Details:

Title	Deformation Techniques for Efficient Polynomial Equation Solving
Author(s)	Joos Heintz, Teresa Krick, Susana Puddu, Juan Sabia, Ariel Waissbein
Type	Article in Journal
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.
Keywords	polynomial equation system, arithmetic circuit, shape (or primitive element) lemma, Newton 3, Hensel iteration
Length	40
Copyright	Academic Press
File
URL	dx.doi.org/10.1006/jcom.1999.0529
Language	English
Journal	Journal of Complexity
Volume	16
Number	1
Pages	70-109
Year	2000
Month	March
Translation	No
Refereed	No