Buchberger Algorithm Research Articles

Binomials Arising from Buchberger Algorithm on Polyomino Ideals

A polyomino is a finite set of unit squares joined side by side on the Cartesian plane. Qureshi introduced an ideal constructed from a polyomino which is called "polyomino ideal". In this paper, we study the binomials arising from Buchberger Algorithm on polyomino ideals. We also introduce socket wrench polyominoes and study the Gr\"{o}bner bases of the ideal $I_{\mathcal P}$ and some algebraic properties of $K[\mathcal P]$.

Signature-based algorithm under non-compatible term orders and its application to change of ordering

The notion of the compatibility between a term order in a polynomial ring R and a module term order in Rl is crucial to ensure the termination of a signature-based algorithm for general input ideals. However, it is shown experimentally that the compatibility does not necessarily imply efficient computation. Our experiments show that combining non-compatible term orders can improve performance for computing Gröbner bases with respect to some term orders. In such cases, we can use the Hilbert function to guarantee the termination. The Hilbert function can be computed by using a Gröbner basis with respect to some term order and thus the resulting algorithm is considered a change of ordering algorithm. In this paper, we give the details of the new change of ordering algorithm and we compare its performance with that of the usual Hilbert-driven Buchberger algorithm and the Gröbner walk algorithm.

A new algorithm for Gröbner bases conversion

A new approach for Gröbner bases conversion of polynomial ideals (over a field) of arbitrary dimension is presented. In contrast to the only other approach based on Gröbner fan and Gröbner walk for positive dimensional ideals, the proposed approach is simpler to understand, prove, and implement. It is based on defining for a given polynomial, a truncated sub-polynomial consisting of all monomials that can possibly become the leading monomial with respect to the target ordering: the monomials between the leading monomial of the target ordering and the leading monomial of the initial ordering.The main ingredient of the new algorithm is the computation of a Gröbner basis with respect to the target ordering for the ideal generated by such truncated parts of the input Gröbner basis. This is done using the extended Buchberger algorithm that also outputs the relationship between the input and output bases. That information is used in attempts to recover a Gröbner basis of the ideal with respect to the target ordering. In general, more than one iteration may be needed to get a Gröbner basis with respect to the target ordering since truncated polynomials may miss some leading monomials.The new algorithm has been implemented in Maple and its operation is illustrated using an example. The performance of this implementation is compared with the implementations of other approaches in Maple. In practice, a Gröbner basis with respect to a target ordering can be computed in a single iteration on most examples.Since the proposed basis conversion algorithm uses simple concepts of Gröbner basis theory, it can be easily taught in contrast to methods based on Gröbner walk.

Computing Gröbner bases and free resolutions of OI-modules

Given a sequence of related modules Mn defined over a sequence of related polynomial rings, one may ask how to simultaneously compute a finite Gröbner basis for each Mn. Furthermore, one may ask how to simultaneously compute the module of syzygies of each Mn. In this paper we address both questions. Working in the setting of OI-modules over a Noetherian polynomial OI-algebra, we provide OI-analogues of Buchberger's Criterion, Buchberger's Algorithm for computing Gröbner bases, and Schreyer's Theorem for computing syzygies. We also establish a stabilization result for Gröbner bases.

Signature-based standard basis algorithm under the framework of GVW algorithm

The GVW algorithm, one of the most important so-called signature-based algorithms, is designed to eliminate a large number of useless polynomial reductions from Buchberger's algorithm. The cover theorem serves as the theoretical foundation of the GVW algorithm, and up to now, it applies only to a certain class of monomial orders, namely global orders and a special class of local orders. In this paper we extend this theorem to any semigroup order, which can be either global, local or even mixed. Building upon the pioneering idea of the Mora normal form algorithm, we propose a more comprehensive and general proof for the cover theorem while bypassing the need to choose a minimal element from an infinite set of monomials in all the existing proofs. Therefore, the algorithm for signature-based standard bases is presented for any semigroup order under the framework of the GVW algorithm, and an example is given to provide an illustration of the algorithm.

Axioms for a theory of signature bases

Twenty years after the discovery of the F5 algorithm, Gröbner bases with signatures are still challenging to understand and to adapt to different settings. This contrasts with Buchberger's algorithm, which we can bend in many directions keeping correctness and termination obvious. I propose an axiomatic approach to Gröbner bases with signatures with the purpose of uncoupling the theory and the algorithms, giving general results applicable in many different settings (e.g. Gröbner for submodules, F4-style reduction, noncommutative rings, non-Noetherian settings, etc.), and extending the reach of signature algorithms.

