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.
Highlights
We investigate the problem of finding a parametric positive solution of a system of signed parametric polynomial inequalities, if exists
The problem is important as systems of polynomial inequalities often arise in science and engineering applications, including, e.g., the qualitative analysis of biological or chemical networks [7,20,21,40] or Satisfiability Modulo Theories (SMT) solving [1,22,32]
We reduce the problem to SMT solving over quantifier-free linear real arithmetic (QF LRA)
Summary
We investigate the problem of finding a parametric positive solution of a system of signed parametric polynomial inequalities, if exists. The problem is important as systems of polynomial inequalities often arise in science and engineering applications, including, e.g., the qualitative analysis of biological or chemical networks [7,20,21,40] or Satisfiability Modulo Theories (SMT) solving [1,22,32]. In both these areas, one is often interested in c The Author(s) 2018 V. The problem of finding parametric positive solutions is essentially that of quantifier elimination over the first order theory of real closed fields. Existing general quantifier elimination software is still too inefficient for finding parametric positive solutions with relevant problem sizes in our above-mentioned fields of applications.
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