Polynomial Identities Research Articles

Identities for subspaces of a parametric Weyl algebra

In 2013 Benkart, Lopes and Ondrus introduced and studied in a series of papers the infinite-dimensional unital associative algebra Ah generated by elements x,y, which satisfy the relation yx−xy=h for some 0≠h∈F[x]. In this paper we investigate the standard polynomial identities and minimal identities for certain subspaces of Ah over an infinite field of arbitrary characteristic.

Quasi-Stirling polynomials on multisets

A permutation π of a multiset is said to be a quasi-Stirling permutation if there do not exist four indices i<j<k<ℓ such that πi=πk, πj=πℓ and πi≠πj. For a multiset M, denote by Q‾M the set of quasi-Stirling permutations of M. The quasi-Stirling polynomial on the multiset M is defined by Q‾M(t)=∑π∈Q‾Mtdes(π), where des(π) denotes the number of descents of π. By employing generating function arguments, Elizalde derived an elegant identity involving quasi-Stirling polynomials on the multiset {12,22,…,n2}, in analogy to the identity on Stirling polynomials. In this paper, we derive an identity involving quasi-Stirling polynomials Q‾M(t) for any multiset M, which is a generalization of the identity on Eulerian polynomial and Elizalde's identity on quasi-Stirling polynomials on the multiset {12,22,…,n2}. We provide a combinatorial proof the identity in terms of certain ordered labeled trees. Specializing M={12,22,…,n2} implies a combinatorial proof of Elizalde's identity in answer to the problem posed by Elizalde. As an application, our identity enables us to show that the quasi-Stirling polynomial Q‾M(t) has only real roots and the coefficients of Q‾M(t) are unimodal and log-concave for any multiset M, in analogy to Brenti's result for Stirling polynomials on multisets.

Certain functional identities involving a pair of generalized skew derivations with nilpotent values on Lie ideals

ABSTRACT Let R be a prime ring of characteristic different from 2, Qr its right Martindale quotient ring, L a non-central Lie ideal of R (i.e., ), a fixed integer, F and G two generalized skew derivations of R, associated with the same automorphism α of R. If , for any , then there exists such that , for any , unless when R satisfies the standard polynomial identity .

A new family of two-variable polynomials based on hermite polynomials

Abstract The aim of this paper is to introduce a new two-variable polynomials defined via Hermite polynomials. In order to construct some fundamental properties of these polynomials, we first derive a generating function relation. By using definition and this generating relation, we arrive at several recurrence relations, an integral representation, some implicit summation formulae, a symmetry identity for these new two-variable polynomials. Furthermore, we obtain some results which give various classes of multilinear and multilateral generating functions. Then, some special cases are presented. Finally, we also give a general class of these new polynomials and prove explicit closed-form formulae of them.

Ideal Membership Problem over 3-Element CSPs with Dual Discriminator Polymorphism

In this paper we examine polynomial ideals that are the vanishing ideals of solution sets of combinatorial problems encoded by constraint satisfaction problems over a finite language. We consider a 3-element domain and the dual discriminator polymorphism (constraints under this polymorphism are a generalization of the 2-satisfiability problem). Assuming the graded lexicographic ordering of monomials, we show that the reduced Gröbner basis of ideals whose varieties are closed under this polymorphism can be computed in polynomial time. This proves polynomial time solvability of the ideal membership problem (IMP) with restrictions on degree $d=O(1)$, which we call IMP$_d$, for these constrained problems. It is a first step toward the challenging long-term goal of identifying when IMP$_d$ is polynomial time solvable for a finite domain.

