Solving Degenerate Sparse Polynomial Systems Faster

Consider a system F of n polynomial equations in n unknowns, over an algebraically closed field of arbitrary characteristic. We present a fast method to find a point in every irreducible component of the zero set Z of F. Our techniques allow us to sharpen and lower prior complexity bounds for this problem by fully taking into account the monomial term structure. As a corollary of our development we also obtain new explicit formulae for the exact number of isolated roots of F and the intersection multiplicity of the positive-dimensional part of Z. Finally, we present a combinatorial construction of non-degenerate polynomial systems, with specified monomial term structure and maximally many isolated roots, which may be of independent interest.Comment: This is the final journal version of math.AG/9702222 (``Toric Generalized Characteristic Polynomials''). This final version is a major revision with several new theorems, examples, and references. The prior results are also significantly improve

Sub-cubic Change of Ordering for Gröner Basis: A Probabilistic Approach

International audienceThe usual algorithm to solve polynomial systems using Gröbner bases consists of two steps: first computing the DRL Gröbner basis using the F5 algorithm then computing the LEX Gröbner basis using a change of ordering algorithm. When the Bézout bound is reached, the bottleneck of the total solving process is the change of ordering step. For 20 years, thanks to the FGLM algorithm the complexity of change of ordering is known to be cubic in the number of solutions of the system to solve. We show that, in the generic case or up to a generic linear change of variables, the multiplicative structure of the quotient ring can be computed with no arithmetic operation. Moreover, given this multiplicative structure we propose a change of ordering algorithm for Shape Position ideals whose complexity is polynomial in the number of solutions with exponent ω where 2 ≤ ω < 2.3727 is the exponent in the complexity of multiplying two dense matrices. As a consequence, we propose a new Las Vegas algorithm for solving polynomial systems with a finite number of solutions by using Gröbner basis for which the change of ordering step has a sub-cubic (i.e. with exponent ω) complexity and whose total complexity is dominated by the complexity of the F5 algorithm. In practice we obtain significant speedups for various polynomial systems by a factor up to 1500 for specific cases and we are now able to tackle some instances that were intractable

Asymptotic acceleration of solving multivariate polynomial systems of equations

Award 668365) We propose new Las Vegas randomized algorithms for the solution of a multivariate generic or sparse polynomial system of equations. The algorithms use O ( ( +4 n)3 nD2 log b) arithmetic operations to approximate all real roots of the system as well as all roots lying in a fixed n-dimensional box or disc. Here D is an upper bound on the number of all the roots of the system, is the number of real roots or the roots lying in the box or disc, =2;b is the required upper bound on the output errors, and O (s) stands for O(s log c s), c being a constant independent of s. We also yield the bounds O (12 nD2) for the complexity of counting the numbers of all roots in a fixed box (disc) and all real roots and O (12 nD2 log b) for the complete solution of generic system. For a large clas