Polynomial Ideal Theory Research Articles

Gradings and graded linear maps on algebras

Abstract Let A be a superalgebra over a field F of characteristic zero. We prove tight relations between graded automorphisms, pseudoautomorphisms, superautomorphisms and K-gradings on A, where K is the Klein group. Moreover, we investigate the consequences of such connections within the theory of polynomial identities. In the second part we focus on the superalgebra U ⁢ T n ⁢ ( F ) {UT_{n}(F)} of n × n {n\times n} upper triangular matrices by completely classifying the graded-pseudo-super automorphism that one can define on it. Finally, we compute the ideals of identities of U ⁢ T n ⁢ ( F ) {UT_{n}(F)} endowed with a graded or a pseudo automorphism, for any n, and the ideals of identities with superautomorphism in the cases n = 2 {n=2} and n = 3 {n=3} .

On minimal varieties of superalgebras with superautomorphism

Let A be a finite-dimensional superalgebra with superautomorphism φ over a field of characteristic zero. In [11] the authors gave a positive answer to the Amitsur's conjecture in this setting showing that the φ-exponent of A exists and it is an integer. In the present paper we extend the notion of minimal variety to the context of superalgebras with superautomorphism and prove that a variety is minimal of fixed φ-exponent if, and only if, it is generated by a subalgebra of an upper block triangular matrix algebra equipped with a suitable elementary Z2-grading and superautomorphism. Along the way, we give a contribution on the isomorphism question within the theory of polynomial identities.

Types of fuzzy graph colouring and polynomial ideal theory

The graph colouring problem (GCP) is one of the most important optimisation problems in graph theory. In real life scenarios, many applications of graph colouring are fuzzy in nature. Fuzzy set and fuzzy graph can manage the uncertainty, associated with the information of a problem, where conventional mathematical models/graph may fail to reveal satisfactory result. To include those fuzzy properties in solving those types of problems, we have extended the various types of classical graph colouring methods to fuzzy graph colouring methods. In this study, we describe three basic types of fuzzy graph colouring methods namely, fuzzy vertex colouring, fuzzy edge colouring and fuzzy total colouring. We introduce a method to colour the vertices of the fuzzy graph using the polynomial ideal theory and find the fuzzy vertex chromatic number of the fuzzy graph. A practical example of scheduling committees meeting is given to demonstrate our proposed algorithm.

On H-simple not necessarily associative algebras

An algebra [Formula: see text] with a generalized [Formula: see text]-action is a generalization of an [Formula: see text]-module algebra where [Formula: see text] is just an associative algebra with [Formula: see text] and a relaxed compatibility condition between the multiplication in [Formula: see text] and the [Formula: see text]-action on [Formula: see text] holds. At first glance, this notion may appear too general, however, it enables to work with algebras endowed with various kinds of additional structures (e.g. comodule algebras over Hopf algebras, graded algebras, algebras with an action of a semigroup by anti-endomorphisms). This approach proves to be especially fruitful in the theory of polynomial identities. We show that if [Formula: see text] is a finite dimensional (not necessarily associative) algebra over a field of characteristic [Formula: see text] and [Formula: see text] is simple with respect to a generalized [Formula: see text]-action, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of polynomial [Formula: see text]-identities of [Formula: see text]. In particular, if [Formula: see text] is a finite dimensional (not necessarily group graded) graded-simple algebra, then there exists [Formula: see text] where [Formula: see text] is the sequence of codimensions of graded polynomial identities of [Formula: see text]. In addition, we study the free-forgetful adjunctions corresponding to (not necessarily group) gradings and generalized [Formula: see text]-actions.

Dimension and depth dependent upper bounds in polynomial ideal theory

We improve certain upper bounds for the degree of Gröbner bases and the Castelnuovo-Mumford regularity of polynomial ideals. For the degree of Gröbner bases, we exclusively work in deterministically verifiable and achievable generic positions of a combinatorial nature, namely either strongly stable position or quasi stable position. Furthermore, we exhibit new dimension and depth depending upper bounds for the Castelnuovo-Mumford regularity and the degrees of the elements of the reduced Gröbner basis (w.r.t. the degree reverse lexicographical ordering) of a homogeneous ideal in these positions. Finally, it is shown that similar upper bounds hold in positive characteristic.

