Positive Solutions of Systems of Signed Parametric Polynomial Inequalities

Hoon Hong 1 Thomas Sturm 2, 3
3 VERIDIS - Modeling and Verification of Distributed Algorithms and Systems
MPII - Max-Planck-Institut für Informatik, Inria Nancy - Grand Est, LORIA - FM - Department of Formal Methods
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.
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Hoon Hong, Thomas Sturm. Positive Solutions of Systems of Signed Parametric Polynomial Inequalities. CASC 2018 - International Workshop on Computer Algebra in Scientific Computing, Sep 2018, Lille, France. pp.238 - 253, ⟨10.1007/978-3-319-99639-4_17⟩. ⟨hal-01889827⟩

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