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Abstract

We consider systems of strict multivariate polynomial inequalities over the
reals. All polynomial coefficients are parameters ranging over the reals, where
for each coefficient we prescribe its sign. We are interested in the existence
of positive real solutions of our system for all choices of coefficients
subject to our sign conditions. We give a decision procedure for the existence
of such solutions. In the positive case our procedure yields a parametric
positive solution as a rational function in the coefficients. Our framework
allows to reformulate heuristic subtropical approaches for non-parametric
systems of polynomial inequalities that have been recently used in qualitative
biological network analysis and, independently, in satisfiability modulo theory
solving. We apply our results to characterize the incompleteness of those
methods.