This ACM volume deals with tackling problems that can be represented by data structures which are essentially matrices with polynomial entries, mediated by the disciplines of commutative algebra and algebraic geometry. The discoveries stem from an interdisciplinary branch of research which has been growing steadily over the past decade. The author covers a wide range, from showing how to obtain deep heuristics in a computation of a ring, a module or a morphism, to developing means of solving nonlinear systems of equations - highlighting the use of advanced techniques to bring down the cost of computation. Although intended for advanced students and researchers with interests both in algebra and computation, many parts may be read by anyone with a basic abstract algebra course.

Computational methods in commutative algebra and algebraic geometry

Algorithmic algebra

Straight--Line Programs in Geometric Elimination Theory

In this paper we show, by generalizing Hermite’s theorem to the multivariate setting, how to count the number of real or complex points of a discrete algebraic set which lie within some algebraic constraint region. We introduce a family of quadratic forms determined by the algebraic constraints and defined in terms of the trace from the coordinate ring of the variety to the ground field, and we show that the rank and signature of these forms are sufficient to determine the number of real points lying within a constraint region. In all cases we count geometric points, which is to say, we count points without multiplicity. The theoretical results on these quadratic forms are more or less classical, but forgotten too, and can be found also in [3].

Counting real zeros in the multivariate case

We present a new method for solving symbolically zero-dimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of an alternative data structure: arithmetic networks and straight-line programs with FOR gates. For sequential time complexity measured by the size of these networks we obtain the following result: it is possible to solve any affine or toric zero-dimensional equation system in non-uniform sequential time which is polynomial in the length of the input description and the “geometric degree” of the equation system. Here, the input is thought to be given by a straight-line program (or alternatively in sparse representation), and the length of the input is measured by number of variables, degree of equations and size of the program (or sparsity of the equations). Geometric degree has to be adequately defined. It is always bounded by the algebraic-combinatoric “Bezout number” of the system which is given by the Hilbert function of a suitable homogeneous ideal. However, in many important cases, the value of the geometric degree is much smaller than the Bezout number since it does not take into account multiplicities or degrees of extraneous components (which are at infinity in the affine case or contained in some coordinate hyperplane in the toric case).

When Polynomial Equation Systems Can Be Solved Fast

The membership problem for unmixed polynomial ideals is solvable in single exponential time

Canonical representatives in moderate cohomology

Let V be a n-dimensional Stein manifold, I be a closed ideal of holomorphic functions on V. It was proved by Roger Gay that, given an analytic functional T such that hT=0 (as a functional) for any h∈I, one can find some (n, n)_compactly supported current\(\tilde T\), such that\(\tilde T(\varphi ) = 0\) for any ϕ ∈Iɛ 0,0(V) and T(h)=\(\tilde T(h)\) for any h analytic on V. In this paper, we give some explicit construction of Ť in terms of residual currents when I is defined as a complete intersection or is locally Cohen-Macaulay. Moreover, by means of integral representation formulas of the Andersson-Berndtsson-Passare type, we also study the non complete intersection case in order to represent analytic functionals orthogonal to the ideal in terms of currents annihilated (as currents) by some power (less than n) of the local integral closure ofIɛ0,0.

Analytic functionals annihilated by ideals

The classical problem of deciding membership to arbitrary polynomial ideals is EXPSPACE complete. Moreover, the problem of finding a representation of a polynomial by generators of a given ideal may involve doubly exponential (in the number of variables) degrees ([16]). The same difficulty arises when computing Groebner bases of arbitrary polynomial ideals ([11]). This means that all known techniques to decide membership and to find representations of polynomials with respect to a given ideal lead to doubly exponential (sequential time) worst case complexities. However, if the geometry of the underlying algebraic variety is particularly simple, e.g. if the given ideal is zero dimensional or complete intersection, algorithms of considerably lower complexity can be found (see e.g. [7], [9]). The improvements are due to recent progress concerning affine versions of the effective Nullstellensatz (compare [18] and the references given there).

Carmen Sessa

Papers

The membership problem for unmixed polynomial ideals is solvable in single exponential time

Canonical representatives in moderate cohomology

Analytic functionals annihilated by ideals

Duality methods for the membership problem

An effective residual criterion for the membership problem in C[z1,…,zn]