On the Complexity of Toric Ideals

2012

The method of Buchberger allows to effectively solve the membership problem in polynomial ideals and many other interesting problems. Mayr and Meyer showed that this is very expensive in the worst case. So the problem has to be specialized for more efficient computations. As previous results show, the complexity of the membership problem is mainly related to the degrees of the representation problem and Grobner bases which are studied in the first part of the thesis. The main contributions are upper and lower bounds for Grobner bases depending on the ideal dimension and some results for toric ideals. In the second part, these findings are applied to questions of complexity. The presentation comprises an incremental space-efficient algorithm for the computation of Grobner bases, an algorithm in polylogarithmic space for the membership problem in toric ideals and the space-efficient computation of the radicals of low-dimensional ideals.

SIAM Journal on Discrete Mathematics, 2016

Chordal structure and bounded treewidth allow for efficient computation in numerical linear algebra, graphical models, constraint satisfaction, and many other areas. In this paper, we begin the study of how to exploit chordal structure in computational algebraic geometry-in particular, for solving polynomial systems. The structure of a system of polynomial equations can be described in terms of a graph. By carefully exploiting the properties of this graph (in particular, its chordal completions), more efficient algorithms can be developed. To this end, we develop a new technique, which we refer to as chordal elimination, that relies on elimination theory and Gröbner bases. By maintaining graph structure throughout the process, chordal elimination can outperform standard Gröbner bases algorithms in many cases. The reason is because all computations are done on "smaller" rings of size equal to the treewidth of the graph (instead of the total number of variables). In particular, for a restricted class of ideals, the computational complexity is linear in the number of variables. Chordal structure arises in many relevant applications. We demonstrate the suitability of our methods in examples from graph colorings, cryptography, sensor localization, and differential equations.

Journal of Symbolic Computation, 1999

Toric ideals are binomial ideals which represent the algebraic relations of sets of power products. They appear in many problems arising from different branches of mathematics. In this paper, we develop new theories which allow us to devise a parallel algorithm and an efficient elimination algorithm. In many respects they improve existing algorithms for the computation of toric ideals.

Theory of Computing Systems, 2021

Let F[X] be the polynomial ring over the variables X = {x 1 , x 2 ,. .. , x n }. An ideal I = p 1 (x 1),. .. , p n (x n) generated by univariate polynomials {p i (x i)} n i=1 is a univariate ideal. We study the ideal membership problem for the univariate ideals and show the following results. Let f (X) ∈ F[ 1 ,. .. , r ] be a (low rank) polynomial given by an arithmetic circuit where i : 1 ≤ i ≤ r are linear forms, and I = p 1 (x 1),. .. , p n (x n) be a univariate ideal. Given α ∈ F n , the (unique) remainder f (X) (mod I) can be evaluated at α in deterministic time d O(r) • poly(n), where d = max{deg(f), deg(p 1). .. , deg(p n)}. This yields a randomized n O(r) algorithm for minimum vertex cover in graphs with rank-r adjacency matrices. It also yields an n O(r) algorithm for evaluating the permanent of a n × n matrix of rank r, over any field 2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms

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

Extending the notion of indispensable binomials of a toric ideal ((14), (7)), we define indispensable monomials of a toric ideal and establish some of their properties. They are useful for searching indispensable binomials of a toric ideal and for proving the existence or non-existence of a unique minimal system of binomials generators of a toric ideal. Some examples of indispensable monomials from statistical models for contingency tables are given.

Journal of Pure and Applied Algebra

The computation of Gröbner bases is an established hard problem. By contrast with many other problems, however, there has been little investigation of whether this hardness is robust. In this paper, we frame and present results on the problem of approximate computation of Gröbner bases. We show that it is NP-hard to construct a Gröbner basis of the ideal generated by a set of polynomials, even when the algorithm is allowed to discard a 1 − ǫ fraction of the generators, and likewise when the algorithm is allowed to discard variables (and the generators containing them). Our results shows that computation of Gröbner bases is robustly hard even for simple polynomial systems (e.g. maximum degree 2, with at most 3 variables per generator). We conclude by greatly strengthening results for the Strong c-Partial Gröbner problem posed by De Loera et al. [10]. Our proofs also establish interesting connections between the robust hardness of Gröbner bases and that of SAT variants and graph-coloring.

Journal of Symbolic Computation, 2004

We encode the binomials belonging to the toric ideal I A associated with an integral d × n matrix A using a short sum of rational functions as introduced by Barvinok Barvinok (1994); Barvinok and Woods (2003). Under the assumption that d, n are fixed, this representation allows us to compute the Graver basis and the reduced Gröbner basis of the ideal I A , with respect to any term order, in time polynomial in the size of the input. We also derive a polynomial time algorithm for normal form computation which replaces in this new encoding the usual reductions typical of the division algorithm. We describe other applications, such as the computation of Hilbert series of normal semigroup rings, and we indicate further connections to integer programming and statistics.

Journal of Symbolic Computation, 2007

Let K be an arbitrary field and {d1, . . . , dn} a set of all-different positive integers. The aim of this work is to propose and evaluate an algorithm for checking whether or not the toric ideal of the affine monomial curve

Journal of Symbolic Computation, 2011

We construct an explicit minimal strong Gröbner basis of the ideal of vanishing polynomials in the polynomial ring over Z/m for m ≥ 2. The proof is done in a purely combinatorial way. It is a remarkable fact that the constructed Gröbner basis is independent of the monomial order and that the set of leading terms of the constructed Gröbner basis is unique, up to multiplication by units. We also present a fast algorithm to compute reduced normal forms, and furthermore, we give a recursive algorithm for building a Gröbner basis in Z/m[x1, x2, . . . , xn] along the prime factorization of m. The obtained results are not only of mathematical interest but have immediate applications in formal verification of data paths for microelectronic systems-on-chip.