Non-commutative Polynomial Ring Research Articles

The structure of matrix polynomial algebras

This work formally introduces and starts investigating the structure of matrix polynomial algebra extensions of a coefficient algebra by (elementary) matrix-variables over a ground polynomial ring in not necessary commuting variables. These matrix subalgebras of full matrix rings over polynomial rings show up in noncommutative algebraic geometry. We carefully study their (one-sided or bilateral) noetherianity, obtaining a precise lift of the Hilbert Basis Theorem when the ground ring is either a commutative polynomial ring, a free noncommutative polynomial ring or a skew polynomial ring extension by a free commutative term-ordered monoid. We equally address the natural but rather delicate question of recognising which matrix polynomial algebras are Cayley-Hamilton algebras, which are interesting noncommutative algebras arising from the study of $\mathrm{Gl}_{n}$-varieties.

Irreducible integrable modules for the full toroidal Lie algebras coordinated by rational quantum torus

Let Cq be the non-commutative Laurent polynomial ring associated with a (n+1)×(n+1) rational quantum matrix q. Let sld(Cq)⊕HC1(Cq) be the universal central extension of Lie subalgebra sld(Cq) of gld(Cq). We take the Lie algebra τ=gld(Cq)⊕HC1(Cq). Let Der(Cq) be the Lie algebra of all derivations of Cq and consider the Lie algebra τ˜=τ⋊Der(Cq), called the full toroidal Lie algebra coordinated by rational quantum tori. In this paper we get a classification of irreducible integrable modules with finite dimensional weight spaces for τ˜ with nonzero central action on the modules.

Trivial multiple zeta values in Tate algebras

AbstractWe study trivial multiple zeta values in Tate algebras. These are particular examples of the multiple zeta values in Tate algebras introduced by the second author. If the number of variables involved is “not large” in a way that is made precise in the paper, we can endow the set of trivial multiple zeta values with a structure of module over a non-commutative polynomial ring with coefficients in the rational fraction field over ${\mathbb{F}}_q$. We determine the structure of this module in terms of generators and we show how in many cases, this is sufficient for the detection of linear relations between Thakur’s multiple zeta values.

On Finite Noncommutative Gröbner Bases

It is well known that in the noncommutative polynomial ring in serveral variables Buchberger’s algorithm does not always terminate. Thus, it is important to characterize noncommutative ideals that admit a finite Gröbner basis. In this context, Eisenbud, Peeva and Sturmfels defined a map γ from the noncommutative polynomial ring k⟨X1, …, Xn⟩ to the commutative one k[x1, …, xn] and proved that any ideal [Formula: see text] of k⟨X1, …, Xn⟩, written as [Formula: see text] = γ−1([Formula: see text]) for some ideal [Formula: see text] of k[x1, …, xn], amits a finite Gröbner basis with respect to a special monomial ordering on k⟨X1, …, Xn⟩. In this work, we approach the opposite problem. We prove that under some conditions, any ideal [Formula: see text] of k⟨X1, …, Xn⟩ admitting a finite Gröbner basis can be written as [Formula: see text] = γ−1([Formula: see text]) for some ideal [Formula: see text] of k[x1, …, xn].

Lower Bounds and PIT for Non-commutative Arithmetic Circuits with Restricted Parse Trees

We investigate the power of Non-commutative Arithmetic Circuits, which compute polynomials over the free non-commutative polynomial ring $${\mathbb{F}\langle{x_1,\ldots,x_N\rangle}}$$ , where variables do not commute. We consider circuits that are restricted in the ways in which they can compute monomials: this can be seen as restricting the families of parse trees that appear in the circuit. Such restrictions capture essentially all non-commutative circuit models for which lower bounds are known. We prove several results about such circuits.

