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 zerodimensional 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. Academic Press

Algorithms for the universal decomposition algebra

“…; see for instance [26,25,31,24,23,19]. We can avoid working with m-variate rational function coefficients, as the formula above implies that we can obtain Si as follows.…”

Modular equations for hyperelliptic curves

“…We estimate that this polynomial has several million monomials, so new techniques will be needed to store it, relying on the evaluation philosophy of [16].…”

Deformation Techniques for Sparse Systems

“…The complexity of such algorithms is usually determined by geometric invariants associated to the family of systems under consideration (see, e.g., [16], [25], [53], [44], [24], [27], [13], [50], [31], [39]), typically in the form of a suitable (arithmetic or geometric) Bézout number (see [36], [25], [33], [46], [26], [23], [43]). …”

“…The complexity of this procedure can be roughly estimated by the product of two geometric invariants: the degree of the morphism π and the degree of the curve W . The algorithm is nearly optimal in worst case [13], and has good performance over certain well-posed families of polynomial systems of practical interest (see [24], [50], [7], [12]). …”