Parametrized systems of generalized polynomial equations: first applications to fewnomials

Abstract

We consider positive solutions to parametrized systems of generalized polynomial equations (with real exponents and positive parameters). By a fundamental result obtained in parallel work, polynomial systems are determined by geometric objects, rather than matrices: a polytope $P$ (arising from the coefficient matrix) and two subspaces representing monomial differences and dependencies (arising from the exponent matrix). The dimension of the latter subspace, the monomial dependency $d$, is crucial. Indeed, we rewrite $\textit{polynomial}$ equations in terms of $d$ $\textit{binomial}$ equations on the coefficient polytope $P$, involving $d$ monomials in the parameters. We further study the solution set on $P$ using methods from analysis such as sign-characteristic functions and Wronskians. In this work, we present first applications to fewnomial systems through five (classes of) examples. In particular, we study (i) $n$ trinomials involving ${n+2}$ monomials in $n$ variables, having dependency $d=1$, and (ii) one trinomial and one $t$-nomial (with $t\ge3$) in two variables, having $d=t-1\ge2$. For (i), we bound the number of positive solutions using the number of roots of a univariate polynomial of degree at most $n$. We also show that this number is always less than or equal to the number of sign changes in an optimal Descartes' rule given in Bihan et al. (2021). For (ii), we improve upper bounds given in Li et al. (2003) and Koiran et al. (2015). Further, for two trinomials ($t=3$), we refine the known upper bound of five in terms of the exponents, and we find an example with five positive solutions that is even simpler than the smallest "Haas system".