Commutative Polynomial Research Articles

Ore Principal Subresultant Coefficients in Solutions

The principal subresultant coefficients of polynomials play a fundamental role in elimination theory and computer algebra. Recently they have been extended to Ore polynomials. They are defined by an expression in the coefficients of Ore polynomials. In this paper, we provide another expression for them. This expression is written in terms of the “solutions” of Ore polynomials (in “generic” case). It can be viewed as a generalization of the well known expression for resultants of two commutative polynomials: the product of the pair-wise differences of their roots.

Associated primes in commutative polynomial rings

The aim of this paper is to give a new and direct proof of the theorem.

Residue Complex for Sklyanin Algebras of Dimension Three

We describe the residue complex for three-dimensional Sklyanin algebras, which are the interesting special cases of quantum polynomial rings in three variables. In particular, we show that the multiplicities of the point modules in the residue complex are all one, just as in the classical case of commutative polynomial rings in three variables. We explain why the residue complex in the quantum case has the same multiplicities of point modules as that in the commutative case (even if there are fewer point modules in the quantum case) by pointing out two quantum anomalies.

Test de primauté dans une extension de ore itérée

In this paper we prove a generalization of the primality test in [3] for a wide class of iterated Ore extensions of a commutative polynomial ring.

Monomial Orderings, Rewriting Systems, and Gröbner Bases for the Commutator Ideal of a Free Algebra

In this paper we consider a free associative algebra on three generators over an arbitrary fieldK. Given a term ordering on the commutative polynomial ring on three variables overK, we construct uncountably many liftings of this term ordering to a monomial ordering on the free associative algebra. These monomial orderings are total well orderings on the set of monomials, resulting in a set of normal forms. Then we show that the commutator ideal has an infinite reduced Gröbner basis with respect to these monomial orderings, and all initial ideals are distinct. Hence, the commutator ideal has at least uncountably many distinct reduced Gröbner bases. A Gröbner basis of the commutator ideal corresponds to a complete rewriting system for the free commutative monoid on three generators; our result also shows that this monoid has at least uncountably many distinct minimal complete rewriting systems.The monomial orderings we use are not compatible with multiplication, but are sufficient to solve the ideal membership problem for a specific ideal, in this case the commutator ideal. We propose that it is fruitful to consider such, more general, monomial orderings in non-commutative Gröbner basis theory.

Primitive and Poisson Spectra of Twists of Polynomial Rings

A family of flat deformations of a commutative polynomial ring S on n generators is considered, where each deformation B is a twist of S by a semisimple, linear automorphism σ of ℙ n−1, such that a Poisson bracket is induced on S. We show that if the symplectic leaves associated with this Poisson structure are algebraic, then they are the orbits of an algebraic group G determined by the Poisson bracket. In this case, we prove that if σ is 'generic enough', then there is a natural one-to-one correspondence between the primitive ideals of B and the symplectic leaves if and only if σ has a representative in GL(ℂ n ) which belongs to G. As an example, the results are applied to the coordinate ring $$\mathcal{O}_q (M_2 )$$ of quantum 2 × 2 matrices which is not a twist of a polynomial ring, although it is a flat deformation of one; if q is not a root of unity, then there is a bijection between the primitive ideals of $$\mathcal{O}_q (M_2 )$$ and the symplectic leaves.

Algorithmic Properties of Polynomial Rings

In this paper we investigate how algorithms for computing heights, radicals, unmixed and primary decompositions of ideals can be lifted from a Noetherian commutative ringRto polynomial rings overR.

The Hecke algebras for the Fricke groups

We discuss the Hecke algebra \({\cal H}(\Gamma_0(N),GL_2^+({\rm Q}))\)⋌e for the standard subgroup Г0 (N) of S L2(ℤ). In particular the center of this Hecke algebra is determined. The results are applied to the Fricke group, which is a normal and maximal discrete extension of Г0 (N), and to the attached Hecke algebra, which turns out to be a commutative polynomial ring. In this case the Hecke groups \(G(\sqrt 2)\)⋌e and \(G(\sqrt 3)\)⋌e are included. Finally we give a description of modular forms for the Fricke groups in terms of new-forms in the sense of Atkin-Lehner.

Algebras associated to elliptic curves

This paper completes the classification of Artin-Schelter regular algebras of global dimension three. For algebras generated by elements of degree one this has been achieved by Artin, Schelter, Tate and Van den Bergh. We are therefore concerned with algebras which are not generated in degree one. We show that there exist some exceptional algebras, each of which has geometric data consisting of an elliptic curve together with an automorphism, just as in the case where the algebras are assumed to be generated in degree one. In particular, we study the elliptic algebras A ( + ) A(+) , A ( − ) A(-) , and A ( a ) A({\mathbf {a}}) , where a ∈ P 2 {\mathbf {a}}\in \mathbb {P}^{2} , which were first defined in an earlier paper. We omit a set S ⊂ P 2 S\subset \mathbb {P}^2 consisting of 11 specified points where the algebras A ( a ) A({\mathbf {a}}) become too degenerate to be regular. Theorem. Let A A represent A ( + ) A(+) , A ( − ) A(-) or A ( a ) A({\mathbf {a}}) , where a ∈ P 2 ∖ S {\mathbf {a}} \in \mathbb {P}^2\setminus S . Then A A is an Artin-Schelter regular algebra of global dimension three. Moreover, A A is a Noetherian domain with the same Hilbert series as the (appropriately graded) commutative polynomial ring in three variables. This, combined with our earlier results, completes the classification.

