Commutative Polynomial Research Articles

Inner isotopes associated with automorphisms of commutative associative algebras

The principal observation of the present paper is that an inner isotopy (i.e. a principal isotopy defined by an algebra endomorphism) is a very helpful instrument in constructing and studying interesting classes of nonassociative algebras. By using methods developed in the paper, we define a new class of commutative nonassociative algebras obtained by inner isotopy from commutative associative polynomial algebras. There is a natural bijection between isomorphism classes of our algebras and integer partitions of the algebra dimensions. Among the interesting features of the nonassociative algebras constructed are that these algebras are generic, some of examples are axial and metrized algebras. We completely describe both the set of algebra idempotents and their spectra.

Binomial ideals in quantum tori and quantum affine spaces

The article targets binomial ideals in quantum tori and quantum affine spaces. First, noncommutative analogs of known results for commutative (Laurent) polynomial rings are obtained, including the following: Under the assumption of an algebraically closed base field, it is proved that primitive ideals are binomial, as are radicals of binomial ideals and prime ideals minimal over binomial ideals. In the case of a quantum torus Tq, the results are strongest: In this situation, the binomial ideals are parametrized by characters on sublattices of the free abelian group whose group algebra is the center of Tq; the sublattice-character pairs corresponding to primitive ideals as well as to radicals and minimal primes of binomial ideals are determined. As for occurrences of binomial ideals in quantum algebras: It is shown that cocycle-twisted group algebras of finitely generated abelian groups are quotients of quantum tori modulo binomial ideals. Another appearance is as follows: Cocycle-twisted semigroup algebras of finitely generated commutative monoids, as well as quantum affine toric varieties, are quotients of quantum affine spaces modulo certain types of binomial ideals.

A Bounded Below, Noncontractible, Acyclic Complex Of Projective Modules

We construct examples of bounded below, noncontractible, acycliccomplexes of finitely generated projective modules over some rings S,as well as bounded above, noncontractible, acyclic complexes ofinjective modules. The rings S are certain rings of infinite matrices with entries inthe rings of commutative polynomials or formal power series ininfinitely many variables. In the world of comodules or contramodules over coalgebras overfields, similar examples exist over the cocommutative symmetriccoalgebra of an infinite-dimensional vector space. A simpler, universal example of a bounded below, noncontractible,acyclic complex of free modules with one generator, communicated tothe author by Canonaco, is included at the end of the paper.

Bases of fixed point subalgebras on nilpotent Leibniz algebras

Let K be a field of characteristic zero, X={x_(1,) x_2,…,x_n} and R_m={r_(1,) ,…,r_m} be two sets of variables, F be the free left nitpotent Leibniz algebra generated by X, and K[R_m ] be the commutative polynomial algebra generated by R_m over the base field K. The fixed point subalgebra of an automorphism φ is the subalgebra of F consisting of elements that are invariant under the automorphism. In this work, we consider specific automorphisms of F and determine the fixed point subalgebras of these automorphisms. Then, we find bases of these fixed point subalgebras. In addition, we get generators of these subalgebras as a free K[R_m ] -module.

THE NOWICKI CONJECTURE FOR BICOMMUTATIVE ALGEBRAS

Let K be a field of characteristic zero, and K[Xn,Yn ] be the commutative associative unitary polynomial algebra of rank 2n generated by the set Xn∪Yn={x1,…,xn,y1,…,yn }. It is well known that the algebra K[Xn,Yn ]^δ of constants of the locally nilpotent linear derivation δ of K[Xn,Yn ] sending yi to xi, and xi to 0, is generated by x1,…,xn and the determinants of the form xi yj-xj yi; that was first conjectured by Nowicki in 1994, and later proved by several authors. Bicommutative algebras are nonassociative noncommutative algebras satisfying the identities (xy)z=(xz)y and x(yz)=y(xz). In this study, we work in the 2n generated free bicommutative algebra as a noncommutative nonassociative analogue of the Nowicki conjecture, and find the generators of the algebra of constants in this algebra.

Generalized commutative quaternion polynomials of the Fibonacci type

Generalized commutative quaternions is a number system which generalizes elliptic, parabolic and hyperbolic quaternions, bicomplex numbers, complex hyperbolic numbers and hyperbolic complex numbers. In this paper we introduce and study generalized commutative quaternion polynomials of the Fibonacci type.

