How to compute the Chow form of an unmixed polynomial ideal in single exponential time

Abstract

Abstract. Let K be a field and F1 ..... F,. homogeneous polynomials in the indeterminates X o ..... X, with coefficients in K. We describe an efficiently parallelizable single exponential time algorithm which computes the Chow form of the ideal I : = (F 1 ..... F,,), provided that I is unmixed. This algorithm requires only linear algebra computations over K. Let g2 be an algebraically closed field and V a closed algebraic subvariety of the n-dimensional projective space P" over .Q. Assume that V is equidimensional, i.e. assume that all the irreducible components of V have the same dimension dim (V). Then the intersection of V with a "generic" linear subvariety E of P" of codimension codim (E):= n -dim (E) is:In fact, V c~ E is not empty whenever E is a linear subvariety, generic or not, such that dim (V) > codim (E). Moreover this property may be taken as the definition of dimension of algebraic varieties. On the other hand, the number of points in the intersection of V with a "generic" linear subvariety E such that dim (V) = codim (E) does not depend on E; it is known as the degree of V and plays a central role in intersection theory. We write deg(V) for the degree of V.Let r:= d i m ( V ) + 1. Translating linear subvarieties of P" to homogeneous linear equation systems, we may interpret the set of all linear subvarieties E of P" with codim (E) = r as a quasiprojective variety whose projective closure is the set of linear subvarieties with codimension < r. The set consisting of those linear subvarieties which have non-empty intersection with V is a hypersurface of this

The Computational Complexity of the Chow Form

“…The previous algorithms for the computation of the Chow forms of the equidimensional components of a positive-dimensional variety ( [36], [8], [22], [44]) have an essentially worse complexity than ours, with the exception of the one due to G. Jeronimo, S. Puddu and J. Sabia ( [33]), which computes the Chow form of the component of maximal dimension of an algebraic variety within complexity (s d n ) O (1) . Here, not only we compute the Chow form of all of the equidimensional components but we also replace the Bézout number d n by d δ, where δ denotes the geometric degree.…”

Computing Chow Forms and Some Applications

“…Note that this algorithm solves the same task as the algorithms in [2] and [12] within lower complexity bounds and without further information on V . The significant reduction in complexity is partly due to the different way of coding polynomials with respect to [2] and [5]: the polynomials involved are not only encoded as vectors of coefficients but also as arithmetic circuits (straight-line programs) as well.…”

“…This is the reason several algorithms to compute the Chow form of a projective variety have been constructed (see, for example, [2,5,12]). The algorithm described in [12] only deals with irreducible varieties and the one shown in [2] deals with equidimensional varieties (that is, one has to know in advance whether the variety considered is irreducible or equidimensional, respectively). In [5], the algorithm yields an equidimensional decomposition of a projective variety and computes the Chow form of every component.…”

“…In [5], the algorithm yields an equidimensional decomposition of a projective variety and computes the Chow form of every component. If the variety V is given by s homogeneous polynomials in n + 1 variables of degrees bounded by d, the sequential complexities of the algorithms in [5], [2], and [12] respectively, where r is the dimension of V . In this paper we show the existence of a well-parallelizable algorithm which, given as input s homogeneous polynomials in n + 1 variables of degrees bounded by d ≥ n which define a projective variety V , produces as output the Chow form of the equidimensional component of V of the greatest dimension.…”

Sparse Systems of Parameters for Determinantal Varieties

“…. , X , and then consider systems of parameters 1 2 N expressed in terms of these coordinates. Of course, one could always perform a linear change of coordinates so as to assume that X X , X X , .…”