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

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- by Elisenda Feliu and AmirHosein Sadeghimanesh PDF
- Math. Comp.
**91**(2022), 2739-2769 Request permission

## 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.## References

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## Additional Information

**Elisenda Feliu**- Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
- MR Author ID: 865998
- ORCID: 0000-0001-7205-6511
- Email: efeliu@math.ku.dk
**AmirHosein Sadeghimanesh**- Affiliation: Centre for Computational Science & Mathematical Modelling (CSM), Innovation Village 10, Cheetah Road, CV1 2TL Coventry, UK
- MR Author ID: 1311158
- ORCID: 0000-0002-6945-3118
- Email: ad6397@coventry.ac.uk
- Received by editor(s): October 26, 2020
- Received by editor(s) in revised form: September 24, 2021, and May 3, 2022
- Published electronically: August 11, 2022
- Additional Notes: The authors acknowledge funding from the Independent Research Fund of Denmark.
- © Copyright 2022 American Mathematical Society
- Journal: Math. Comp.
**91**(2022), 2739-2769 - MSC (2020): Primary 14Q30; Secondary 13P15
- DOI: https://doi.org/10.1090/mcom/3760
- MathSciNet review: 4473102