Polynomial Identities Research Articles

On a general notion of a polynomial identity and codimensions

Using the braided version of Lawvere's algebraic theories and Mac Lane's PROPs, we introduce polynomial identities for arbitrary algebraic structures in a braided monoidal category C as well as their codimensions in the case when C is linear over some field. The new cases include coalgebras, bialgebras, Hopf algebras, braided vector spaces, Yetter–Drinfel'd modules, etc. We find bases for polynomial identities and calculate codimensions in some important particular cases.

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.

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} .

A Deterministic Method for Computing Bertini Type Invariants of Parametric Ideals

A new framework is proposed for computing general numerical invariants, called Bertini type invariants, associated to a family of polynomial ideals depending on deformation parameters. A deterministic algorithm is introduced for computing parameter dependency of Bertini type invariants. As an application, an algorithm for computing Chern–Schwartz–MacPherson classes of a family of singular projective hypersurfaces is obtained.

On the identities and cocharacters of the algebra of 3 × 3 matrices with orthosymplectic superinvolution

Let M1,2(F) be the algebra of 3×3 matrices with orthosymplectic superinvolution ⁎ over a field F of characteristic zero. We study the ⁎-identities of this algebra through the representation theory of the group Hn=(Z2×Z2)∼Sn. We decompose the space of multilinear ⁎-identities of degree n into the sum of irreducibles under the Hn-action in order to study the irreducible characters appearing in this decomposition with non-zero multiplicity. Moreover, by using the representation theory of the general linear group, we determine all the ⁎-polynomial identities of M1,2(F) up to degree 3.

Identities of Hecke–Kiselman monoids

AbstractIt is shown that the Hecke–Kiselman monoid $${\text {HK}}_{\Theta }$$ HK Θ associated to a finite oriented graph $$\Theta $$ Θ satisfies a semigroup identity if and only if $${\text {HK}}_{\Theta }$$ HK Θ does not have free noncommutative subsemigroups. It follows that this happens exactly when the semigroup algebra $$K[{\text {HK}}_{\Theta }]$$ K [ HK Θ ] over a field K satisfies a polynomial identity. The latter is equivalent to a condition expressed in terms of the graph $$\Theta $$ Θ . The proof allows to derive concrete identities satisfied by such monoids $${\text {HK}}_{\Theta }$$ HK Θ .

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.

SIGACT News Complexity Theory Column 121

This month, Pranjal Dutta and Sumanta Ghosh contributed an in-depth survey on the Polynomial Identity Testing problem, and the efforts made towards obtaining deterministic algorithms for testing identities for various models of algebraic circuits. I wish to thank Pranjal and Sumanta for their thorough work. I hope you enjoy reading their article. On the next issues of SIGACT News, you can expect columns by Dean Doron (probably on probably on binary codes and the GV bound, but stay tuned!), by Stacey Jeffrey (on Quantum Las Vegas Algorithms and Composition), by Igor Carboni Oliveira (on Complexity Theory in Bounded Arithmetic: Recent Results and Current Frontiers) and by Noam Mazor (topic TBA).

Machine learning parameter systems, Noether normalisations and quasi-stable positions

We discuss the use of machine learning models for finding “good coordinates” for polynomial ideals. Our main goal is to put ideals into quasi-stable position, as this generic position shares most properties of the generic initial ideal, but can be deterministically reached and verified. Furthermore, it entails a Noether normalisation and provides us with a system of parameters. Traditional approaches use either random choices which typically destroy all sparsity or rather simple human heuristics which are only moderately successful. Our experiments show that machine learning models provide us here with interesting alternatives that most of the time make nearly optimal choices.

Specht property for the graded identities of the pair (M2(D),sl2(D))

Let D be a Noetherian infinite integral domain, denote by M2(D) and by sl2(D) the 2×2 matrix algebra and the Lie algebra of the traceless matrices in M2(D), respectively. In this paper we study the weak polynomial identities for the natural grading by the cyclic group Z2 of order 2 on M2(D) and on sl2(D). We describe a finite basis of the graded polynomial identities for the pair (M2(D),sl2(D)). Moreover we prove that the ideal of the graded identities for this pair satisfies the Specht property, that is every ideal of graded identities of pairs (associative algebra, Lie algebra), satisfying the graded identities for (M2(D),sl2(D)), is finitely generated. The polynomial identities for M2(D) are known if D is any field of characteristic different from 2. The identities for the Lie algebra sl2(D) are known when D is an infinite field. The identities for the pair we consider were first described by Razmyslov when D is a field of characteristic 0, and afterwards by the second author when D is an infinite field. The graded identities for the pair (M2(D),gl2(D)) were also described, by Krasilnikov and the second author.In order to obtain these results we use certain graded analogues of the generic matrices, and also techniques developed by G. Higman concerning partially well ordered sets.

Certain functional identities on division rings of characteristic two

Let D be a noncommutative division ring. In a recent paper, Lee and Lin proved that if charD≠2, the only solution of additive maps f,g on D satisfying the identity f(x)=xng(x−1) on D∖{0} with n≠2 a positive integer is the trivial case, that is, f=0 and g=0. Applying Hua's identity and the theory of functional and generalized polynomial identities, we give a complete solution of the same identity for any nonnegative integer n if charD=2.

Kummer–Witt–Jackson algebras

This paper is concerned with the construction of a small, but non-trivial, example of a polynomial identity algebra, which we call the Jackson algebra, that will be used in sequels to this paper to study non-commutative arithmetic geometry. In this paper this algebra is studied from a ring-theoretic and geometric viewpoint. Among other things it turns out that this algebra is a “non-commutative family” of central simple algebras and thus parameterizes Brauer classes over extensions of the base.

