Kac-Rice formulas and the number of solutions of parametrized systems of polynomial equations. (English) Zbl 1502.14145

Consider a parametrized system of \(n\) polynomial equations in \(n\) variables \(f_{\kappa}(t)=0,\ t\in A,\ \kappa\in B\) with \(A\subseteq \mathbb{R}^n\) and \(B\subseteq \mathbb{R}^m.\) The problem approached in this paper is to partition the parameter space \(B\) into regions where the number of solutions to the system in \(A\) is \(0,1,2,\dots,\infty.\)
A case of interest is when such a system comes from the equilibrium solutions of a system of ODEs, for instance a system of ODEs modeling a network of chemical reactions. One can, in theory, resolve this problem employing quantifier elimination or Cylindrical Algebraic Decomposition, but this is already computationally unfeasible for small systems with three or four parameters and three or four variables. A numerical approach is welcome.
The authors propose and describe in details an approach using Kac-Rice formula. With this approach they are able to compute an approximated partition of the parameter space in boxes using Monte Carlo integrals. This technique is very useful and successful in concrete examples, as one can start with a very coarse partition and keep refining it where it is still computationally feasible in an iterative matter. A more refined partition “just” requires more computation.
The strategy can handle cases with a much larger number of parameters than the exact methods and the authors provide clear and concrete examples. The algorithms are implemented in several distinct programming languages and they discuss some of the computational difficulties and limitations.


14Q30 Computational real algebraic geometry
13P15 Solving polynomial systems; resultants
65C05 Monte Carlo methods
65D30 Numerical integration


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