Computing the Dimension of a Polynomial Ideal

Applicable Algebra in Engineering, Communication and Computing, 2017

In this paper, we study first the relationship between Pommaret bases and Hilbert series. Given a finite Pommaret basis, we derive new explicit formulas for the Hilbert series and for the degree of the ideal generated by it which exhibit more clearly the influence of each generator. Then we establish a new dimension depending Bézout bound for the degree and use it to obtain a dimension depending bound for the ideal membership problem.

Journal of Complexity, 1997

In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)-tuple P = ( f, g 1 , g 2 , . . . , g w ) where f and the g i are multivariate polynomials, and the problem is to determine whether f is in the ideal generated by the g i . For polynomials over the integers or rationals, this problem is known to be exponential space complete. We discuss further complexity results for problems related to polynomial ideals, like the word and subword problems for commutative semigroups, a quantitative version of Hilbert's Nullstellensatz in a complexity theoretic version, and problems concerning the computation of reduced polynomials and Gröbner bases. © 1997 Academic Press

Journal of Symbolic Computation, 2000

Recent theoretical advances in elimination theory use straight-line programs as a datastructure to represent multivariate polynomials. We present here the Projective Noether Package which is a Maple implementation of one of these new algorithms, yielding as a byproduct a computation of the dimension of a projective variety. Comparative results on benchmarks for time and space of several families of multivariate polynomial equation systems are given and we point out both weaknesses and advantages of different approaches.

Let B be a one dimensional k-subalgebra of the polynomial ring k[X] := k[X 1 , . . . , X n ], where k is a field of characteristic zero. We describe an algorithm which decides if there exists a k-derivation D on k[X] such that B = k[X] (=the kernel of the derivation D). In case B is a ring of constants the algorithm also gives such a derivation.

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, 2019

We present a deterministic polynomial time approximation scheme (PTAS) for computing the algebraic rank of a set of bounded degree polynomials. The notion of algebraic rank naturally generalizes the notion of rank in linear algebra, i.e., instead of considering only the linear dependencies, we also consider higher degree algebraic dependencies among the input polynomials. More specifically, we give an algorithm that takes as input a set f := {f 1 ,. .. , f n } ⊂ F[x 1 ,. .. , x m ] of polynomials with degrees bounded by d, and a rational number > 0 and runs in time O((nmd) O(d 2) • M (n)), where M (n) is the time required to compute the rank of an n × n matrix (with field entries), and finally outputs a number r, such that r is at least (1 −) times the algebraic rank of f. Our key contribution is a new technique which allows us to achieve the higher degree generalization of the results by Bläser, Jindal, Pandey (CCC'17) who gave a deterministic PTAS for computing the rank of a matrix with homogeneous linear entries. It is known that a deterministic algorithm for exactly computing the rank in the linear case is already equivalent to the celebrated Polynomial Identity Testing (PIT) problem which itself would imply circuit complexity lower bounds (Kabanets, Impagliazzo, STOC'03). Such a higher degree generalization is already known to a much stronger extent in the noncommutative world, where the more general case in which the entries of the matrix are given by poly

Journal of Symbolic Computation, 2000

This paper presents an algorithm for the Quillen-Suslin Theorem for quotients of polynomial rings by monomial ideals, that is, quotients of the form A = k x 0 ; :::;xn]=I, with I a monomial ideal and k a eld. T. Vorst proved that nitely generated projective modules over such algebras are free. Given a nitely generated module P, described by generators and relations, the algorithm tests whether P is projective, in which case it computes a free basis for P.

Journal of Symbolic Computation, 2009

The Buchberger-Möller algorithm is a well-known efficient tool for computing the vanishing ideal of a finite set of points. If the coordinates of the points are (imprecise) measured data, the resulting Gröbner basis is numerically unstable. In this paper we introduce a numerically stable Approximate Vanishing Ideal (AVI) Algorithm which computes a set of polynomials that almost vanish at the given points and almost form a border basis. Moreover, we provide a modification of this algorithm which produces a Macaulay basis of an approximate vanishing ideal. We also generalize the Border Basis Algorithm ([Kehrein, A., Kreuzer, M., 2006. Computing border bases. J. Pure Appl. Algebra 205, 279-295]) to the approximate setting and study the approximate membership problem for zero-dimensional polynomial ideals. The algorithms are then applied to actual industrial problems.

1999

Modern computer algebra systems allow the computation of complicated examples in Commutative Algebra, Algebraic Geometry and Arithmetic Geometry. During the last couple of years such computations have helped to predict and check many theorems. Vice versa, inspired by complicated examples coming from theory, computer algebra developers have refined their algorithms and implementations.

Journal of Pure and Applied Algebra, 2000

We show several arithmetic estimates for Hilbert's Nullstellensatz. This includes an algorithmic procedure computing the polynomials and constants occurring in a BÃ ezout identity, whose complexity is polynomial in the geometric degree of the system. Moreover, we show for the ÿrst time height estimates of intrinsic type for the polynomials and constants appearing, again polynomial in the geometric degree and linear in the height of the system. These results are based on a suitable representation of polynomials by straight-line programs and duality techniques using the Trace Formula for Gorenstein algebras. As an application we show more precise upper bounds for the function S (x) counting the number of primes yielding an inconsistent modular polynomial equation system. We also give a computationally interesting lower bound for the density of small prime numbers of controlled bit length for the reduction to positive characteristic of inconsistent systems. Again, this bound is given in terms of intrinsic parameters. : S 0 0 2 2 -4 0 4 9 ( 9 8 ) 0 0 1 4 8 -0 S := {X d 1 ; X 1 − X d 2 ; : : : ; X n−2 − X d n−1 ; 1 − X n−1 X d−1 n }:

Gröbner Bases, Coding, and Cryptography, 2009