An improved finiteness test and a systematic procedure to compute the strong [formula omitted] norm of differential algebraic systems with multiple delays

We study the strong ℋ2 norm of systems modeled by semi-explicit Delay Differential Algebraic Equations (DDAEs). We recall that the finiteness of the strong ℋ2 norm is linked to an algebraic decision problem that can be solved by checking a finite numbers of equalities. We first improve the verification of the finiteness condition. In particular, the complexity of our new condition removes a dependency on the number of delays. We also show that, without imposing further conditions on the system, the number of checks cannot be further reduced. The methodology relies on interpreting the verification of the finiteness conditions in terms of a Polynomial Identity Testing problem. Second we show, in a constructive way, that if the strong ℋ2 norm is finite, the system can always be transformed into a regular neutral-type system with the same ℋ2 norm, without derivatives in the input or in the output equations. This result closes a gap in the literature as such a transformation was known to exist only under additional assumptions on the system. The transformation enables the computation of the strong ℋ2 norm using delay Lyapunov matrices. Illustrative examples are provided throughout the paper.

Finite groups admitting a coprime automorphism satisfying an additional polynomial identity

Abstract It is known that a finite group with an automorphism 𝜑 of coprime order has a soluble radical of ( | φ | , | C G ⁢ ( φ ) | ) (\lvert\varphi\rvert,\lvert C_{G}(\varphi)\rvert) -bounded Fitting height and index. We extend this classical result as follows. Let f ⁢ ( x ) = a 0 + a 1 ⋅ x + ⋯ + a d ⋅ x d ∈ Z ⁢ [ x ] f(x)=a_{0}+a_{1}\cdot x+\cdots+a_{d}\cdot x^{d}\in\mathbb{Z}[x] be a primitive polynomial, and let 𝐺 be a finite group with an automorphism 𝜑 of coprime order satisfying g a 0 ⋅ φ ⁢ ( g ) a 1 ⁢ ⋯ ⁢ φ d ⁢ ( g ) a d = 1 g^{a_{0}}\cdot\varphi(g)^{a_{1}}\cdots\varphi^{d}(g)^{a_{d}}=1 for all g ∈ G g\in G . Then the soluble radical of 𝐺 has ( d , | C G ⁢ ( φ ) | ) (d,\lvert C_{G}(\varphi)\rvert) -bounded Fitting height and index. The bounds are made explicit and are particularly good for small values of the degree 𝑑.

Fitting height of finite groups admitting a fixed-point-free automorphism satisfying an additional polynomial identity

Let f(x) be a non-zero polynomial with integer coefficients. An automorphism φ of a group G is said to satisfy the elementary abelian identity f(x) if the linear transformation induced by φ on every characteristic elementary abelian section of G is annihilated by f(x). We prove that if a finite (soluble) group G admits a fixed-point-free automorphism φ satisfying an elementary abelian identity f(x), where f(x) is a primitive polynomial, then the Fitting height of G is bounded in terms of deg⁡(f(x)). We also prove that if f(x) is any non-zero polynomial and G is a σ′-group for a finite set of primes σ=σ(f(x)) depending only on f(x), then the Fitting height of G is bounded in terms of the number irr(f(x)) of different irreducible factors in the decomposition of f(x). These bounds for the Fitting height are stronger than the well-known bounds in terms of the composition length α(|φ|) of 〈φ〉 when deg⁡f(x) or irr(f(x)) is small in comparison with α(|φ|).

Application of LaSalle’s Invariance Principle on Polynomial Differential Equations Using Quantifier Elimination

LaSalle&#x2019;s invariance principle is a commonly used extension of Lyapunov&#x2019;s second method to study asymptotic stability of nonlinear systems. If the system can be written in polynomial form, the examination can be automated using algebraic geometry and quantifier elimination. This article addresses this automated examination using a method relying on polynomial ideals and applies it on some example systems. In addition, some properties of these special ideals are derived that allow to reduce the computational effort significantly.

A Polynomial Degree Bound on Equations for Non-rigid Matrices and Small Linear Circuits

