces a positive zero, our formulas give sign conditions for the existence of positive solutions, as well as parity statements for their number.