A “pseudo-polynomial” algorithm for the Frobenius number and Gröbner basis

Given n⩾2 and a1,…,an∈N. Let S=〈a1,…,an〉 be a semigroup. The aim of this paper is to give an effective pseudo-polynomial algorithm on a1, which computes the Apéry set and the Frobenius number of S. We also find the Gröbner basis of the toric ideal defined by S, for the weighted degree reverse lexicographical order ≺w to x1,…,xn, without using Buchberger's algorithm. As an application we introduce and study some special classes of semigroups. Namely, when S is generated by generalized arithmetic progressions and generalized almost arithmetic progressions with the ratio a positive or a negative number. We determine symmetric and almost symmetric semigroups generated by a generalized arithmetic progression.

Learning a performance metric of Buchberger’s algorithm

What can be (machine) learned about the complexity of Buchberger's algorithm? Given a system of polynomials, Buchberger's algorithm computes a Gr\"obner basis of the ideal these polynomials generate using an iterative procedure based on multivariate long division. The runtime of each step of the algorithm is typically dominated by a series of polynomial additions, and the total number of these additions is a hardware independent performance metric that is often used to evaluate and optimize various implementation choices. In this work we attempt to predict, using just the starting input, the number of polynomial additions that take place during one run of Buchberger's algorithm. Good predictions are useful for quickly estimating difficulty and understanding what features make Gr\"obner basis computation hard. Our features and methods could also be used for value models in the reinforcement learning approach to optimize Buchberger's algorithm introduced in [Peifer, Stillman, and Halpern-Leistner, 2020]. We show that a multiple linear regression model built from a set of easy-to-compute ideal generator statistics can predict the number of polynomial additions somewhat well, better than an uninformed model, and better than regression models built on some intuitive commutative algebra invariants that are more difficult to compute. We also train a simple recursive neural network that outperforms these linear models. Our work serves as a proof of concept, demonstrating that predicting the number of polynomial additions in Buchberger's algorithm is a feasible problem from the point of view of machine learning.

Recursive structures in involutive bases theory

We study characterisations of involutive bases using a recursion over the variables in the underlying polynomial ring and corresponding completion algorithms. Three key ingredients are (i) an old result by Janet recursively characterising Janet bases for which we provide a new and simpler proof, (ii) the Berkesch–Schreyer variant of Buchberger's algorithm and (iii) a tree representation of sets of terms also known as Janet trees. We start by extending Janet's result to a recursive criterion for minimal Janet bases leading to an algorithm to minimise any given Janet basis. We then extend Janet's result also to Janet-like bases as introduced by Gerdt and Blinkov. Next, we design a novel recursive completion algorithm for Janet bases. We study then the extension of these results to Pommaret bases. It yields a novel recursive characterisation of quasi-stability which we use for deterministically constructing “good” coordinates more efficiently than in previous works. A small modification leads to a novel deterministic algorithm for putting an ideal into Nœther position. Finally, we provide a general theory of involutive-like bases with special emphasis on Pommaret-like bases and study the syzygy theory of Janet-like and Pommaret-like bases.

Galerkin approach to approximate solutions of some boundary value problems

This paper uses the Galerkin method to find approximate solutions of some boundary value problems. The solving process requires to solve a system of algebraic equations, which are large and difficult to be solved. According to the Groebner bases theory, an improved Buchberger's algorithm is proposed to solve the algebraic system. The results show that the Galerkin approach is simple and efficient.

Some improvements for the algorithm of Gröbner bases over dual valuation domain

<abstract><p>As a special ring with zero divisors, the dual noetherian valuation domain has attracted much attention from scholars. This article aims at to improve the Buchberger's algorithm over the dual noetherian valuation domain. We present some criterions that can be applied in the algorithm for computing Gröbner bases, and the criterions may drastically reduce the number of S-polynomials in the course of the algorithm. In addition, we clearly demonstrate the improvement with an example.</p></abstract>

Signature Gröbner bases, bases of syzygies and cofactor reconstruction in the free algebra