İZİ SIFIR OLAN JENERİK MATRİSLER CEBİRİ İÇİN NOWICKI SANISI

A locally nilpotent linear derivation δ of the commutative polynomial algebra K[Xd ]=K[x1,…,xd ] of rank d is called Weitzenböck. It is well known that the subalgebra K[Xd ]^δ of K[Xd ] consisting of polynomials which are sent to zero by δ is finitely generated. Let the Weitzenböck derivation δ act on K[Xd,Yd ] such that δ(yi )=xi, δ(xi )=0, i=1,…,d. The explicit form of generators of the algebra K[Xd,Yd ]^δ was conjectured by Nowicki in 1994. In this study, we consider the Nowicki conjecture in the algebra W generated by two traceless generic matrices with entries from commutative associative unitary polynomial algebra with six variables, and obtain the free generators of the algebra W^δ of constants in this algebra.

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.

Iterated Hopf Ore extensions in positive characteristic

Iterated Hopf Ore extensions (IHOEs) over an algebraically closed base field \Bbbk of positive characteristic p are studied. We show that every IHOE over \Bbbk satisfies a polynomial identity (PI), with PI-degree a power of p , and that it is a filtered deformation of a commutative polynomial ring. We classify all 2 -step IHOEs over \Bbbk , thus generalising the classification of 2 -dimensional connected unipotent algebraic groups over \Bbbk . Further properties of 2 -step IHOEs are described: for example their simple modules are classified, and every 2 -step IHOE is shown to possess a large Hopf center and hence an analog of the restricted enveloping algebra of a Lie \Bbbk -algebra. As one of a number of questions listed, we propose that such a restricted Hopf algebra may exist for every IHOE over \Bbbk .

Signature Gröbner bases, bases of syzygies and cofactor reconstruction in the free algebra

Signature-based algorithms have become a standard approach for computing Gröbner bases in commutative polynomial rings. However, so far, it was not clear how to extend this concept to the setting of noncommutative polynomials in the free algebra. In this paper, we present a signature-based algorithm for computing Gröbner bases in precisely this setting. The algorithm is an adaptation of Buchberger's algorithm including signatures. We prove that our algorithm correctly enumerates a signature Gröbner basis as well as a Gröbner basis of the module generated by the leading terms of the generators' syzygies, and that it terminates whenever the ideal admits a finite signature Gröbner basis. Additionally, we adapt well-known signature-based criteria eliminating redundant reductions, such as the syzygy criterion, the F5 criterion and the singular criterion, to the case of noncommutative polynomials. We also generalise reconstruction methods from the commutative setting that allow to recover, from partial information about signatures, the coordinates of elements of a Gröbner basis in terms of the input polynomials, as well as a basis of the syzygy module of the generators. We have written a toy implementation of all the algorithms in the Mathematica package OperatorGB and we compare our signature-based algorithm to the classical Buchberger algorithm for noncommutative polynomials.

Generalized Typical Dimension of a Graded Module

In this paper, we prove an upper bound for the leading coefficient of the characteristic polynomial of a graded ideal in a ring of generalized polynomials. Examples of such rings are the rings of commutative polynomials (for which the classical Bézout theorem holds), as well as some rings of differential operators. For a system of generalized homogeneous equations in small codimensions we obtain exact estimates that are polynomial in d. In the general case, the estimate is double exponential in τ : O(d2τ−1), where d is the maximal degree of generators of a graded ideal and τ is its codimension. For systems of linear differential equations, bounds of the same asymptotics, but by other methods, were obtained by D. Grigoriev.

A construction of swap or switch polynomials

We discuss several constructions of swap polynomials that is non commutative polynomials in matrix variables xi∈Md(Q) with values in Md(Q)⊗2 which are multiples of the transposition operator (1,2).

Degree bounds for Hopf actions on Artin–Schelter regular algebras

We study semisimple Hopf algebra actions on Artin–Schelter regular algebras and prove several upper bounds on the degrees of the minimal generators of the invariant subring, and on the degrees of syzygies of modules over the invariant subring. These results are analogues of results for group actions on commutative polynomial rings proved by Noether, Fogarty, Fleischmann, Derksen, Sidman, Chardin, and Symonds.

Computation of Koszul homology and application to involutivity of partial differential systems

The formal integrability of systems of partial differential equations plays a fundamental role in different analysis and synthesis problems for both linear and nonlinear differential control systems. Following Spencer's theory, to test the formal integrability of a system of partial differential equations, we must study when the symbol of the system, namely, the top-order part of the linearization of the system, is 2-acyclic or involutive, i.e., when certain Spencer cohomology groups vanish. Combining the fact that Spencer cohomology is dual to Koszul homology and symbolic computation methods, we show how to effectively compute the homology modules defined by the Koszul complex of a finitely presented module over a commutative polynomial ring. These results are implemented using the OREMORPHISMS package. We then use these results to effectively characterize 2-acyclicity and involutivity of the symbol of a linear system of partial differential equations. Finally, we show explicit computations on two standard examples.

