A Gröbner Free Alternative for Polynomial System Solving

Abstract

Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial, and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold Kronecker and has been revisited many times in computer algebra. We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton's iterator. Roughly speaking the complexity of our algorithm is polynomial in some kind of degree of the system, in its height, and linear in the complexity of evaluation of the system. We present our implementation in the Magma system which is called Kronecker in homage to his method for solving systems of polynomial equations. We show that the theoretical complexity of our algorithm is well reflected in practice and we exhibit some cases for which our program is more efficient than the other available software. Academic Press

The Computational Complexity of the Chow Form

“…We refer to [7] for a rather complete account of the subject or to [27] for a brief survey of the existing literature. Let R denote a commutative Q-algebra and let d ∈ N. Now we are going to consider some procedures involving polynomials encoded by slp's.…”

“…The situation in the present work is not much different from that in e.g. [31], [27]. Hence we just describe this procedure in order to adapt it to our setting and to estimate its complexity; its correctness follows directly from [31, Section 2] and the arguments therein.…”

Bit complexity for multi-homogeneous polynomial system solving—Application to polynomial minimization

“…This data structure has a long history, going back to work of Kronecker and Macaulay [31,34], and has been used in a host of algorithms in effective algebra [17,19,1,20,18,40,21,33].…”

Sparse Rational Univariate Representation

“…We find this representation and its variants with many different names in literature. For example, it appears as Kronecker representation, since Kronecker initiated it, as rational parametrization, or Geometric Resolution [28], see also [32,38], rational univariate representation (RUR) [35], see also [1], or primitive element [10].…”

“…The existing estimates on the degree and the height of the polynomial involved in the representation are based on total degree or ISSAC '17, July [25][26][27][28]2017 Bézout bounds, on the height theory of varieties and on the theory of Chow forms, [1,15,35]. This representation is related to the arithmetic Nullstellensätz [18,30,39,40], see also [33] for the most recent approach, and the separation bounds of the polynomial systems [21].…”