Bounds on numers of vectors of multiplicities for polynomials which are easy to compute (original) (raw)

Let F be an algebraically closed eld of zero characteristic, a polynomial ' 2 F X1; : : : ; Xn] have a multiplicative complexity r and f1; : : : ; fk 2 F X1; : : : ; Xn] be some polynomials of degrees not exceeding d, such that ' = f1 = = fk = 0 has a nite number of roots. We show that the number of possible distinct vectors of multiplicities of these roots is small when r; d and k are small. As technical tools we design algorithms which produce Gr obner bases and vectors of multiplicities of the roots for a parametric zerodimensional system. The complexities of these algorithms are singly exponential. We also describe an algorithm for parametric absolute factorization of multivariate polynomials. This algorithm has subexponential complexity in the case of a small (relative to the number of variables) degree of the polynomials.