The primeness of noncommutative polynomials on prime rings

We study the primeness of noncommutative polynomials on prime rings. Let [Formula: see text] be a prime ring with extended centroid [Formula: see text], [Formula: see text] a right ideal of [Formula: see text], [Formula: see text] a noncommutative polynomial over [Formula: see text], which is not a polynomial identity (PI) for [Formula: see text], and [Formula: see text]. Then [Formula: see text] for all [Formula: see text] if and only if one of the following holds: (i) [Formula: see text]; (ii) [Formula: see text] for some idempotent [Formula: see text] and [Formula: see text] such that either [Formula: see text] is a PI for [Formula: see text] or [Formula: see text] is central-valued on [Formula: see text] and [Formula: see text]. We then apply the result to higher commutators of right ideals. Some results of the paper are also studied from the view of point of the notion of [Formula: see text]-primeness of rings.

On the structure of prime and semiprime rings with generalized skew derivations

Let [Formula: see text] be a ring of characteristic different from [Formula: see text], [Formula: see text] fixed positive integers, [Formula: see text] a noncentral Lie ideal of [Formula: see text] and [Formula: see text] a nonzero generalized skew derivation of [Formula: see text]. We prove the following results: (a) If [Formula: see text] is prime and there exists [Formula: see text] such that [Formula: see text] then [Formula: see text], the [Formula: see text] matrix ring over a field [Formula: see text]. (b) If [Formula: see text] is semiprime and [Formula: see text] then either [Formula: see text] centralizes a nonzero ideal of [Formula: see text] or [Formula: see text] is a polynomial identity for [Formula: see text].

Representation theoretic interpretation and interpolation properties of inhomogeneous spin q-Whittaker polynomials

We establish new properties of inhomogeneous spin q-Whittaker polynomials, which are symmetric polynomials generalizing t=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t=0$$\\end{document} Macdonald polynomials. We show that these polynomials are defined in terms of a vertex model, whose weights come not from an R-matrix, as is often the case, but from other intertwining operators of Uq′(sl^2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U'_q({\\widehat{\\mathfrak {sl}}}_2)$$\\end{document}-modules. Using this construction, we are able to prove a Cauchy-type identity for inhomogeneous spin q-Whittaker polynomials in full generality. Moreover, we are able to characterize spin q-Whittaker polynomials in terms of vanishing at certain points, and we find interpolation analogues of q-Whittaker and elementary symmetric polynomials.

Weak polynomial identities of small degree for the Weyl algebra

In this paper we investigate weak polynomial identities for the Weyl algebra $\mathsf{A}_1$ over an infinite field of arbitrary characteristic. Namely, we describe weak polynomial identities of the minimal degree, which is three, and of degrees 4 and 5. We also describe weak polynomial identities is two variables.

SIGACT News Complexity Theory Column 120

Hamed Hatami and Pooya Hatami, our guest columnists in this issue, take us to a fascinating tour through communication complexity classes, and their relations to structural properties of Boolean matrices. Thanks a lot to Hamed and Pooya for their contribution! Coming up on the next issues, we will have columns by Pranjal Dutta and Sumanta Ghosh (on Polynomial identity testing), by Dean Doron (topic TBA), by Stacey Jeffrey (on Quantum Las Vegas Algorithms and Composition) and by Igor Carboni Oliveira (on Complexity Theory in Bounded Arithmetic: Recent Results and Current Frontiers). The future looks promising and exciting.

Quantum Heisenberg enveloping algebras at roots of unity

In this article, the two-parameter quantum Heisenberg enveloping algebra, which serves as a model for certain quantum generalized Heisenberg algebras, has been studied at roots of unity. In this context, the quantum Heisenberg enveloping algebra becomes a polynomial identity algebra, and the dimension of simple modules is bounded by its PI degree. The PI degree, center, and complete classification of simple modules up to isomorphism are explicitly presented. We work over a field of arbitrary characteristic, although our results concerning the representations require that it is algebraically closed.

On Matrix Multiplication and Polynomial Identity Testing

On Matrix Multiplication and Polynomial Identity Testing

Integer-valued polynomials on subsets of upper triangular matrix rings

S. Frisch, showed that the integer-valued polynomials on upper triangular matrix ring Int T n ( K ) ( T n ( D ) ) : = { f ∈ T n ( K ) [ x ] | f ( T n ( D ) ) ⊆ T n ( D ) } is a ring, where D is an integral domain with field of fractions K. Let R 1 ⊆ R 2 be commutative rings with identity. In this paper, we study the set Int T n ( R 2 ) ( Ω , T n ( R 1 ) ) : = { f ∈ T n ( R 2 ) [ x ] | f ( Ω ) ⊆ T n ( R 1 ) } for some subsets Ω ⊆ T n ( R 1 ) . We generalize Frisch’s result and show that Int T n ( R 2 ) ( T n ( R 1 ) ) : = Int T n ( R 2 ) ( T n ( R 1 ) , T n ( R 1 ) ) is a ring. We state a lower bound for the Krull dimension of the integer-valued polynomials on upper triangular matrix rings. Finally, we state the concept of Skolem closure of an ideal of the integer-valued polynomials on upper triangular matrix rings and as a consequence, we obtain a classification of maximal ideals of the integer-valued polynomials on upper triangular matrix rings.