Bounds on numers of vectors of multiplicities for polynomials which are easy to compute
Dima Grigoriev, Nicolai Vorobjov · 2000
Let F be an algebraically closed field of zero characteristic, a polynomial @@@@ ∈ F[X1, … , Xn have a multiplicative complexity r and ƒ1, … ƒk ∈ F[X1, … , Xn] be some polynomials of degrees not exceeding d, such that @@@@ = ƒ1 = ··· = ƒk = 0 has a finite 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 Gröbner bases and vectors of multiplicities of the roots for a parametric zero-dimensional 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.