KAC-RICE FORMULAS AND THE NUMBER OF SOLUTIONS OF PARAMETRIZED SYSTEMS OF POLYNOMIAL EQUATIONS
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KAC-RICE FORMULAS AND THE NUMBER OF SOLUTIONS OF PARAMETRIZED SYSTEMS OF POLYNOMIAL EQUATIONS. / Feliu, Elisenda; Sadeghimanesh, Amirhosein.
I: Mathematics of Computation, Bind 91, Nr. 338, 2022, s. 2739-2769.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - KAC-RICE FORMULAS AND THE NUMBER OF SOLUTIONS OF PARAMETRIZED SYSTEMS OF POLYNOMIAL EQUATIONS
AU - Feliu, Elisenda
AU - Sadeghimanesh, Amirhosein
N1 - Publisher Copyright: © 2022 American Mathematical Society
PY - 2022
Y1 - 2022
N2 - 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
AB - 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
KW - Kac-Rice formula
KW - Monte Carlo integration
KW - multistationarity
KW - parameter region
KW - polynomial system
U2 - 10.1090/mcom/3760
DO - 10.1090/mcom/3760
M3 - Journal article
AN - SCOPUS:85139238286
VL - 91
SP - 2739
EP - 2769
JO - Mathematics of Computation
JF - Mathematics of Computation
SN - 0025-5718
IS - 338
ER -
ID: 342674952