Towards an effective study of the algebraic parameter estimation problem

The paper aims at developing the first steps toward a symbolic computation approach to the algebraic parameter estimation problem defined by Fliess and Sira-Ramirez. In this paper, within the algebraic analysis approach, we first give a general formulation of the algebraic parameter estimation for signals which are defined by ordinary differential equations with polynomial coefficients such as the standard orthogonal polynomials (e.g., Chebyshev or Hermite, polynomials). Based on a result on holonomic functions, we show that the algebraic parameter estimation problem for a truncated expansion of a function into an orthogonal basis of L2 defined by orthogonal polynomials can be studied similarly. Then, using symbolic computation methods such as Gröbner basis techniques for (noncommutative) polynomial rings, we first show how to compute ordinary differential operators which annihilate a given polynomial and which contain only certain parameters in their coefficients. Then, we explain how to compute the intersection of the annihilator ideals of two polynomials and characterize the ordinary differential operators which annihilate a first polynomial but not a second one. These results, at the core of the algebraic parameter estimation, are implemented in the NonA package.

NLControl – a Mathematica Package for Nonlinear Control Systems

The paper describes a symbolic computation software package NLControl, dedicated to the study of various modelling, analysis, synthesis problems for nonlinear control systems. Most of the developed functions are based on algebraic approach of differential forms, and on the theory of non-commutative polynomial rings. For a few control problems an alternative solution is implemented in terms of functions algebra. The package addresses both continuous-and discrete-time systems as well as systems, defined on homogeneous time scales. Comparison of NLControl and Mathematica built-in nonlinear control functions is given.

Realisation of linear time-varying systems

ABSTRACTIn this paper, the realisation problem of linear multi-input multi-output, time-varying systems is studied. The approach, based on the theory of non-commutative polynomial rings, yields explicit and simple formulas for computation of the state coordinates as well as for state equations in observable canonical form. The formulas are based on (left) Euclidean polynomial division.

A New Perspective of Cyclicity in Convolutional Codes

In this paper, we propose a new way of providing cyclic structures to convolutional codes. We define the skew cyclic convolutional codes as left ideals of a quotient ring of a suitable non-commutative polynomial ring. In contrast to the previous approaches to cyclicity for convolutional codes, we use Ore polynomials with coefficients in a field (the rational function field over a finite field), so their arithmetic is very well known and we may proceed similarly to cyclic block codes. In particular, we show how to obtain easily skew cyclic convolutional codes of a given dimension, and we compute an idempotent generator of the code and its dual.

Integrability for Nonlinear Time-Delay Systems

In this note, the notion of integrability is defined for 1-forms defined in the time-delay context. While in the delay-free case, a set of 1-forms defines a vector space, it is shown that 1-forms computed for time-delay systems have to be viewed as elements of a module over a certain non-commutative polynomial ring. Two notions of integrability are defined, strong and weak integrability, which coincide in the delay-free case. Necessary and sufficient conditions are given to check if a set of 1-forms is strongly or weakly integrable. To show the importance of the topic, integrability of 1-forms is used to characterize the accessibility property for nonlinear time-delay systems. The possibility of transforming a system into a certain normal form is also considered.

A primer on ideal theoretical operation in non-commutative polynomial rings

Buchberger Theory provides, for commutative polynomial rings, computational tools allowing to perform ideal theoretical operations. Such techniques are easily adapted to deal with similar problems in the non-commutative setting. I give here an elementary primer.

Dual codes of product semi-linear codes

Let Fq be a finite field with q elements and denote by θ:Fq→Fq an automorphism of Fq. In this paper, we deal with linear codes of Fqn invariant under a semi-linear map T:Fqn→Fqn for some n≥2. In particular, we study three kinds of their dual codes, some relations between them and we focus on codes which are products of module skew codes in the non-commutative polynomial ring as a subcase of linear codes invariant by a semi-linear map T. In this setting we give also an algorithm for encoding, decoding and detecting errors and we show a method to construct codes invariant under a fixed T.

