Buchberger Algorithm Research Articles

Algorithmic operator algebras via normal forms in tensor rings

We propose a general algorithmic approach to noncommutative operator algebras generated by additive operators using quotients of tensor rings that are defined by tensor reduction systems. Skew polynomials are a well-established tool covering many cases arising in applications. However, integro-differential operators over an arbitrary integro-differential algebra do not fit this structure, for example. Instead of using parametrized Gröbner bases in free algebras, as has been used so far in the literature, we use Bergman's basis-free analog in tensor rings. Since reduction rules are given by module homomorphisms, the tensor setting often allows for a finite reduction system. A confluent tensor reduction system enables effective computations based on normal forms. Using tensor rings, we can also model integro-differential operators with matrix coefficients, where constants are not commutative.To have smaller reduction systems, we develop a generalization of Bergman's setting. It allows overlapping domains of reduction homomorphisms, which also make the algorithmic verification of the confluence criterion more efficient. Moreover, we discuss a heuristic approach to complete a given reduction system to a confluent one in analogy to Buchberger's algorithm and Knuth–Bendix completion. Integro-differential operators are used to illustrate the tensor setting, verification of confluence, and completion of tensor reduction systems. We also introduce a confluent reduction system and normal forms for integro-differential operators with linear substitutions, which have applications in delay differential equations. Verification of the confluence criterion and completion based on S-polynomial computations is supported by the Mathematica package TenReS.

A new idea on the interval-symbol method with correct zero rewriting for reducing exact computations

The ISCZ method (Interval-Symbol method with Correct Zero rewriting) was proposed in [2] based on Shirayanagi-Sweedler stabilization theory ([3]), to reduce the amount of exact computations as much as possible to obtain the exact results. The authors of [2] applied this method to Buchberger's algorithm which computes a Gröbner basis, but the effectiveness was not achieved except for a few examples. This is not only because the complex structure of Buchberger's algorithm causes symbols to significantly grow but also because we naively implemented the ISCZ method without any particular devices. In this poster, we propose a new idea for efficiency of the ISCZ method and show its effect by applying it to calculation of Frobenius canonical form of square matrices. Jordan canonical form is also well-known, but it requires an extension of the field containing the roots of its characteristic polynomial. On the other hand, Frobenius canonical form can be computed by using only basic arithmetic operations, but nevertheless has almost the same information as Jordan canonical form.

The Buchberger resolution

We define the Buchberger resolution, which is a graded free resolution of a monomial ideal in a polynomial ring. Its construction uses a generalization of the Buchberger graph and encodes much of the combinatorics of the Buchberger algorithm. The Buchberger resolution is a cellular resolution that coincides with the Scarf resolution for generic monomial ideals, which is the case when it is minimal. The simplicial complex underlying the Buchberger resolution is of interest for its own sake and its combinatorics is not fully understood. We close with a conjecture on the clique complex of the Buchberger graph.

Androids Armed With Poisoned Chocolate Squares: Ideal Nim and Its Relatives

SummaryWe describe two new combinatorial games. The first, Ideal Nim, both generalizes the well-known game Nim and its relative Chomp, and provides a recreational perspective on some important ideas of commutative algebra; for instance, the fact that the game is guaranteed to end is equivalent to Dickson's lemma, a well-known fact of commutative algebra. This relationship leads to a game-based proof of Dickson's lemma. The second game, Gröbner Nim, is really a variant of Ideal Nim that illustrates Buchberger's algorithm to compute a Gröbner basis. We conclude by describing the relationship between Gröbner Nim and polynomial rings.

The (Ordinary) Generating Functions Enumerating 123-Avoiding Words with r Occurrences of Each of 1, 2, . . . , n Are Always Algebraic

The set of 123-avoiding permutations (alias words in $${\{1, \ldots, n\}}$$ with exactly 1 occurrence of each letter) is famously enumerated by the ubiquitous Catalan numbers, whose generating function C(x) famously satisfies the algebraic equation $${C(x) = 1 + xC(x)^{2}}$$ . Recently, Bill Chen, Alvin Dai, and Robin Zhou found (and very elegantly proved) an algebraic equation satisfied by the generating function enumerating 123-avoiding words with two occurrences of each of $${\{1, \ldots, n\}}$$ . Inspired by the Chen-Dai-Zhou result, we present an algorithm for finding such an algebraic equation for the ordinary generating function enumerating 123- avoiding words with exactly r occurrences of each of $${\{1, \ldots, n\}}$$ for any positive integer r, thereby proving that they are algebraic, and not merely D-finite (a fact that is promised by WZ theory). Our algorithm consists of presenting an algebraic enumeration scheme, combined with the Buchberger algorithm.