Types of fuzzy graph coloring and polynomial ideal theory

Types of fuzzy graph coloring and polynomial ideal theory

Rationality of Hilbert series in noncommutative invariant theory

It is a fundamental result in commutative algebra and invariant theory that a finitely generated graded module over a commutative finitely generated graded algebra has a rational Hilbert series, and consequently the Hilbert series of the algebra of polynomial invariants of a group of linear transformations is rational, whenever this algebra is finitely generated. This basic principle is applied here to prove rationality of Hilbert series of algebras of invariants that are neither commutative nor finitely generated. Our main focus is on linear groups acting on certain factor algebras of the tensor algebra that arise naturally in the theory of polynomial identities.

Gröbner Bases Techniques for an $S$-Packing $k$-Coloring of a Graph

In this paper, polynomial ideal theory is used to deal with the problem of the $S$-packing coloring of a finite undirected and unweighted graph by introducing a family of polynomials encoding the problem. A method to find the $S$-packing colorings of the graph is presented and illustrated by examples.

A Deductive Approach towards Reasoning about Algebraic Transition Systems

Algebraic transition systems are extended from labeled transition systems by allowing transitions labeled by algebraic equations for modeling more complex systems in detail. We present a deductive approach for specifying and verifying algebraic transition systems. We modify the standard dynamic logic by introducing algebraic equations into modalities. Algebraic transition systems are embedded in modalities of logic formulas which specify properties of algebraic transition systems. The semantics of modalities and formulas is defined with solutions of algebraic equations. A proof system for this logic is constructed to verify properties of algebraic transition systems. The proof system combines with inference rules decision procedures on the theory of polynomial ideals to reduce a proof-search problem to an algebraic computation problem. The proof system proves to be sound but inherently incomplete. Finally, a typical example illustrates that reasoning about algebraic transition systems with our approach is feasible.

Discovering non-terminating inputs for multi-path polynomial programs

This paper investigates the termination problems of multi-path polynomial programs (MPPs) with equational loop guards. To establish sufficient conditions for termination and nontermination simultaneously, the authors propose the notion of strong/weak non-termination which under/over-approximates non-termination. Based on polynomial ideal theory, the authors show that the set of all strong non-terminating inputs (SNTI) and weak non-terminating inputs (WNTI) both correspond to the real varieties of certain polynomial ideals. Furthermore, the authors prove that the variety of SNTI is computable, and under some sufficient conditions the variety of WNTI is also computable. Then by checking the computed SNTI and WNTI varieties in parallel, termination properties of a considered MPP can be asserted. As a consequence, the authors establish a new framework for termination analysis of MPPs.

Using Elimination Theory to Construct Rigid Matrices

The rigidity of a matrix A for target rank r is the minimum number of entries of A that must be changed to ensure that the rank of the altered matrix is at most r. Since its introduction by Valiant (1977), rigidity and similar rank-robustness functions of matrices have found numerous applications in circuit complexity, communication complexity, and learning complexity. Almost all nxn matrices over an infinite field have a rigidity of (n-r)^2. It is a long-standing open question to construct infinite families of explicit matrices even with superlinear rigidity when r = Omega(n). In this paper, we construct an infinite family of complex matrices with the largest possible, i.e., (n-r)^2, rigidity. The entries of an n x n matrix in this family are distinct primitive roots of unity of orders roughly exp(n^2 log n). To the best of our knowledge, this is the first family of concrete (but not entirely explicit) matrices having maximal rigidity and a succinct algebraic description. Our construction is based on elimination theory of polynomial ideals. In particular, we use results on the existence of polynomials in elimination ideals with effective degree upper bounds (effective Nullstellensatz). Using elementary algebraic geometry, we prove that the dimension of the affine variety of matrices of rigidity at most k is exactly n^2-(n-r)^2+k. Finally, we use elimination theory to examine whether the rigidity function is semi-continuous.

Cohen and multiple Cohen strongly summing multilinear operators

Considering the successful theory of multiple summing multilinear operators as a prototype, we introduce the classes of multiple Cohen strongly p-summing multilinear operators and polynomials. The adequacy of these classes under the viewpoint of the theory of multilinear and polynomial ideals and holomorphy types is discussed in detail.

