Kac-Rice formulas and the number of solutions of parametrized systems of polynomial equations

Elisenda Feliu, AmirHosein Sadeghimanesh

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2 Citations (Scopus)
121 Downloads (Pure)

Abstract

Kac-Rice formulas express the expected number of elements a fiber of a random field has in terms of a multivariate integral. We consider here parametrized systems of polynomial equations that are linear in enough parameters, and provide a Kac-Rice formula for the expected number of solutions of the system when the parameters follow continuous distributions. Combined with Monte Carlo integration, we apply the formula to partition the parameter region according to the number of solutions or find a region in parameter space where the system has the maximal number of solutions. The motivation stems from the study of steady states of chemical reaction networks and gives new tools for the open problem of identifying the parameter region where the network has at least two positive steady states. We illustrate with numerous examples that our approach successfully handles a larger number of parameters than exact methods.
Original languageEnglish
Pages (from-to)2739-2769
Number of pages31
JournalMathematics of Computation (MCOM)
Volume91
Issue number338
DOIs
Publication statusPublished - 11 Aug 2022

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Keywords

  • Kac-Rice formula
  • polynomial system
  • parameter region
  • Monte Carlo integration
  • multistationarity

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