Pré-Publication, Document De Travail Année : 2020

Solving parametric systems of polynomial equations over the reals through Hermite matrices

Huu Phuoc Le
Mohab Safey El Din

Résumé

We design a new algorithm for solving parametric systems of equations having finitely many complex solutions for generic values of the parameters. More precisely, let f=(f1,,fm)Q[y][x] with y=(y1,,yt) and x=(x1,,xn), VCt×Cn be the algebraic set defined by the simultaneous vanishing of the fi's and π be the projection (y,x)y. Under the assumptions that f admits finitely many complex solutions when specializing y to generic values and that the ideal generated by f is radical, we solve the following algorithmic problem. On input f, we compute {\em semi-algebraic formulas} defining open semi-algebraic sets S1,,S in the parameters' space Rt such that i=1Si is dense in Rt and, for 1i, the number of real points in Vπ1(η) is invariant when η ranges over Si. This algorithm exploits special properties of some well chosen monomial bases in the quotient algebra Q(y)[x]/I where IQ(y)[x] is the ideal generated by f in Q(y)[x] as well as the specialization property of the so-called Hermite matrices which represent Hermite's quadratic forms. This allows us to obtain ``compact'' representations of the semi-algebraic sets Si by means of semi-algebraic formulas encoding the signature of a given symmetric matrix. When f satisfies extra genericity assumptions (such as regularity), we use the theory of Gr\"obner bases to derive complexity bounds both on the number of arithmetic operations in Q and the degree of the output polynomials. More precisely, letting d be the maximal degrees of the fi's and D=n(d1)dn, we prove that, on a generic input f=(f1,,fn), one can compute those semi-algebraic formulas with O ˜ ((t+Dt) 23t n2t+1d3nt+2(n+t)+1) arithmetic operations in Q and that the polynomials involved in these formulas have degree bounded by D. We report on practical experiments which illustrate the efficiency of this algorithm, both on generic parametric systems and parametric systems coming from applications since it allows us to solve systems which are out of reach on the current state-of-the-art.
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Dates et versions

hal-03029441 , version 1 (28-11-2020)
hal-03029441 , version 2 (16-12-2021)

Identifiants

  • HAL Id : hal-03029441 , version 1

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Huu Phuoc Le, Mohab Safey El Din. Solving parametric systems of polynomial equations over the reals through Hermite matrices. 2020. ⟨hal-03029441v1⟩
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