On the robust hardness of Gröbner basis computation (original) (raw)

Extended Hardness Results for Approximate Gröbner Basis Computation

ArXiv, 2016

Two models were recently proposed to explore the robust hardness of Gr\"obner basis computation. Given a polynomial system, both models allow an algorithm to selectively ignore some of the polynomials: the algorithm is only responsible for returning a Gr\"obner basis for the ideal generated by the remaining polynomials. For the qqq-Fractional Gr\"obner Basis Problem the algorithm is allowed to ignore a constant (1−q)(1-q)(1−q)-fraction of the polynomials (subject to one natural structural constraint). Here we prove a new strongest-parameter result: even if the algorithm is allowed to choose a (3/10−epsilon)(3/10-\epsilon)(3/10−epsilon)-fraction of the polynomials to ignore, and need only compute a Gr\"obner basis with respect to some lexicographic order for the remaining polynomials, this cannot be accomplished in polynomial time (unless P=NPP=NPP=NP). This statement holds even if every polynomial has maximum degree 3. Next, we prove the first robust hardness result for polynomial systems of maximum de...

Degree Bounds and Complexity of Gröbner Bases of Important Classes of Polynomial 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.

Gröbner Bases and Polynomial Equations

2016

Let S = k[x1, x2, . . . , xn] denote a polynomial ring over a field k where x1, x2, . . . , xn are indeterminates. A Gröbner basis is a set of polynomials in S which has several remarkable properties which enable us to carry out standard operations on ideals, rings and modules in an algorithmic way. Every set of polynomials in S can be transformed into a Gröbner basis. This process generalises three important algorithms: (1) Gauss elimination method for solving a system of linear equations, (2) Euclid’s algorithm for finding the greatest common divisor and (3) The simplex method of linear programming. One of the goals of these two lectures is to explain how to reduce the problem of solving a system of polynomial equations to a problem of finding eigenvalues of commuting matrices. We will introduce term orders first on the set of monomials in S and define the concept of Gróbner basis of an ideal. Term orders on monomials in k[x1, x2, . . . , xn] The set of monomials in the polynomial...

Algorithmic Algebraic Combinatorics and Gröbner Bases

Algorithmic Algebraic Combinatorics and Gröbner Bases, 2009

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Gröbner Bases: An Introduction and Some Applications

International Virtual FDP on “Frontiers of Mathematics”, 2020

* Introduction ? Varieties ? Ideals ? Linear Case ? Polynomials of One Variable * Gröbner Bases ? Term Orders ? S-Polynomials ? Buchberger’s Algorithm ? Sample Computations * Some Application of Gröbner Bases ? The 3-Color Problem ? Automatic Geometric Theorem Proving ? Other Applications * References

The Gröbner basis of the ideal of vanishing polynomials

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.

New developments in the theory of Gröbner bases and applications to formal verification

Journal of Pure and Applied Algebra, 2009

We present foundational work on standard bases over rings and on Boolean Gröbner bases in the framework of Boolean functions. The research was motivated by our collaboration with electrical engineers and computer scientists on problems arising from formal verification of digital circuits. In fact, algebraic modelling of formal verification problems is developed on the word-level as well as on the bit-level. The word-level model leads to Gröbner basis in the polynomial ring over Z/2 n while the bit-level model leads to Boolean Gröbner bases. In addition to the theoretical foundations of both approaches, the algorithms have been implemented. Using these implementations we show that special data structures and the exploitation of symmetries make Gröbner bases competitive to state-of-the-art tools from formal verification but having the advantage of being systematic and more flexible.

SOME APPLICATIONS OF GRÖBNER BASES

In this paper we will introduce a brief introduction to Gröbner bases theory and some applications of Gröbner bases. 1. The 3-Colorable Problem 2. Application in Cryptography 3. Automatic Geometric Theorem Proving

Exploiting Chordal Structure in Polynomial Ideals: A Gröbner Bases Approach

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.