Signature-based algorithms have become a standard approach for computing Gröbner bases in commutative polynomial rings. However, so far, it was not clear how to extend this concept to the setting of noncommutative polynomials in the free algebra. In this paper, we present a signature-based algorithm for computing Gröbner bases in precisely this setting. The algorithm is an adaptation of Buchberger's algorithm including signatures. We prove that our algorithm correctly enumerates a signature Gröbner basis as well as a Gröbner basis of the module generated by the leading terms of the generators' syzygies, and that it terminates whenever the ideal admits a finite signature Gröbner basis. Additionally, we adapt well-known signature-based criteria eliminating redundant reductions, such as the syzygy criterion, the F5 criterion and the singular criterion, to the case of noncommutative polynomials. We also generalise reconstruction methods from the commutative setting that allow to recover, from partial information about signatures, the coordinates of elements of a Gröbner basis in terms of the input polynomials, as well as a basis of the syzygy module of the generators. We have written a toy implementation of all the algorithms in the Mathematica package OperatorGB and we compare our signature-based algorithm to the classical Buchberger algorithm for noncommutative polynomials.

Right Buchberger Algorithm over Bijective Skew PBW Extensions

In this paper we present a right version of the Buchberger algorithm over skew Poincaré-Birkhoff-Witt extensions (skew PBW extensions for short) defined by Gallego and Lezama [5]. This algorithm is an adaptation of the left case given in [3]. In particular, we developed a right version of the division algorithm and from this we built the right Grbner bases theory over bijective skew PBW extensions. The algorithms were implemented in the SPBWE library developed in Maple, this paper includes an application of these to the membership problem. The theory developed here is fundamental to complete the SPBWE library and thus be able to implement various homological applications that arise as result of obtaining the right Grbner bases over skew PBW extensions.

Threaded Gröbner bases : a Macaulay2 package

The complexity of Gr\"{o}bner computations has inspired many improvements to Buchberger's algorithm over the years. Looking for further insights into the algorithm's performance, we offer a threaded implementation of classical Buchberger's algorithm in {\it Macaulay2}. The output of the main function of the package includes information about {\it lineages} of non-zero remainders that are added to the basis during the computation. This information can be used for further algorithm improvements and optimization.

The Investigation of Nonlinear Polynomial Control Systems

The paper considers methods for estimating stability using Lyapunov functions, which are used for nonlinear polynomial control systems. The apparatus of the Gro¨bner basis method is used to assess the stability of a dynamical system. A description of the Gro¨bner basis method is given. To apply the method, the canonical relations of the nonlinear system are approximated by polynomials of the components of the state and control vectors. To calculate the Gro¨bner basis, the Buchberger algorithm is used, which is implemented in symbolic computation programs for solving systems of nonlinear polynomial equations. The use of the Gro¨bner basis for finding solutions of a nonlinear system of polynomial equations is considered, similar to the application of the Gauss method for solving a system of linear equations. The equilibrium states of a nonlinear polynomial system are determined as solutions of a nonlinear system of polynomial equations. An example of determining the equilibrium states of a nonlinear polynomial system using the Gro¨bner basis method is given. An example of finding the critical points of a nonlinear polynomial system using the Gro¨bner basis method and the Wolfram Mathematica application software is given. The Wolfram Mathematica program uses the function of determining the reduced Gro¨bner basis. The application of the Gro¨bner basis method for estimating the attraction domain of a nonlinear dynamic system with respect to the equilibrium point is considered. To determine the scalar potential, the vector field of the dynamic system is decomposed into gradient and vortex components. For the gradient component, the scalar potential and the Lyapunov function in polynomial form are determined by applying the homotopy operator. The use of Gro¨bner bases in the gradient method for finding the Lyapunov function of a nonlinear dynamical system is considered. The coordination of input-output signals of the system based on the construction of Gro¨bner bases is considered.

English

In this paper, an original reduction algorithm for solving simultaneous multivariate polynomial equations is presented. The algorithm is exponential in complexity, but the well-known algorithms, such as the extended Euclidean algorithm and Buchberger's algorithm, are superexponential. The superexponential complexity of the well-known algorithms is due to their not being "minimal" in a certain sense. Buchberger's algorithm produces a Groebner basis. The proposed original reduction algorithm achieves the required task, via computation of determinants of parametric Sylvester matrices, and produces a Rabin basis, which is shown to be minimal, when two multivariate polynomials are reduced at a time. The minimality of Rabin basis allows us to prove exponential lower bounds for the space complexity of an algebraic proof of certification, for a specific computational problem in the computational complexity class PSPACE, showing that the complexity classes PSPACE and P cannot be the same. By the same reasoning, it follows that Co-NP is not the same as either of the complexity classes NP and P, and that the polynomial time hierarchy does not collapse. It is also shown that the class BPP of languages decidable by bounded error probabilistic algorithms with (probabilistic) polynomial time proofs for the membership of input words is not the same as any one of the complexity classes P, NP and Co-NP. It follows again, from the discussions, that the complexity classes NP and P are not the same, by relativization of BPP, with respect to P and NP.