We show that there is an equation of degree at most poly( n ) for the (Zariski closure of the) set of the non-rigid matrices: That is, we show that for every large enough field 𝔽, there is a non-zero n 2 -variate polynomial P ε 𝔽[ x 1, 1 , ..., x n, n ] of degree at most poly( n ) such that every matrix M that can be written as a sum of a matrix of rank at most n /100 and a matrix of sparsity at most n 2 /100 satisfies P(M) = 0. This confirms a conjecture of Gesmundo, Hauenstein, Ikenmeyer, and Landsberg [ 9 ] and improves the best upper bound known for this problem down from exp ( n 2 ) [ 9 , 12 ] to poly( n ). We also show a similar polynomial degree bound for the (Zariski closure of the) set of all matrices M such that the linear transformation represented by M can be computed by an algebraic circuit with at most n 2 /200 edges (without any restriction on the depth). As far as we are aware, no such bound was known prior to this work when the depth of the circuits is unbounded. Our methods are elementary and short and rely on a polynomial map of Shpilka and Volkovich [ 21 ] to construct low-degree “universal” maps for non-rigid matrices and small linear circuits. Combining this construction with a simple dimension counting argument to show that any such polynomial map has a low-degree annihilating polynomial completes the proof. As a corollary, we show that any derandomization of the polynomial identity testing problem will imply new circuit lower bounds. A similar (but incomparable) theorem was proved by Kabanets and Impagliazzo [ 11 ].

𝐹-thresholds and test ideals of Thom-Sebastiani type polynomials

We provide a formula for F F -thresholds of a Thom-Sebastiani type polynomial over a perfect field of prime characteristic. We also compute the first test ideal of Thom-Sebastiani type polynomials. Finally, we apply our results to find hypersurfaces where the log canonical thresholds equal the F F -pure thresholds for infinitely many prime numbers.

KdV-type Wronskian rational solutions to the (4+1)-dimensional Fokas equation

The purpose of this paper is to discuss the (4+1)-dimensional Fokas equation, which has been investigated as an integrable generalization of the KP equation and the DS equation. A general Wronskian structure is set up and the involved functions for Wronskian entries satisfy a system of combined linear partial differential equations, based on the Wronskian conditions of the KdV equation. The resulting Wronskian formulation gives a comprehensive way to build rational solutions for the (4+1)-dimensional Fokas equation. Moreover, the linear superposition principle for two Wronskian rational solutions is also obtained by employing a polynomial identity. The presented results show that the exact solutions of the (4+1)-dimensional Fokas equation can be reduced to the exact solutions of the KdV equation.

Germs of operator and polynomial ideals with applications to composition operators

Germs of operator and polynomial ideals with applications to composition operators

Graded linear maps on superalgebras

Let A be an associative algebra over a fixed field F of characteristic zero. In this paper we focus our attention on those algebras A graded by Z2, the cyclic group of order 2. In this case A is said to be a superalgebra and it can be decomposed in the direct sum of homogeneous subspaces: A=A0⊕A1. The main goal of this paper is to prove tight relations between some graded linear maps that can be defined on superalgebras, namely involutions, superinvolutions and pseudoinvolutions. Along the way, we shall present a classification of the pseudoinvolutions that one can define on the algebra UTn(F) of n×n upper-triangular matrices. In the final part of the paper we shall also give some consequences of these results in the context of the theory of polynomial identities.

Varieties of algebras with pseudoinvolution: Codimensions, cocharacters and colengths

Let A be a finitely generated superalgebra with pseudoinvolution ⁎ over an algebraically closed field F of characteristic zero. In this paper we develop a theory of polynomial identities for this kind of algebras . In particular, we shall consider three sequences that can be attached to Id ⁎ ( A ) , the T 2 ⁎ -ideal of identities of A : the sequence of ⁎-codimensions c n ⁎ ( A ) , the sequence of ⁎-cocharacter χ 〈 n 〉 ⁎ ( A ) and the ⁎-colength sequence l n ⁎ ( A ) . Our purpose is threefold. First we shall prove that the ⁎-codimension sequence is eventually non-decreasing, i.e., c n ⁎ ( A ) ≤ c n + 1 ⁎ ( A ) , for n large enough. Secondly, we study superalgebras with pseudoinvolution having the multiplicities of their ⁎-cocharacter bounded by a constant. Among them, we characterize the ones with multiplicities bounded by 1. Finally, we classify superalgebras with pseudoinvolution A such that l n ⁎ ( A ) is bounded by 3. In the last section we relate the ⁎-colengths with the polynomial growth of the ⁎-codimensions: we show that l n ⁎ ( A ) is bounded by a constant if and only if c n ⁎ ( A ) grows at most polynomially.

