Bner Basis 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 $K[\mathcal P]$.

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.

New non-naturally reductive Einstein metrics on exceptional simple Lie groups

In this article, we achieved several non-naturally reductive Einstein metrics on exceptional simple Lie groups, which are formed by the decomposition arising from general Wallach spaces. By using the decomposition corresponding to the two involutive automorphisms, we calculated the non-zero coefficients in the expression for the components of Ricci tensor with respect to the given metrics. The Einstein metrics are obtained as solutions of systems polynomial equations, which we manipulate by symbolic computations using Gr\"{o}bner bases.

Computing automorphisms of Mori dream spaces

We present an algorithm to compute the automorphism group of a Mori dream space. As an example calculation, we determine the automorphism groups of singular cubic surfaces with general parameters. The strategy is to study graded automorphisms of an affine algebra graded by a finitely generated abelian group and apply the results to the Cox ring. Besides the application to Mori dream spaces, our results could be used for symmetry based computing, e.g., for Gröbner bases or tropical varieties.

Analysis of all time-delay stability for biological systems using symbolic computation methods

All time-delay stability for biological systems shows that the time-delay system possess good reliability,so this issue has always been the highlight of the scholars research. However,researchers usually adopt the traditional mathematical methods or numerical calculation methods. Based on Hurwitz criterion and polynomial complete discriminant system,a sufficient and necessary algebraic criterion of all time-delay stability for nonlinear biological systems with parameters was introduced. By using symbolic computation methods,such as the methods of Grbner basis,triangular decomposition and real solution classification,a systematic and algorithmic approach for automatically analyzing all time-delay stability of biological systems with parameters was proposed. All the computations in our approach are all exact,which may help biologists and engineers to perform algebraic analysis for certain biological models. The successful experiments on the all time-delay stability analysis of several biological models,such as time delayed Lotka-Volterra systems and SIR epidemic models with time delay,showed the feasibility of our algebraic approach and also the superiority of symbolic computation methods compared with traditional mathematic methods.

Signed Tilings by Ribbon L n-Ominoes, n Even, via Gröbner Bases

Let Tn be the set of ribbon L-shaped n-ominoes for some n≥4 even, and let T+n be Tn with an extra 2 x 2 square. We investigate signed tilings of rectangles by Tn and T+n . We show that a rectangle has a signed tiling by Tn if and only if both sides of the rectangle are even and one of them is divisible by n, or if one of the sides is odd and the other side is divisible by . We also show that a rectangle has a signed tiling by T+n, n≥6 even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by . Our proofs are based on the exhibition of explicit GrÖbner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in Nitica, V. (2015) Every tiling of the first quadrant by ribbon L n-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, 5, 11-25, cannot be obtained from coloring invariants.

Markov Bases and Generalized Lawrence Liftings

Minimal Markov bases of configurations of integer vectors correspond to minimal binomial generating sets of the assocciated lattice ideal. We give necessary and sufficient conditions for the elements of a minimal Markov basis to be (a) inside the universal Gr{\" o}bner basis and (b) inside the Graver basis. We study properties of Markov bases of generalized Lawrence liftings for arbitrary matrices $A\in\mathcal{M}_{m\times n}(\Bbb{Z})$ and $B\in\mathcal{M}_{p\times n}(\Bbb{Z})$ and show that in cases of interest the {\em complexity} of any two Markov bases is the same.

Tropical noetherity and Gröbner bases

A set that is a Gröbner basis for an ideal with respect to every Gröbner ordering is called a universal Gröbner basis for that ideal. In the paper, it is proved that there exists a universal Gröbner basis in which the polynomials have controlled degrees. The main result is the theorem on the

Gröbner bases and gradings for partial difference ideals

In this paper we introduce a working generalization of the theory of Gröbner bases for algebras of partial difference polynomials with constant coefficients. One obtains symbolic (formal) computation for systems of linear or non-linear partial difference equations arising, for instance, as discrete models or by the discretization of systems of differential equations. From an algebraic viewpoint, the algebras of partial difference polynomials are free objects in the category of commutative algebras endowed with the action by endomorphisms of a monoid isomorphic to $\mathbb {N}^r$. Then, the investigation of Gröbner bases in this context contributes also to the current research trend consisting in studying polynomial rings under the action of suitable symmetries that are compatible with effective methods. Since the algebras of difference polynomials are not Noetherian, we propose in this paper a theory for grading them that provides a Noetherian subalgebra filtration. This implies that the variants of Buchberger’s algorithm we developed for difference ideals terminate in the finitely generated graded case when truncated up to some degree. Moreover, even in the non-graded case, we provide criterions for certifying completeness of eventually finite Gröbner bases when they are computed within sufficiently large bounded degrees. We generalize also the concepts of homogenization and saturation, and related algorithms, to the context of difference ideals. The feasibility of the proposed methods is shown by an implementation in Maple that is the first to provide computations for systems of non-linear partial difference equations. We make use of a test set based on the discretization of concrete systems of non-linear partial differential equations.