A local-global principle for linear dependence of noncommutative polynomials

A set of polynomials in noncommuting variables is called locally linearly dependent if their evaluations at tuples of matrices are always linearly dependent. By a theorem of Camino, Helton, Skelton and Ye, a finite locally linearly dependent set of polynomials is linearly dependent. In this short note an alternative proof based on the theory of polynomial identities is given. The method of the proof yields generalizations to directional local linear dependence and evaluations in general algebras over fields of arbitrary characteristic. A main feature of the proof is that it makes it possible to deduce bounds on the size of the matrices where the (directional) local linear dependence needs to be tested in order to establish linear dependence.

Geometry of integral polynomials, M-ideals and unique norm preserving extensions

We use the Aron–Berner extension to prove that the set of extreme points of the unit ball of the space of integral k-homogeneous polynomials over a real Banach space X is { ± ϕ k : ϕ ∈ X ⁎ , ‖ ϕ ‖ = 1 } . With this description we show that, for real Banach spaces X and Y, if X is a nontrivial M-ideal in Y, then ⊗ ˆ ε k , s k , s X (the k-th symmetric tensor product of X endowed with the injective symmetric tensor norm) is never an M-ideal in ⊗ ˆ ε k , s k , s Y . This result marks up a difference with the behavior of nonsymmetric tensors since, when X is an M-ideal in Y, it is known that ⊗ ˆ ε k k X (the k-th tensor product of X endowed with the injective tensor norm) is an M-ideal in ⊗ ˆ ε k k Y . Nevertheless, if X is also Asplund, we prove that every integral k-homogeneous polynomial in X has a unique extension to Y that preserves the integral norm. Other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given.

Recognizing Graph Theoretic Properties with Polynomial Ideals

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect $k$-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.

Mechanism and model of the oscillatory electro-oxidation of methanol

A mechanism for the kinetic instabilities observed in the galvanostatic electro-oxidation of methanol is suggested and a model developed. The model is investigated using stoichiometric network analysis as well as concepts from algebraic geometry (polynomial rings and ideal theory) revealing the occurrence of a Hopf and a saddle-node bifurcation. These analytical solutions are confirmed by numerical integration of the system of differential equations.

On a combinatorial theorem of Macaulay and its applications to the theory of polynomial ideals

On a combinatorial theorem of Macaulay and its applications to the theory of polynomial ideals

Light-cone Yang-Mills mechanics: SU(2) vs. SU(3)

We investigate the light-cone SU(n) Yang-Mills mechanics formulated as the leading order of the long-wavelength approximation to the light-front SU(n) Yang-Mills theory. In the framework of the Dirac formalism for degenerate Hamiltonian systems, for models with the structure groups SU(2) and SU(3), we determine the complete set of constraints and classify them. We show that the light-cone mechanics has an extended invariance: in addition to the local SU(n) gauge rotations, there is a new local two-parameter Abelian transformation, not related to the isotopic group, that leaves the Lagrangian system unchanged. This extended invariance has one profound consequence. It turns out that the light-cone SU(2) Yang-Mills mechanics, in contrast to the well-known instant-time SU(2) Yang-Mills mechanics, represents a classically integrable system. For calculations, we use the technique of Grobner bases in the theory of polynomial ideals.

Parallel algorithms for Gröbner-basis construction

The problem of the Grobner-basis construction is important both from the theoretical and applied points of view. As examples of applications of Grobner bases, one can mention the consistency problem for systems of nonlinear algebraic equations and the determination of the number of solutions to a system of nonlinear algebraic equations. The Grobner bases are actively used in the constructive theory of polynomial ideals and at the preliminary stage of numerical solution of systems of nonlinear algebraic equations. Unfortunately, many real examples cannot be processed due to the high computational complexity of known algorithms for computing the Grobner bases. However, the efficiency of the standard basis construction can be significantly increased in practice. In this paper, we analyze the known algorithms for constructing the standard bases and consider some methods for increasing their efficiency. We describe a technique for estimating the efficiency of paralleling the algorithms and present some estimates.

Contributions to constructive polynomial ideal theory XXIII

In a 1941 paper (which is condensed here), N. M. Gjunter refers to some of his unknown papers from 1910, 1913 and 1925, which are partially written in context with the 1941 paper. Thus Gjunter had already pursued a constructive theory of polynomial ideals in 1913. Among other results, he proves the inequality attributed to Macaulay/Sperner in 1913 (from 1927 and 1930, respectively). Also discussed are results analogous to more recent work. The author hereby finishes his sequence of articles.