Polynomial Identities Research Articles

Joins, ears and Castelnuovo–Mumford regularity

We introduce a new class of polynomial ideals associated to a simple graph, G. Let K[EG] be the polynomial ring on the edges of G and K[VG] the polynomial ring on the vertices of G. We associate to G an ideal, I(XG), defined as the preimage of (xi2−xj2:i,j∈VG)⊆K[VG] by the map K[EG]→K[VG] which sends a variable, te, associated to an edge e={i,j}, to the product xixj of the variables associated to its vertices. We show that K[EG]/I(XG) is a one-dimensional, Cohen–Macaulay, graded ring and we relate its degree with the number of vertices and connected components of the graph. We show that I(XG) is a binomial ideal and that, with respect to a fixed monomial order, its initial ideal has a generating set independent of the field K. We focus on the Castelnuovo–Mumford regularity of I(XG) providing the following sharp upper and lower bounds:μ(G)≤regI(XG)≤|VG|−b0(G)+1, where μ(G) is the maximum vertex join number of the graph and b0(G) is the number of its connected components. We show that the lower bound is attained for a bipartite graph and use this to derive a new combinatorial result on the number of even length ears of nested ear decomposition.

Graded polynomial identities for the upper triangular matrix algebra over a finite field

Let K be a finite field and let UTn(K) be the algebra of n×n upper triangular matrices over K. In this paper we describe the set of all G-graded polynomial identities of UTn(K), where G is any group. Moreover, we describe a linear basis for the corresponding relatively free graded algebra.

On the factorability of polynomial identities of upper block triangular matrix algebras graded by cyclic groups

Let F be an algebraically closed field of characteristic zero and G be an arbitrary finite cyclic group. In this paper, given an m-tuple (A1,…,Am) of finite dimensional G-simple algebras, we focus on the study of the factorability of the TG-ideals IdG((UT(A1,…,Am),α˜)) of the G-graded upper block triangular matrix algebras UT(A1,…,Am) endowed with elementary G-gradings induced by some maps α˜.When G is a cyclic p-group we prove that the factorability of the ideal IdG((UT(A1,…,Am),α˜) is equivalent to the G-regularity of all (except for at most one) the G-simple components A1,…,Am, as well to the existence of a unique isomorphism class of α˜-admissible elementary G-gradings for UT(A1,…,Am). Moreover, we present some necessary and sufficient conditions to the factorability of IdG((UT(A1,A2),α˜)), even in case G is not a p-group, with some stronger assumptions on the gradings of the algebras A1 and A2.

Computing the multilinear factors of lacunary polynomials without heights

We present a deterministic algorithm which computes the multilinear factors of multivariate lacunary polynomials over number fields. Its complexity is polynomial in ℓn where ℓ is the lacunary size of the input polynomial and n its number of variables, that is in particular polynomial in the logarithm of its degree. We also provide a randomized algorithm for the same problem of complexity polynomial in ℓ and n.Over other fields of characteristic zero and finite fields of large characteristic, our algorithms compute the multilinear factors having at least three monomials of multivariate polynomials. Lower bounds are provided to explain the limitations of our algorithm. As a by-product, we also design polynomial-time deterministic polynomial identity tests for families of polynomials which were not known to admit any.Our results are based on so-called Gap Theorem which reduce high-degree factorization to repeated low-degree factorizations. While previous algorithms used Gap Theorems expressed in terms of the heights of the coefficients, our Gap Theorems only depend on the exponents of the polynomials. This makes our algorithms more elementary and general, and faster in most cases.

Emptiness problems for integer circuits

We study the computational complexity of emptiness problems for circuits over sets of natural numbers with the operations union, intersection, complement, addition, and multiplication. For most settings of allowed operations we precisely characterize the complexity in terms of completeness for classes like NL, NP, and PSPACE. The case where intersection, addition, and multiplication is allowed turns out to be equivalent to the complement of polynomial identity testing (PIT).Our results imply the following improvements and insights on problems studied in earlier papers. We improve the bounds for the membership problem MC(∪,∩,x‾,+,×) studied by McKenzie and Wagner 2007 and for the equivalence problem EQ(∪,∩,x‾,+,×) studied by Glaßer et al. 2010. Moreover, it turns out that the following problems are equivalent to PIT, which shows that the challenge to improve their bounds is just a reformulation of a well-studied, major open problem in algebraic computing complexity:•membership problem MC(∩,+,×) studied by McKenzie and Wagner 2007•integer membership problems MCZ(+,×), MCZ(∩,+,×) studied by Travers 2006•equivalence problem EQ(+,×) studied by Glaßer et al. 2010.