UNIQUE DECODING OF PLANE AG CODES REVISITED

We reformulate an interpolation-based unique decoding algorithm of AG codes, using the theory of Gr<TEX>$\ddot{o}$</TEX>bner bases of modules on the coordinate ring of the base curve. The conceptual description of the reformulated algorithm lets us better understand the majority voting procedure, which is central in the interpolation-based unique decoding. Moreover the smaller Gr<TEX>$\ddot{o}$</TEX>bner bases imply smaller space and time complexity of the algorithm.

FAST UNIQUE DECODING OF PLANE AG CODES

An interpolation-based unique decoding algorithm of Algebraic Geometry codes was recently introduced. The algorithm iteratively computes the sent message through a majority voting procedure using the Gr<TEX>$\ddot{o}$</TEX>bner bases of interpolation modules. We now combine the main idea of the Guruswami-Sudan list decoding with the algorithm, and thus obtain a hybrid unique decoding algorithm of plane AG codes, significantly improving the decoding speed.

Grobner Bases for Lattices and an Algebraic Decoding Algorithm

In this paper we present Grobner bases for lattices given in a general form, including integer and non-integer lattices. Grdot{o}bner bases for binary linear codes were introduced by Borges-Quintana et al. . We extend their work to non-binary group block codes. Then, given a lattice Λ and its associated label code L, which is a group code, we define an ideal for L. A Grobner basis is assigned to Λ as the Grobner basis of its label code L. Since the associated label code for integer and non-integer lattices are group codes, the assigned Grobner bases can be obtained for both cases. Using this Grobner basis an algebraic decoding algorithm is introduced. We provide an example of the decoding method for a lower dimension lattice. We explain that the complexity of this decoding method depends on the division algorithm and show this decoding method has polynomial time complexity. Experiments for some versions of root lattices (E_7 and E_8) show that for low SNR the performance of these lattices is near to the lower bounds given in .

A Gröbner basis for Kazhdan-Lusztig ideals

{\it Kazhdan-Lusztig ideals}, a family of generalized determinantal ideals investigated in [Woo-Yong~'08], provide an explicit choice of coordinates and equations encoding a neighborhood of a torus-fixed point of a Schubert variety on a type $A$ flag variety. Our main result is a Gr\"{o}bner basis for these ideals. This provides a single geometric setting to transparently explain the naturality of pipe dreams on the {\it Rothe diagram of a permutation}, and their appearance in: \begin{itemize} \item combinatorial formulas [Fomin-Kirillov '94] for Schubert and Grothendieck polynomialsof [Lascoux-Sch\"{u}tzenberger '82]; \item the equivariant $K$-theory specialization formula of [Buch-Rim\'{a}nyi '04]; and \item a positive combinatorial formula for multiplicities of Schubert varieties in good cases, including those for which the associated Kazhdan-Lusztig ideal is homogeneous under the standard grading. \end{itemize} Our results generalize (with alternate proofs) [Knutson-Miller '05]'s Gr\"{o}bner basis theorem for Schubert determinantal ideals and their geometric interpretation of the monomial positivity of Schubert polynomials. We also complement recent work of [Knutson '08 $\&amp;$ '09] on degenerations of Kazhdan-Lusztig varieties in general Lie type, as well as work of [Goldin '01] on equivariant localization and of [Lakshmibai-Weyman '90], [Rosenthal-Zelevinsky '01], and [Krattenthaler '01] on Grassmannian multiplicity formulas.