An Axiomatic Approach to Gröbner Basis Theory by Examining Several Reduction Principles

Several variants of the classical theory of Grobner bases can be found in the literature. They come, depending on the structure they operate on, with their own specific peculiarity. Setting up an expedient reduction concept depends on the arithmetic equipment that is provided by the structure in question. Often it is necessary to introduce a term order that can be used for determining the orientation of the reduction, the choice of which might be a delicate task. But there are other situations where a different type of structure might give the appropriate basis for formulating adequate rewrite rules. In this paper we have tried to find a unified concept for dealing with such situations. We develop a global theory of Grobner bases for modules over a large class of rings. The method is axiomatic in that we demand properties that should be satisfied by a reduction process. Reduction concepts obeying the principles formulated in the axioms are then guaranteed to terminate. The class of rings we consider is large enough to subsume interesting candidates. Among others this class contains rings of differential operators, Ore-algebras and rings of difference-differential operators. The theory is general enough to embrace the well-known classical Grobner basis concepts of commutative algebra as well as several modern approaches for modules over relevant noncommutative rings. We start with introducing the appropriate axioms step by step, derive consequences from them and end up with the Buchberger Algorithm, that makes it possible to compute a Grobner basis. At the end of the paper we provide a few examples to illustrate the abstract concepts in concrete situations.

An efficient reduction strategy for signature-based algorithms to compute Gröbner basis

This paper introduces a strategy for signature-based algorithms to compute Gröbner basis. In comparison with Buchberger algorithm, signature-based algorithms generate S-pairs instead of S-polynomials, and operate s -reductions instead of usual reductions which is used in Buchberger algorithm. There are two strategies for s -reductions: one is only-top reduction strategy which is the way that only leading monomials are s -reduced. The other is full reduction strategy which is the way that all monomials are s -reduced. New strategy for s -reduction of S-pairs, which we call selective-full strategy, is introduced in this paper. The idea of the strategy is that when we have computed a signature Gröbner basis, there are unnecessary elements for a minimal Gröbner basis, so candidates of elements included in a minimal Gröbner basis should be sufficiently reduced. According to the experimental result, proposed strategy is efficient for computing the reduced Gröbner basis. For computing a signature Gröbner basis, it's the most efficient or not the worst of the three strategies.

Gröbner bases over fields with valuations

Let $K$ be a field with a valuation and let $S$ be the polynomial ring $S:= K[x_1, \dots , x_n]$. We discuss the extension of Gröbner theory to ideals in $S$, taking the valuations of coefficients into account, and describe the Buchberger algorithm in this context. In addition we discuss some implementation and complexity issues. The main motivation comes from tropical geometry, as tropical varieties can be defined using these Gröbner bases, but we also give examples showing that the resulting Gröbner bases can be substantially smaller than traditional Gröbner bases. In the case $K =\mathbb Q$ with the $p$-adic valuation the algorithms have been implemented in a Macaulay 2 package.

Минимальный базис модуля сизигий старших членов

Systems of polynomial equations are one of the most universal mathematical objects. Almost all the problems of cryptographic analysis can be reduced to finding solutions to systems of polynomial equations. The corresponding direction of research is called algebraic cryptanalysis. In terms of computational complexity, systems of polynomial equations cover the entire range of possible options, from algorithmic insolubility of Diophantine equations to well-known efficient methods for solving linear systems. The method of Buchberger [ 5] brings a system of algebraic equations to the system of a special type defined by the Grobner original system of equations, allowing the use of the exception of the dependent variables. The basis for determining the Groebner basis is the permissible ordering on the set of terms. The set of admissible orderings on the set of terms is infinite and even continuum. The most time-consuming step in finding the Groebner basis using the Buchberger algorithm is to prove that all S-polynomials representing a system of generators of K[X]-module S-polynomials. There is a natural problem of finding such a minimal system of generators. The existence of such a system of generators follows from Nakayama's theorem. An algorithm for constructing such a basis for any ordering is proposed.