On Serre Reduction of Multidimensional Systems

Serre reduction of a system plays a key role in the theory of Multidimensional systems, which has a close connection with Serre reduction of polynomial matrices. In this paper, we investigate the Serre reduction problem for two kinds of nD polynomial matrices. Some new necessary and sufficient conditions about reducing these matrices to their Smith normal forms are obtained. These conditions can be easily checked by existing Gröbner basis algorithms of polynomial ideals.

Integrability approach to Fehér-Némethi-Rimányi-Guo-Sun type identities for factorial Grothendieck polynomials

Recently, Guo and Sun derived an identity for factorial Grothendieck polynomials which is a generalization of the one for Schur polynomials by Fehér, Némethi and Rimányi. We analyze the identity from the point of view of quantum integrability, based on the correspondence between the wavefunctions of a five-vertex model and the Grothendieck polynomials. We give another proof using the quantum inverse scattering method. We also apply the same idea and technique to derive an identity for factorial Grothendieck polynomials for rectangular Young diagrams. Combining with the Guo-Sun identity, we get a duality formula. We also discuss a q-deformation of the Guo-Sun identity.

A polynomial identity implying Schur’s partition theorem

We propose and prove a new polynomial identity that implies Schur's partition theorem. We give combinatorial interpretations of some of our expressions in the spirit of Kursungoz. We also present some related polynomial and $q$-series identities.

Weighted enumeration of Bruhat chains in the symmetric group

We use the recently introduced padded Schubert polynomials to prove a common generalization of the fact that the weighted number of maximal chains in the strong Bruhat order on the symmetric group is ( n 2 ) ! {n \choose 2}! for both the code weights and the Chevalley weights, generalizing a result of Stembridge. We also define weights which give a one-parameter family of strong order analogues of Macdonald’s well-known reduced word identity for Schubert polynomials.

Corrigendum and addendum to “The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang–Baxter equation”

One of the main results stated in Theorem 4.4 of our article, which appears in Trans. Amer. Math. Soc. 372 (2019), no. 10, 7191–7223, is that the structure algebra K [ M ( X , r ) ] K[M(X,r)] , over a field K K , of a finite bijective left non-degenerate solution ( X , r ) (X,r) of the Yang–Baxter equation is a module-finite central extension of a commutative affine subalgebra. This is proven by showing that the structure monoid M ( X , r ) M(X,r) is central-by-finite. This however is not true, even in case ( X , r ) (X,r) is a (left and right) non-degenerate involutive solution. The proof contains a subtle mistake. However, it turns out that the monoid M ( X , r ) M(X,r) is abelian-by-finite and thus the conclusion that K [ M ( X , r ) ] K[M(X,r)] is a module-finite normal extension of a commutative affine subalgebra remains valid. In particular, K [ M ( X , r ) ] K[M(X,r)] is Noetherian and satisfies a polynomial identity. The aim of this paper is to give a proof of this result. In doing so, we also strengthen Lemma 5.3 (and its consequences, namely Lemma 5.4 and Proposition 5.5) showing that these results on the prime spectrum of the structure monoid hold even if the assumption that the solution ( X , r ) (X,r) is square-free is omitted.

Stably Noetherian Algebras of Polynomial Growth

Let A be a right noetherian algebra over a field k. If the base field extension A ⊗kK remains right noetherian for all extension fields K of k, then A is called stably right noetherian over k. We develop an inductive method to show that certain algebras of finite Gelfand-Kirillov dimension are stably noetherian, using critical composition series. We use this to characterize which algebras satisfying a polynomial identity are stably noetherian. The method also applies to many $\mathbb {N}$ -graded rings of finite global dimension; in particular, we see that a noetherian Artin-Schelter regular algebra must be stably noetherian. In addition, we study more general variations of the stably noetherian property where the field extensions are restricted to those of a certain type, for instance purely transcendental extensions.

Bifurcations of Periodic Solutions of a Hamiltonian System with a Discrete Symmetry Group

An autonomous Hamiltonian system with two degrees of freedom that is invariant under the Klein four-group of linear canonical automorphisms of the extended phase space of the system is studied. A sequence of symplectic transformations of the monodromy matrix of the symmetric periodic solution of this system is constructed. Using these transformations, the structure and bifurcation of the phase flow in the vicinity of this solution is investigated. It is shown that the bifurcations corresponding to the multiple period increase are different for the solutions with double symmetry and the solutions with a single symmetry. An example of critical periodic solutions of the family of doubly symmetric orbits of the plane circular Hill problem is discussed. The majority of tedious analytical calculations are performed using packages for the computation of Grobner bases and for the work with polynomial ideals in the computer algebra system Maple.