Generalizing Tanisaki’s ideal via ideals of truncated symmetric functions

We define a family of ideals $I_h$ in the polynomial ring $\mathbb{Z}[x_1,...,x_n]$ that are parametrized by Hessenberg functions $h$ (equivalently Dyck paths or ample partitions). The ideals $I_h$ generalize algebraically a family of ideals called the Tanisaki ideal, which is used in a geometric construction of permutation representations called Springer theory. To define $I_h$, we use polynomials in a proper subset of the variables ${x_1,...,x_n}$ that are symmetric under the corresponding permutation subgroup. We call these polynomials {\em truncated symmetric functions} and show combinatorial identities relating different kinds of truncated symmetric polynomials. We then prove several key properties of $I_h$, including that if $h>h'$ in the natural partial order on Dyck paths then $I_{h} \subset I_{h'}$, and explicitly construct a Gr\"{o}bner basis for $I_h$. We use a second family of ideals $J_h$ for which some of the claims are easier to see, and prove that $I_h = J_h$. The ideals $J_h$ arise in work of Ding, Develin-Martin-Reiner, and Gasharov-Reiner on a family of Schubert varieties called partition varieties. Using earlier work of the first author, the current manuscript proves that the ideals $I_h = J_h$ generalize the Tanisaki ideals both algebraically and geometrically, from Springer varieties to a family of nilpotent Hessenberg varieties.

Gröbner Basis Attacks on Lightweight RFID Authentication Protocols

Since security and privacy problems in RFID systems have attracted much attention, numerous RFID authentication protocols have been suggested. One of the various design approaches is to use light-weight logics such as bitwise Boolean operations and addition modulo <TEX>$2^m$</TEX> between m-bits words. Because these operations can be implemented in a small chip area, that is the major requirement in RFID protocols, a series of protocols have been suggested conforming to this approach. In this paper, we present new attacks on these lightweight RFID authentication protocols by using the Gr<TEX>$\ddot{o}$</TEX>bner basis. Our attacks are superior to previous ones for the following reasons: since we do not use the specific characteristics of target protocols, they are generally applicable to various ones. Furthermore, they are so powerful that we can recover almost all secret information of the protocols. For concrete examples, we show that almost all secret variables of six RFID protocols, LMAP, <TEX>$M^2AP$</TEX>, EMAP, SASI, Lo et al.'s protocol, and Lee et al.'s protocol, can be recovered within a few seconds on a single PC.

Finite generation of symmetric ideals

Let $A$ be a commutative Noetherian ring, and let $R = A[X]$ be the polynomial ring in an infinite collection $X$ of indeterminates over $A$. Let ${\mathfrak S}_{X}$ be the group of permutations of $X$. The group ${\mathfrak S}_{X}$ acts on $R$ in a natural way, and this in turn gives $R$ the structure of a left module over the group ring $R[{\mathfrak S}_{X}]$. We prove that all ideals of $R$ invariant under the action of ${\mathfrak S}_{X}$ are finitely generated as $R[{\mathfrak S}_{X}]$-modules. The proof involves introducing a certain well-quasi-ordering on monomials and developing a theory of Gröbner bases and reduction in this setting. We also consider the concept of an invariant chain of ideals for finite-dimensional polynomial rings and relate it to the finite generation result mentioned above. Finally, a motivating question from chemistry is presented, with the above framework providing a suitable context in which to study it.

General Jacobi Identity Revisited Again

Synthetic differential geometry occupies a unique position in topos-theoretic physics. Nevertheless it has appeared somewhat too conceptual to physicists in general, partly because it has appeared to lack computational aspects. Its computational facets are really concerned with computation of the quasi-colimit of a finite diagram of infinitesimal spaces, or equivalently, with computation of the limit of a finite diagram of Weil algebras. Indeed we have been forced to do a highly involved computation of the above kind by hand in our previous papers (Nishimura, H. in Int. J. Theor. Phys. 36:1099–1131, 1997 and Nishimura, H. in Int. J. Theor. Phys. 38:2163–2174, 1999). The principal objective in this paper is to show that Gro bner bases techniques provide us with means that relegate such computations to computers.