Framed graphs and the non-local ideal in the knot Floer cube of resolutions

This article addresses the two significant aspects of Ozsv\'ath and Szab\'o's knot Floer cube of resolutions that differentiate it from Khovanov and Rozansky's HOMFLY-PT chain complex: (1) the use of twisted coefficients and (2) the appearance of a mysterious non-local ideal. Our goal is to facilitate progress on Rasmussen's conjecture that a spectral sequence relates the two knot homologies. We replace the language of twisted coefficients with the more quantum topological language of framings on trivalent graphs. We define a homology theory for framed trivalent graphs with boundary that -- for a particular non-blackboard framing -- specializes to the homology of singular knots underlying the knot Floer cube of resolutions. For blackboard framed graphs, our theory conjecturally recovers the graph homology underlying the HOMFLY-PT chain complex. We explain the appearance of the non-local ideal by expressing it as an ideal quotient of an ideal that appears in both the HOMFLY-PT and knot Floer cubes of resolutions. This result is a corollary of our main theorem, which is that closing a strand in a braid graph corresponds to taking an ideal quotient of its non-local ideal. The proof is a Gr\"obner basis argument that connects the combinatorics of the non-local ideal to those of Buchberger's Algorithm.

Extended letterplace correspondence for nongraded noncommutative ideals and related algorithms

Let K〈xi〉 be the free associative algebra generated by a finite or a countable number of variables xi. The notion of "letterplace correspondence" introduced in [R. La Scala and V. Levandovskyy, Letterplace ideals and non-commutative Gröbner bases, J. Symbolic Comput. 44(10) (2009) 1374–1393; R. La Scala and V. Levandovskyy, Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra, J. Symbolic Comput. 48 (2013) 110–131] for the graded (two-sided) ideals of K〈xi〉 is extended in this paper also to the nongraded case. This amounts to the possibility of modelizing nongraded noncommutative presented algebras by means of a class of graded commutative algebras that are invariant under the action of the monoid ℕ of natural numbers. For such purpose we develop the notion of saturation for the graded ideals of K〈xi,t〉, where t is an extra variable and for their letterplace analogues in the commutative polynomial algebra K[xij, tj], where j ranges in ℕ. In particular, one obtains an alternative algorithm for computing inhomogeneous noncommutative Gröbner bases using just homogeneous commutative polynomials. The feasibility of the proposed methods is shown by an experimental implementation developed in the computer algebra system Maple and by using standard routines for the Buchberger algorithm contained in Singular.

Noncommutative Gröbner Bases over Rings

In this work, we propose a method for computing noncommutative Gröbner bases over a noetherian valuation ring. We have generalized the fundamental theorem on normal forms over an arbitrary ring. The classical method of dynamical commutative Gröbner bases is generalized to Buchberger's algorithm over R = 𝒱 ⟨ x 1,…, x m ⟩, a free associative algebra with noncommuting variables, where 𝒱 = ℤ/nℤ and 𝒱 = ℤ. The proposed process generalizes previous known techniques for the computation of commutative Gröbner bases over a nætherian valuation ring and/or noncommutative Gröbner bases over a field.

Verification of Galois field based circuits by formal reasoning based on computational algebraic geometry

Algebraic error correcting codes (ECC) are widely used to implement reliability features in modern servers and systems and pose a formidable verification challenge. We present a novel methodology and techniques for provably correct design of ECC logics. The methodology is comprised of a design specification method that directly exposes the ECC algorithm's underlying math to a verification layer, encapsulated in a tool "BLUEVERI", which establishes the correctness of the design conclusively by using an apparatus of computational algebraic geometry (Buchberger's algorithm for Grobner basis construction). We present results from its application to example circuits to demonstrate the effectiveness of the approach. The methodology has been successfully applied to prove correctness of large error correcting circuits on IBM's POWER systems to protect memory storage and processor to memory communication, as well as a host of smaller error correcting circuits.

A non-commutative [formula omitted] algorithm with an application to the computation of Loewy layers

We provide a non-commutative version of the F5 algorithm, namely for right-modules over path algebra quotients. It terminates, if the path algebra quotient is a basic algebra. We show that the signatures used in the F5 algorithm allow to read off a basis for each Loewy layer, provided that a negative degree monomial ordering is used. As a byproduct, Gröbner bases in this setting can be computed more efficiently with the F5 algorithm than with Buchberger's algorithm.

On the S-operation in polynomial rings

S-polynomials used in the Buchberger's algorithm for Gröbner basis, are introduced in a more general situation. For polynomials viewed as mappings from a monoid to a ring, we define the S-operation and derive its properties. We also demonstrate its application on polynomials on a ring which is not a field.

Computing dimension of affine varieties using Groebner basis approach

Given a set of polynomials F ⊆ k X , what is the dimension of the affine variety V F ? Giving an affirmative answer to this question was never so easier in algebraic geometry until the development of Groebner basis. Groebner bases are nice because we can compute these; and this is Buchberger algorithm which makes Groebner bases so fruitful. Given a polynomial collection F ⊆ k X , Groebner basis helps in computing free set in k X with respect to the ideal F . Then dimension of V F equals the cardinality of the free set with respect to F . Here we describe an algorithm for computing such free set and answer the aforementioned question from computational point of view. We see that, in case of finding dimension of affine variety, computational technique is much more informative and motivating than theoretical method.

