The Computational Complexity of the Chow Form

Abstract

Abstract. We present a bounded probability algorithm for the computation of the Chow forms of the equidimensional components of an algebraic variety. Its complexity is polynomial in the length and in the geometric degree of the input equation system defining the variety. In particular, it provides an alternative algorithm for the equidimensional decomposition of a variety. As an application we obtain an algorithm for the computation of a subclass of sparse resultants, whose complexity is polynomial in the dimension and the volume of the input set of exponents. As a further application, we derive an algorithm for the computation of the (unique) solution of a generic over-determined equation system.

Counting Irreducible Components of Complex Algebraic Varieties

“…Together with later algorithms it can be used to count irreducible components in parallel polynomial time. Other algorithms for the equidimensional decomposition are given by Lecerf (2000Lecerf ( , 2003, Jeronimo & Sabia (2002), Jeronimo et al (2004), but these are all ran-…”

Sparse Differential Resultant for Laurent Differential Polynomials

“…In recent years, the multivariate resultant has emerged as one of the most powerful computational tools in elimination theory due to its ability to eliminate several variables simultaneously without introducing many extraneous solutions. Many algorithms with best complexity bounds for problems such as polynomial equation solving and first-order quantifier elimination are strongly based on the multivariate resultant [4,5,15,16,26,38].…”

Newton–Hensel Interpolation Lifting

“…Since then, "Newton-Hensel lifting" in its non-Archimedean versions is primordially present in exact symbolic computation: for example, in univariate rational polynomial factorization in [38] and the breakthrough LLL-polynomial time factorization algorithm [30], and in multivariate polynomial factorization [8], [7], [17], [23]. Also for multivariate polynomial systems solving in the Gröbner basis setting in [36], [37] and in the primitive element setting in [9], [7], [17] and in [16], [15], [21], [22].…”