Polynomial accessibility condition for the multi-input multi-output nonlinear control system

The paper presents a computation-oriented necessary and sufficient accessibility condition for the set of nonlinear higher- order input-output differential equations. The condition is presented in terms of the greatest common left divisor of two polynomial matrices, associated with the system of input-output equations. The basic difference from the linear case is that the elements of the polynomial matrices belong to a non-commutative polynomial ring. The condition found provides a basis for finding the accessible representation of the set of input-output equations, which is a suitable starting point for the construction of an observable and accessible state space realization. Moreover, the condition allows us to check the transfer equivalence of two nonlinear systems.

A Gröbner-Shirshov basis approach to Hua’s identity

Hua’s identity states for any \(a, \, b\) in an associative unitary ring \(R\) that \((a+ab^{-1}a)^{-1}+(a+b)^{-1}=a^{-1}\) provided \(a\), \(b\) and \(a+b\) themselves do have inverses. We find it amusing to derive this identity by studying the Grobner-Shirshov basis of a particular ideal in a suitable non-commutative polynomial ring.

Ancestors graph and an upper bound for the subword complexity function

A general upper bound for the subword complexity function of a fixed point of a primitive substitution is presented. The approach relies on the introduction and discussion of the ancestors graph of a substitution. More precisely, we characterize the language of a fixed point of a primitive substitution as the strongly connected component of the letters in its ancestors graph; we propose algorithms for the local construction of this graph; we give an upper bound for the subword complexity function in terms of the Hilbert series of some quotient of a noncommutative polynomial ring by a sum of automatic bilateral ideals.

Hochschild cohomology of group extensions of quantum symmetric algebras

Quantum symmetric algebras (or noncommutative polynomial rings) arise in many places in mathematics. In this article we find the multiplicative structure of their Hochschild cohomology when the coefficients are in an arbitrary bimodule algebra. When this bimodule algebra is a finite group extension (under a diagonal action) of a quantum symmetric algebra, we give explicitly the graded vector space structure. This yields a complete description of the Hochschild cohomology ring of the corresponding skew group algebra.

Bernoulli-type relations in some noncommutative polynomial ring

We find particular relations which we call “Bernoulli-type” in some noncommutative polynomial ring with a single nontrivial relation. More precisely, our ring is isomorphic to the universal enveloping algebra of a two-dimensional non-abelian Lie algebra. From these Bernoulli-type relations in our ring, we can obtain a representation on a certain left ideal with the Bernoulli numbers as structure constants.

Non-commutative Reidemeister torsion and Morse-Novikov theory

Given a circle-valued Morse function of a closed oriented manifold, we prove that Reidemeister torsion over a non-commutative formal Laurent polynomial ring equals the product of a certain non-commutative Lefschetz-type zeta function and the algebraic torsion of the Novikov complex over the ring. This paper gives a generalization of the result of Hutchings and Lee on abelian coefficients to the case of skew fields. As a consequence we obtain a Morse theoretical and dynamical description of the higher-order Reidemeister torsion.

Skew codes of prescribed distance or rank

In this paper we generalize the notion of cyclic code and construct codes as ideals in finite quotients of non-commutative polynomial rings, so called skew polynomial rings of automorphism type. We propose a method to construct block codes of prescribed rank and a method to construct block codes of prescribed distance. Since there is no unique factorization in skew polynomial rings, there are much more ideals and therefore much more codes than in the commutative case. In particular we obtain a [40, 23, 10]4 code by imposing a distance and a [42,14,21]8 code by imposing a rank, which both improve by one the minimum distance of the previously best known linear codes of equal length and dimension over those fields. There is a strong connection with linear difference operators and with linearized polynomials (or q-polynomials) reviewed in the first section.

Non-commutative multivariable Reidemester torsion and the Thurston norm

Given a 3‐manifold the second author defined functions nW H 1 .MIZ/! N, generalizing McMullen’s Alexander norm, which give lower bounds on the Thurston norm. We reformulate these invariants in terms of Reidemeister torsion over a noncommutative multivariable Laurent polynomial ring. This allows us to show that these functions are semi-norms. 57M27; 57N10