Juan Sabia

Deformation techniques for efficient polynomial equation solving

Abstract Suppose we are given a parametric polynomial equation system encoded by an arithmetic circuit, which represents a generically flat and unramified family of zero-dimensional algebraic varieties. Let us also assume that there is given the complete description of the solution of a particular unramified parameter instance of our system. We show that it is possible to “move” the given particular solution along the parameter space in order to reconstruct—by means of an arithmetic circuit—the coordinates of the solutions of the system for an arbitrary parameter instance. The underlying algorithm is highly efficient, i.e., polynomial in the syntactic description of the input and the following geometric invariants: the number of solutions of a typical parameter instance and the degree of the polynomials occurring in the output. In fact, we prove a slightly more general result, which implies the previous statement by means of a well-known primitive element algorithm. We produce an efficient algorithmic description of the hypersurface obtained projecting polynomially the given generically flat family of varieties into a suitable affine space.

On the efficiency of effective Nullstellensa¨tze

AbstractLetk be an infinite and perfect field,x1, ...,xn indeterminates overk and letf1, ...,fs be polynomials ink[x1, ...,xn] of degree bounded by a given numberd, which satisfiesd≥n. We prove an effective affine Nullstellensatz of the following particular form:For arbitrary given parametersd, s, n there exists a probabilistic (randomized) arithmetic network overk of sizesO(1)dO(n) and depthO(n4log2sd) solving the following task:It decides whether the ideal generated byf1, ...,fs ink[x1, ...,xn] is trivial and, if this is the case, it produces a straight-line program of sizesO(1)dO(n) and depthO(n4log2sd) in the function fieldk(x1, ...,xn) which computes polynomialsp1, ...,ps ofk[x1, ...,xn] of degree

Bounds for traces in complete intersections and degrees in the Nullstellensatz

d^{O(n^2 )}

Effective equidimensional decomposition of affine varieties

satisfying

An effective algorithm for quantifier elimination over algebraically closed fields using straight line programs

1 = \mathop \sum \limits_{1 \leqslant j \leqslant s} p_j f_j .