Solving parametric systems of polynomial equations over the reals through Hermite matrices
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), V⊂Ct×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,…,Sℓ in the parameters' space Rt such that ∪ℓi=1Si is dense in
Rt and, for 1≤i≤ℓ, the number of real points in V∩π−1(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 I⊂Q(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(d−1)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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