Finite factorization properties in commutative monoid rings with zero divisors

Several different generalizations of finite factorization domains (i.e., integral domains where every nonzero nonunit has only finitely many divisors up to associates) have been defined for commutative rings with zero divisors. We study these notions in the context of commutative monoid rings with zero divisors, utilizing semigroup theory to simultaneously generalize and extend many past results about “finite factorization” properties in commutative polynomial rings. Along the way, we expand upon the general theory of factorization in commutative rings with zero divisors, providing new characterizations and results about several kinds of “finite factorization rings.”

Symmetric polynomials of algebras related with 2 × 2 generic traceless matrices

Let [Formula: see text] and [Formula: see text] be two generic traceless matrices of size [Formula: see text] with entries from a commutative associative polynomial algebra over a field [Formula: see text] of characteristic zero. Consider the associative unitary algebra [Formula: see text], and its Lie subalgebra [Formula: see text] generated by [Formula: see text] and [Formula: see text] over the field [Formula: see text]. It is well known that the center [Formula: see text] of [Formula: see text] is the polynomial algebra generated by the algebraically independent commuting elements [Formula: see text], [Formula: see text], [Formula: see text]. We call a polynomial [Formula: see text] symmetric, if [Formula: see text]. The set of symmetric polynomials is equal to the algebra [Formula: see text] of invariants of symmetric group [Formula: see text]. Similarly, we define the Lie algebra [Formula: see text] of symmetric polynomials in the Lie algebra [Formula: see text]. We give the description of the algebras [Formula: see text] and [Formula: see text], and we provide finite sets of free generators for [Formula: see text], and [Formula: see text] as [Formula: see text]-modules.

COMPLETELY PRIME ONE-SIDED IDEALS IN SKEW POLYNOMIAL RINGS

AbstractLet R = K[x, σ] be the skew polynomial ring over a field K, where σ is an automorphism of K of finite order. We show that prime elements in R correspond to completely prime one-sided ideals – a notion introduced by Reyes in 2010. This extends the natural correspondence between prime elements and prime ideals in commutative polynomial rings.

On Authentication Schemes Using Polynomials Over Non Commutative Rings

Authentication is a term very important for data communication security. We see many frauds due to authentication failure. The problem manifolds when communication is over insecure channel. Entity authentication is a term which involves proof of sender’s identity and very useful in various applications like in banking sector and various other client server mechanisms. Availability of quantum computers increases the vulnerability of breaking old protocols. Researchers are finding new platforms to overcome this problem and one such example is non commutative polynomial rings [NCPR]. In 2012, M.R.Vallauri [MRV], in his paper suggested an authentication protocol using NCPR. He has proved security analysis under the assumption that polynomial symmetrical decomposition problem (PSDP) is hard. In this paper we show that the protocol suggested by him is breakable without solving PSDP. We also provide corrected protocol to overcome this problem.

Symmetric Polynomials in the Free Metabelian Lie Algebras

Let $$K[X_n]$$ be the commutative polynomial algebra in the variables $$X_n=\{x_1,\ldots ,x_n\}$$ over a field K of characteristic zero. A theorem from undergraduate course of algebra states that the algebra $$K[X_n]^{S_n}$$ of symmetric polynomials is generated by the elementary symmetric polynomials which are algebraically independent over K. In the present paper, we study a noncommutative and nonassociative analogue of the algebra $$K[X_n]^{S_n}$$ replacing $$K[X_n]$$ with the free metabelian Lie algebra $$F_n$$ of rank $$n\ge 2$$ over K. It is known that the algebra $$F_n^{S_n}$$ is not finitely generated, but its ideal $$(F_n')^{S_n}$$ consisting of the elements of $$F_n^{S_n}$$ in the commutator ideal $$F_n'$$ of $$F_n$$ is a finitely generated $$K[X_n]^{S_n}$$ -module. In our main result, we describe the generators of the $$K[X_n]^{S_n}$$ -module $$(F_n')^{S_n}$$ which gives the complete description of the algebra $$F_n^{S_n}$$ .

Biseparable extensions are not necessarily Frobenius

We give necessary and sufficient conditions on an Ore extension $$A[x;\sigma ,\delta ]$$ , where A is a finite dimensional algebra over a field $${\mathbb {F}}$$ , for being a Frobenius extension of the ring of commutative polynomials $${\mathbb {F}}[x]$$ . As a consequence, as the title of this paper highlights, we provide a negative answer to a problem stated by Caenepeel and Kadison.