COMPLEXITY BOUND FOR THE ABSOLUTE FACTORIZATION OF PARAMETRIC POLYNOMIALS

摘要

An algorithm is constructed for the absolute factorization of polynomials with algebraically independent parametric coefficients. It divides the parameter space into pairwise disjoint pieces such that the absolute factorization of polynomials with coefficients in each piece is given uniformly. Namely, for each piece there exist a positive integer l ≤ d, l variables C_1,..., C_l algebraically independent over the ground field F, and rational functions b_(J,j) of the parameters and of the variables C_1,..., C_l such that for any parametric polynomial f with coefficients in this piece, there exist c_1,..., c_l ∈ F with f = ∏_j. G_j, where G_j = ∑_(|J|) B_(J,j)Z~J is absolutely irreducible. Here Z = (Z_0,..., Z_n) are the variables of f, each B_(J,j) is the value of b_(J,j) at the coefficients of f and c_1,..., c_l, and F denotes the algebraic closureof F.