The package CRACK for solving large overdetermined systems

The program CRACK is a computer algebra package written in REDUCE for the solution of over-determined systems of algebraic, ordinary or partial differential equations with at most polynomial non-linearity. It is available as part of version 3.8 of the REDUCE system (dated April 2004) and the latest development version can be downloaded from http://lie.math.brocku.ca/crack/. The purpose of this poster is to accompany a software demonstration of CRACK at ISSAC 2005. The poster is supposed to give a graphical overview of CRACK, emphasizing features which are special to the package. Those are &amp;bull; a rich interface, with visualization aids for inspecting large systems, including - for each equation its properties, history and investigations that have already been done with the equation, - the occurrence of all derivatives of selected functions in any equation, - a statistical overview of the system (number of equations and functions in dependence of number of independent variables), - a matrix display of occurrences of unknown constants/functions in all equations, - a count of the total number of appearances of each unknown, - the determination of not under-determined subsystems, - a listing of all sub- and sub-sub-.. cases investigated so far, with their assumptions, number of steps and number of solutions, - graphical displays of size related measures of the computation done so far; &amp;bull; a discussion of possibilities to trade interactively or automatically the speed of the solution process versus safety (avoidance of expression swell): - only length-reducing versus complete Gr&amp;ouml;bner basis computation steps. - substitutions in shorter equations only, i.e. only in a sub-system versus substitutions in the complete system, - growth bounded substitutions versus general substitutions, - case splittings (induced by factorizations, substitutions with potentially vanishing coefficients or adhoc case distinctions) versus Gr&amp;ouml;bner basis steps, - an investment in the length reduction of equations to reach sparse systems with multiple benefits; &amp;bull; algorithmic extensions which include - the ability to collect and apply syzygies which result as a by-product in the process of computing a differential Gr&amp;ouml;bner basis to integrate linear PDEs. - the treatment of inequalities: their usage, active collection and derivation, and their constant update in an ongoing reduction process based on newly derived equations. - the capability added by Winfried Neun (ZIB Berlin) to run in a truly parallel mode on a beowulf cluster, recently also ported to 64bit AMD processors. - post-computation procedures, especially the possibility to merge solutions of parametric linear algebraic systems and to automate the production of web-pages for solutions that are found. - the ability to separate expressions with respect to independent variables which occur polynomially but with variable exponents, leading to automatically investigated case distinctions of exponents being equal or not; &amp;bull; safety enhancing measures as - the ability to backup and re-load the whole session, - the automatic storing of the complete keyboard input during a session with the opportunity to feed this stored input into a new session, - the possibility to impose time restrictions of notoriously slow sub-steps, like factorizations and sometimes the computation and reduction of S-polynomials in a Gr&amp;ouml;bner basis computations, - a method to interrupt an ongoing automatic computation and change it to interactive mode The poster in landscape format will display the above four topics in boxes. For some of the sub-items above a screen output will give a visual impression, like the matrix indicating occurrences of unknowns in equations. In the following publications the solution of large overdetermined systems was a crucial ingredient: &amp;bull; the solution of large bi-linear algebraic systems and automatic merging of obtained solutions: - Wolf, T., Efimovskaya, O. V.: Classification of integrable quadratic Hamiltonians on e(3), Regular and Chaotic Dynamics, vol 8, no 2 (2003), p 155--162. - Sokolov, V. V., Wolf, T.: Classification of integrable polynomial vector evolution equations, J. Phys. A: Math. Gen. 34 (2001) 11139--11148. - Tsuchida, T. and Wolf, T.: Classification of integrable coupled systems with one scalar and one vector unknown, preprint nlin.SI/0412003 (2004) 60 pages, to appear in J. Phys. A: Math. Gen. - Sokolov, V. V. and Wolf, T.: Integrable quadratic Hamiltonians on &lt;i&gt;so&lt;/i&gt;(4) and &lt;i&gt;so&lt;/i&gt;(3, 1), preprint (2004) 16 pages, nlin.SI/0405066. - Kiselev, A. and Wolf, T.: New recursive chains of N=1 homogeneous superequations, to appear in proceedings of "Symmetry in Nonlinear Mathematical Physics", Kyiv 2005. &amp;bull; the solution of extensive overdetermined ODE/PDE-systems: - Anco, S. and Wolf, T.: Some symmetry classifications of hyperbolic vector evolution equations, nlin.SI/0412015, JNMP, Volume 12, Supplement 1 (2005), p 13--31.

Parallel computation of Janet and Gr�bner bases over rational numbers

In this paper, a parallel algorithm for computation of polynomial Janet bases is considered. After construction of a Janet basis, a reduced Grobner basis (which is a subset of the Janet basis) is extracted from it without any additional reductions. The algorithm discussed is an improved version of an earlier suggested parallel algorithm. The efficiency of a C implementation of the algorithm and its scalability are illustrated by way of the standard test examples that are often used for comparing various algorithms and codes for computing Grobner bases.

Symmetric and Rees algebras of Koszul cycles and their Gr�bner bases

In this paper we study the symmetric algebra S(Ei) and Rees algebra R(Ei) of the modules Ei of i-cycles of the Koszul complex associated with the sequence of indeterminates \({{x_1,\ldots, x_n}}\) of a polynomial ring \({{K[x_1,\ldots, x_n]}}\). For i=2 and i=n−2 we show that \({{x_1,\ldots, x_n}}\) is a d-sequence on S(Ei) and R(Ei) and we determine Grobner bases and Sagbi bases related to these algebras.