A Signature Based Border Basis Algorithm

The Border Basis Algorithm (BBA) still suffers from the lack of analogues of Buchberger’s criteria for avoiding unnecessary reductions. In this paper we develop a signature based technique which provides a first remedial step: signature bounds allow us to recognize multiple reductions of the same ancestor polynomial. The new signature based algorithm is then combined with the Boolean BBA for ideals of Boolean polynomials. Experiments show that it is at least 5 times faster than the standard Boolean BBA.

Classifying Algebras with Graded Involutions or Superinvolutions with Multiplicities of their Cocharacter Bounded by One

Let A be superalgebra over a field of characteristic zero and let ∗ be either a graded involution or a superinvolution defined on A. In this paper we characterize the ∗-algebras whose ∗-cocharacter has multiplicities bounded by one, showing a set of ∗-polynomial identities satisfied by such algebras.

Note on power sums of the zeros of certain Laguerre and Bessel polynomials

ABSTRACT Let be the sum of th powers of the zeros of the generalized Bessel polynomial . In this note, we generalize the well-known Burchnall's result , . In particular, we find the simple explicit formulae for () with k=2,3,4 and establish the new identity for certain Bessel polynomials.

Injective Polynomial Ideals and The Domination Property

After sketching the basic theory of injective ideals of homogeneous polynomials, we characterize injective polynomial ideals by means of a domination property and applications of this characterization to some classical operator ideals and to composition polynomial ideals are provided.

On subdiagonal rational Padé approximations and the Brenner-Thomée approximation theorem for operator semigroups

The computational powers of Mathematica are used to prove polynomial identities that are essential to obtain growth estimates for subdiagonal rational Pade approximations of the exponential function and to obtain new estimates of the constants of the Brenner-Thomee Approximation Theorem of Semigroup Theory.

Involution on prime rings with endomorphisms

Let $\mathcal{R}$ be a prime ring with involution $'*'$ and $\psi: \mathcal{R} \rightarrow \mathcal{R}$ be an endomorphism on $\mathcal{R}$. In this article, we study the action of involution $'*', $ and the effect of endomorphism $\psi$ satisfying $[\psi(x), \psi(x^*)]-[x, x^*]\in \mathcal{Z}(\mathcal{R})$ for all $x\in \mathcal{R}$. In particular, we prove that any centralizing involution on prime rings with involution of characteristic different from two is of the first kind or $\mathcal{R}$ satisfies $s_4$, the standard polynomial identity in four variables. Further, we establish that if a prime ring $\mathcal{R}$ with involution of characteristic different from two admits a non-trivial endomorphism $\psi$ such that $[\psi(x), \psi(x^*)]-[x, x^*]\in \mathcal{Z}(\mathcal{R})$ for all $x\in \mathcal{R}$, then the involution is of the first kind or $\mathcal{R}$ satisfies $s_4$ and $[\psi(x), x] = 0$ for all $x\in \mathcal{R}$.

Perturbed rank 2 Poisson systems and periodic orbits on Casimir invariant manifolds

A class of n-dimensional Poisson systems reducible to an unperturbed harmonic oscillator shall be considered. In such case, perturbations leaving invariant a given symplectic leaf shall be investigated. Our purpose will be to analyze the bifurcation phenomena of periodic orbits as a result of these perturbations in the period annulus associated to the unperturbed harmonic oscillator. This is accomplished via the averaging theory up to an arbitrary order in the perturbation parameter ε. In that theory we shall also use both branching theory and singularity theory of smooth maps to analyze the bifurcation phenomena at points where the implicit function theorem is not applicable. When the perturbation is given by a polynomial family, the associated Melnikov functions are polynomial and tools of computational algebra based on Gröbner basis are employed in order to reduce the generators of some polynomial ideals needed to analyze the bifurcation problem. When the most general perturbation of the harmonic oscillator by a quadratic perturbation field is considered, the complete bifurcation diagram (except at a high codimension subset) in the parameter space is obtained. Examples are given.

Identities on algebras and combinatorial properties of binary words

We consider polynomial identities and codimension growth of nonassociative algebras over a field of characte-ristics zero. We offer new approach which allows to construct nonassociative algebras starting from a given infinite binary word. The sequence of codimensions of such an algebra is closeely connected with combinatorial complexity of the defining word. These constructions give new examples of algebras with abnormal codimension growth. The first important achievement is that our algebras are finitely generated. The second one is that asymptotic behavior of codimension sequences is quite different unlike all previous examples.