Computing Gröbner bases of pure binomial ideals via submodules of [formula omitted

A binomial ideal is an ideal of the polynomial ring which is generated by binomials. In a previous paper, we gave a correspondence between pure saturated binomial ideals of K[x1,…,xn] and submodules of Zn and we showed that it is possible to construct a theory of Gröbner bases for submodules of Zn. As a consequence, it is possible to follow alternative strategies for the computation of Gröbner bases of submodules of Zn (and hence of binomial ideals) which avoid the use of Buchberger algorithm. In the present paper, we show that a Gröbner basis of a Z-module M⊆Zn of rank m lies into a finite set of cones of Zm which cover a half-space of Zm. More precisely, in each of these cones C, we can find a suitable subset Y(C) which has the structure of a finite abelian group and such that a Gröbner basis of the module M (and hence of the pure saturated binomial ideal represented by M) is described using the elements of the groups Y(C) together with the generators of the cones.

Buchberger's algorithm

Buchberger's algorithm

Gröbner Bases for Ideals of σ-PBW Extensions

In this article, we introduce the σ-PWB extensions and construct the theory of Gröbner bases for the left ideals of them. We prove the Hilbert's basis theorem and the division algorithm for this more general class of Poincaré–Birkhoff–Witt extensions. For the particular case of bijective and quasi-commutative σ-PWB extensions, we implement the Buchberger's algorithm for computing Gröbner bases of left ideals.

Gröbner bases for operads

We define a new monoidal structure on the category of collections (shuffle composition). Monoids in this category (shuffle operads) turn out to bring a new insight in the theory of symmetric operads. For this category, we develop the machinery of Gröbner bases for operads and present operadic versions of Bergman's diamond lemma and Buchberger's algorithm. This machinery can be applied to study symmetric operads. In particular, we obtain an effective algorithmic version of Hoffbeck's Poincaré-Birkhoff-Witt criterion of Koszulness for (symmetric) quadratic operads

Blending implicit interval curves and surfaces

This paper is concerned with blending algebraic implicit curve and surface representation where the functional representations have inaccuracies. Algebraic implicit interval curves and surfaces are first defined as interval algebraic objects. The interval Buchberger algorithm, relying on polynomial interval division, is then outlined. General blending problems for algebraic implicit curves and surfaces are considered and various continuity conditions are applied. Computational cases are developed for specific problems.

Competitive equilibria in semi-algebraic economies

This paper develops a method to compute the equilibrium correspondence for exchange economies with semi-algebraic preferences. Given a class of semi-algebraic exchange economies parameterized by individual endowments and possibly other exogenous variables such as preference parameters or asset payoffs, there exists a semi-algebraic correspondence that maps parameters to positive numbers such that for generic parameters each competitive equilibrium can be associated with an element of the correspondence and each endogenous variable (i.e. prices and consumptions) is a rational function of that value of the correspondence and the parameters. This correspondence can be characterized as zeros of a univariate polynomial equation that satisfy additional polynomial inequalities. This polynomial as well as the rational functions that determine equilibrium can be computed using versions of Buchberger's algorithm which is part of most computer algebra systems. The computation is exact whenever the input data (i.e. preference parameters etc.) are rational. Therefore, the result provides theoretical foundations for a systematic analysis of multiplicity in applied general equilibrium.

Group actions on gröbner bases of saturated zero-dimensional binomial ideals

Fully saturated zero-dimensional binomial ideals (lattice ideals) describe lattices on Z n that are of interest in cryptanalysis. Gröbner basis computation for these ideals are computationally very intensive. We sketch a modified version of Buchberger's algorithm for lattice ideals that are closed with respect to a group action on their indeterminates. The main idea is to exploit the symmetries of such ideals to avoid performing S-polynomial reductions that are "equivalent" with respect to the group action. Using our techniques the number of S-polynomials to be considered can be reduced by a factor up to the order of the group.

Computation of Groebner Basis for Systematic Encoding of Generalized Quasi-Cyclic Codes

Generalized quasi-cyclic (GQC) codes form a wide and useful class of linear codes that includes thoroughly quasi-cyclic codes, finite geometry (FG) low density parity check (LDPC) codes, and Hermitian codes. Although it is known that the systematic encoding of GQC codes is equivalent to the division algorithm in the theory of Grobner basis of modules, there has been no algorithm that computes Grobner basis for all types of GQC codes. In this paper, we propose two algorithms to compute Grobner basis for GQC codes from their parity check matrices: echelon canonical form algorithm and transpose algorithm. Both algorithms require sufficiently small number of finite-field operations with the order of the third power of code-length. Each algorithm has its own characteristic; the first algorithm is composed of elementary methods, and the second algorithm is based on a novel formula and is faster than the first one for high-rate codes. Moreover, we show that a serial-in serial-out encoder architecture for FG LDPC codes is composed of linear feedback shift registers with the size of the linear order of code-length; to encode a binary codeword of length n, it takes less than 2n adder and 2n memory elements. Keywords: automorphism group, Buchberger's algorithm, division algorithm, circulant matrix, finite geometry low density parity check (LDPC) codes.