Ideals of spaces of degenerate matrices

The variety Singn,m consists of all tuples X=(X1,…,Xm) of n×n matrices such that every linear combination of X1,…,Xm is singular. Equivalently, X∈Singn,m if and only if det(λ1X1+…+λmXm)=0 for all λ1,…,λm∈Q. Makam and Wigderson [12] asked whether the ideal generated by these equations is always radical, that is, if any polynomial identity that is valid on Singn,m lies in the ideal generated by the polynomials det(λ1X1+…+λmXm). We answer this question in the negative by determining the vanishing ideal of Sing2,m for all m∈N. Our results exhibit that there are additional equations arising from the tensor structure of X. More generally, for any n and m≥n2−n+1, we prove there are equations vanishing on Singn,m that are not in the ideal generated by polynomials of type det(λ1X1+…+λmXm). Our methods are based on classical results about Fano schemes, representation theory and Gröbner bases.

Superalgebras: Polynomial identities and asymptotics

To any superalgebra A is attached a numerical sequence cnsup(A), n≥1, called the sequence of supercodimensions of A. In characteristic zero its asymptotics are an invariant of the superidentities satisfied by A. It is well-known that for a PI-superalgebra such sequence is exponentially bounded and expsup(A)=limn→∞⁡cnsup(A)n is an integer that can be explicitly computed.Here we introduce a notion of fundamental superalgebra over a field of characteristic zero. We prove that if A is such an algebra, then(1)C1ntexpsup(A)n≤cnsup(A)≤C2ntexpsup(A)n, where C1>0,C2,t are constants and t is a half integer that can be explicitly written as a linear function of the dimension of the even part of A and the nilpotency index of the Jacobson radical. We also give a characterization of fundamental superalgebras through the representation theory of the symmetric group.As a consequence we prove that if A is a finitely generated PI-superalgebra, then the inequalities in (1) still hold and t is a half integer. It follows that limn→∞⁡logn⁡cnsup(A)expsup(A)n exists and is a half integer.

The structure of Lie algebras with a derivation satisfying a polynomial identity

We prove nilpotency results for Lie algebras over an arbitrary field admitting a derivation, which satisfies a given polynomial identity r(t) = 0. In the special case of the polynomial we obtain a uniform bound on the nilpotency class of Lie algebras admitting a periodic derivation of order n. We even find an optimal bound on the nilpotency class in characteristic p if p does not divide a certain invariant ρn .

Algebraic properties of Hermitian sums of squares, II

We study real bihomogeneous polynomials r ( z , z ¯ ) r(z,\bar {z}) in n n complex variables for which r ( z , z ¯ ) ‖ z ‖ 2 r(z,\bar {z}) \left \lVert {z} \right \rVert ^2 is the squared norm of a holomorphic polynomial mapping. Such polynomials are the focus of the Sum of Squares Conjecture, which describes the possible ranks for the squared norm r ( z , z ¯ ) ‖ z ‖ 2 r(z,\bar {z}) \left \lVert {z} \right \rVert ^2 and has important implications for the study of proper holomorphic mappings between balls in complex Euclidean spaces of different dimension. Questions about the possible signatures for r ( z , z ¯ ) r(z,\bar {z}) and the rank of r ( z , z ¯ ) ‖ z ‖ 2 r(z,\bar {z}) \left \lVert {z} \right \rVert ^2 can be reformulated as questions about polynomial ideals. We take this approach and apply purely algebraic tools to obtain constraints on the signature of r r .

Affine representability and decision procedures for commutativity theorems for rings and algebras

We consider applications of a finitary version of the Affine Representability theorem, which follows from recent work of Belov-Kanel, Rowen, and Vishne. Using this result we are able to show that when given a finite set of polynomial identities, there is an algorithm that terminates after a finite number of steps which decides whether these identities force a ring to be commutative. We then revisit old commutativity theorems of Jacobson and Herstein in light of this algorithm and obtain general results in this vein. In addition, we completely characterize the homogeneous multilinear identities that imply the commutativity of a ring.