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

Elisenda Feliu; AmirHosein Sadeghimanesh

doi:10.1090/mcom/3760

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

Elisenda Feliu, AmirHosein Sadeghimanesh

Research Centre for Computational Science and Mathematical Modelling

University of Copenhagen

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

166 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 language	English
Pages (from-to)	2739-2769
Number of pages	31
Journal	Mathematics of Computation (MCOM)
Volume	91
Issue number	338
DOIs	https://doi.org/10.1090/mcom/3760
Publication status	Published - 11 Aug 2022

Bibliographical note

Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.

This document is the author’s post-print version, incorporating any revisions agreed during the peer-review process. Some differences between the published version and this version may remain and you are advised to consult the published version if you wish to cite from it.

Keywords

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

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10.1090/mcom/3760

2010.00804v1Submitted manuscript, 670 KB
Post-printAccepted author manuscript, 965 KB

Cite this

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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. ",

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

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