Parametrized systems of generalized polynomial inequalities via linear algebra and convex geometry

We provide fundamental results on positive solutions to parametrized systems of generalized polynomial <i>inequalities</i> (with real exponents and positive parameters), including generalized polynomial <i>equations</i>. In doing so, we also offer a new perspective on fewnomials and (generalized) mass-action systems. We find that geometric objects, rather than matrices, determine generalized polynomial systems: a bounded set/"polytope" <i>P</i> (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 <i>d</i>, is crucial. As our main result, we rewrite <i>polynomial inequalities</i> in terms of <i>d</i> <i>binomial equations</i> on <i>P</i>, involving <i>d</i> monomials in the parameters. In particular, we establish an explicit bijection between the original solution set and the solution set on <i>P</i> via exponentiation. (i) Our results apply to any generalized polynomial system. (ii) The dependency <i>d</i> and the dimension of <i>P</i> indicate the complexity of a system. (iii) Our results are based on methods from linear algebra and convex/polyhedral geometry, and the solution set on <i>P</i> can be further studied using methods from analysis such as sign-characteristic functions (introduced in this work). We illustrate our results (in particular, the relevant geometric objects) through three examples from real fewnomial and reaction network theory. For two mass-action systems, we parametrize the set of equilibria and the region for multistationarity, respectively, and even for univariate trinomials, we offer new insights: We provide a "solution formula" involving discriminants and "roots".