Subtropical Real Root Finding

We describe a new incomplete but terminating method for real root finding for large multivariate polynomials. We take an abstract view of the polynomial as the set of exponent vectors associated with sign information on the coefficients. Then we employ linear programming to heuristically find roots. There is a specialized variant for roots with exclusively positive coordinates, which is of considerable interest for applications in chemistry and systems biology. An implementation of our method combining the computer algebra system Reduce with the linear programming solver Gurobi has been successfully applied to input data originating from established mathematical models used in these areas. We have solved several hundred problems with up to more than 800000 monomials in up to 10 variables with degrees up to 12. Our method has failed due to its incompleteness in less than 8 percent of the cases

Better Answers to Real Questions

We consider existential problems over the reals. Extended quantifier elimination generalizes the concept of regular quantifier elimination by providing in addition answers, which are descriptions of possible assignments for the quantified variables. Implementations of extended quantifier elimination via virtual substitution have been successfully applied to various problems in science and engineering. So far, the answers produced by these implementations included infinitesimal and infinite numbers, which are hard to interpret in practice. We introduce here a post-processing procedure to convert, for fixed parameters, all answers into standard real numbers. The relevance of our procedure is demonstrated by application of our implementation to various examples from the literature, where it significantly improves the quality of the results

Algebraic Aspects of (Bio) Nano-chemical Reaction Networks and Bifurcations in Various Dynamical Systems

The dynamics of (bio) chemical reaction networks have been studied by different methods. Among these methods, the chemical reaction network theory has been proven to successfully predicate important qualitative properties, such as the existence of the steady state and the asymptotic behavior of the steady state. However, a constructive approach to the steady state locus has not been presented. In this thesis, with the help of toric geometry, we propose a generic strategy towards this question. This theory is applied to (bio)nano particle configurations. We also investigate Hopf bifurcation surfaces of various dynamical systems

Symbolic Methods for Chemical Reaction Networks (Dagstuhl Seminar 12462)

During 11-16 November 2012, the Dagstuhl Seminar 12462 "Symbolic Methods for Chemical Reaction Networks" was held in Schloss Dagstuhl - Leibneiz Center for Informatics. The seminar brought together researchers in symbolic computation, chemical engineering, and systems biology. During the seminar, participants presented five-minute talks introducing their research interests, five participants gave longer talks, and all participants had the opportunity to take part in various discussion groups. Abstracts of presentations and summaries of the discussion groups are compiled in this report

Subtropical Real Root Finding

We describe a new incomplete but terminating method for real root finding for large multivariate polynomials. We take an abstract view of the polynomial as the set of exponent vectors associated with sign information on the coefficients. Then we employ linear programming to heuristically find roots. There is a specialized variant for roots with exclusively positive coordinates, which is of considerable interest for applications in chemistry and systems biology. An implementation of our method combining the computer algebra system Reduce with the linear programming solver Gurobi has been successfully applied to input data originating from established mathematical models used in these areas. We have solved several hundred problems with up to more than 800000 monomials in up to 10 variables with degrees up to 12. Our method has failed due to its incompleteness in less than 8 percent of the cases

A Survey of Some Methods for Real Quantifier Elimination, Decision, and Satisfiability and Their Applications

International audienceEffective quantifier elimination procedures for first-order theories provide a powerful tool for genericallysolving a wide range of problems based on logical specifications. In contrast to general first-order provers, quantifierelimination procedures are based on a fixed set of admissible logical symbolswith an implicitly fixed semantics. Thisadmits the use of sub-algorithms from symbolic computation. We are going to focus on quantifier elimination forthe reals and its applications giving examples from geometry, verification, and the life sciences. Beyond quantifierelimination we are going to discuss recent results with a subtropical procedure for an existential fragment of thereals. This incomplete decision procedure has been successfully applied to the analysis of reaction systems inchemistry and in the life sciences