Solving parametric systems of polynomial equations over the reals through Hermite matrices
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We design a new algorithm for solving parametric systems having finitely many complex solutions for generic values of the parameters. More precisely, let f = (f_1, …, f_m)⊂ℚ[y][x] with y = (y_1, …, y_t) and x = (x_1, …, x_n), V⊂ℂ^t+n be the algebraic set defined by f and π be the projection (y, x) → y. Under the assumptions that f admits finitely many complex roots for generic values of y and that the ideal generated by f is radical, we solve the following problem. On input f, we compute semi-algebraic formulas defining semi-algebraic subsets S_1, …, S_l of the y-space such that ∪_i=1^l S_i is dense in ℝ^t and the number of real points in V∩π^-1(η) is invariant when η varies over each S_i. This algorithm exploits properties of some well chosen monomial bases in the algebra ℚ(y)[x]/I where I is the ideal generated by f in ℚ(y)[x] and the specialization property of the so-called Hermite matrices. This allows us to obtain compact representations of the sets S_i by means of semi-algebraic formulas encoding the signature of a symmetric matrix. When f satisfies extra genericity assumptions, we derive complexity bounds on the number of arithmetic operations in ℚ and the degree of the output polynomials. Let d be the maximal degree of the f_i's and D = n(d-1)d^n, we prove that, on a generic f=(f_1,…,f_n), one can compute those semi-algebraic formulas with O^ ( t+Dt2^3tn^2t+1 d^3nt+2(n+t)+1) operations in ℚ and that the polynomials involved have degree bounded by D. We report on practical experiments which illustrate the efficiency of our algorithm on generic systems and systems from applications. It allows us to solve problems which are out of reach of the state-of-the-art.
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