Segre Product of Artin–Schelter Regular Algebras of Dimension 2 and Embeddings in Quantum [formula omitted]3's

The Segre embedding of P1×P1as a smooth quadricQin P3corresponds to the surjection of the four-dimensional polynomial ring onto the Segre productSof two copies of the homogeneous coordinate ring of P1. We study Segre products of noncommutative algebras. If in particularAandBare two copies of a quantum P1thenS=⊕i(Ai⊗kBi) is a twisted homogeneous coordinate ring of the quadricQ. The main result of this paper is the classification of all embeddings of the Segre product of two quantum planes into so-called quantum P3's. These are (the Proj of) Artin–Schelter regular algebrasRof global dimension four with the Hilbert series of a commutative polynomial ring and which map ontoS. IfRis not a twist of a polynomial ring, then the point scheme ofReither is the union of the quadricQwith a line or is only the quadricQ. In the first case,Ris a central extension of a three-dimensional Artin–Schelter regular algebra and a twist of an algebra mapping onto the (commutative) homogeneous coordinate ring ofQ; in the second case, such an algebraRis the first known example of a four-dimensional Artin–Schelter regular algebra which is not determined by its point scheme.

A Gröbner Approach to Involutive Bases

Recently, Zharkov and Blinkov introduced the notion of involutive bases of polynomial ideals. This involutive approach has its origin in the theory of partial differential equations and is a translation of results of Janet and Pommaret. In this paper we present a pure algebraic foundation of involutive bases of Pommaret type. In fact, they turn out to be generalized left Gröbner bases of ideals in the commutative polynomial ring with respect to a non-commutative grading. The introduced theory will allow not only the verification of the results of Zharkov and Blinkov but it will also provide some new facts.

Synergy in the Theories of Gröbner Bases and Path Algebras

AbstractA general theory for Grôbner basis in path algebras is introduced which extends the known theory for commutative polynomial rings and free associative algebras.

When is 𝐹[𝑥,𝑦] a unique factorization domain?

Although the commutative polynomial ring F [ x , y ] F[x,y] is a unique factorization domain (UFD) and the free associative algebra F ⟨ x , y ⟩ F\langle x,y\rangle is a similarity-UFD when F F is a (commutative) field, it is shown that the polynomial ring F [ x , y ] F[x,y] in two commuting indeterminates is not a UFD in any reasonable sense when F F is the skew field of rational quaternions.

Gr�bner bases in exterior algebra

We show that the Buchberger algorithm for commutative polynomials over a field may be generalised to an algebraic structure which embeds such polymomials, the exterior polynomial algebra, and which is a natural domain for linear geometry. In particular, those finite sets of exterior polynomials which induce confluent reduction relations are characterised, and a means of algorithmically constructing them from a given set presented. A distinguished subset of such bases consists of the exterior algebra version of Grobner bases. We characterise such bases and demonstrate how to construct them algorithmically from a given finite set of exterior polynomials.

Some examples of noncommutative local rings

In this paper we construct examples which answer three questions in the general area of noncommutative Noetherian local rings and rings of finite global dimension. The examples are formed in the same basic way, beginning with a commutative polynomial ring A over a field k and a k-derivation δ of A, taking the skew polynomial ring R = A[x;δ] and localizing at a prime ideal of the form IR, where I is a prime ideal of A invariant under δ. The localization is possible by a result of Sigurdsson [13].

Regularity of trace rings of generic matrices

is called the ring of m generic n x n matrices G,, n. The F-subalgebra of M,,(em,,) generated by G,,. and Tr(G,,.) is the ring of m generic n x n matrices and is denoted by T,, n. These trace rings appear naturally in the study of finite dimensional representations of free algebras and in the invariant theory of n x n matrices, [7]. Unlike rings of generic matrices, T,, n shares some properties with commutative polynomial rings; e.g., they are maximal orders and even unique factorization rings in the sense of Chatters and Jordan. However, their homological properties are far from being understood. The main aim of this paper is to prove the following result.

Theoremes de transfert pour les polynomes partiellement commutatifs

We prove that the properties of a semiring K “to be without zero divisors”, and “to be regular and simplifiable” transfer to the K-algebras of the free partially commutative monoids. We also give an algorithm which tests the divisibility of partially commutative polynomials, and computes the quotient in the case of a positive answer.

Standard bases of perfect homogeneous polynomial ideals of height 2

This paper concerns minimal systems of homogeneous generators (i.e., standard bases) of homogeneous ideals Z in a commutative polynomial ring K with coefficients in a field k. A basic numerical question about a standard basis B of Z is to determine its cardinality or, more generally, the number of the elements in B of degree t, for each t > 0; this number depends only on Z (and not on the choice of B) and will be denoted by v,(Z). In this paper we obtain a sharp estimate for the numbers v,(P), when P is a perfect homogeneous ideal of height 2 in K. Precisely, in (2.1) we provide upper and lower bounds for v,(P) in terms of the Hilbert function H = H(K/P, ) of K/P and of an integer /I, which ranges in an interval defined by H. Then in (3.3) we show the sharpness of these bounds: in fact we prove a more general result by exhibiting examples of ideals P (with given H and /I) having, for each degree t, one of the possible values v,(P) expected from (2.1). Finally, in (4.1) we modify the preceding examples and obtain, under certain hypotheses, radical ideals with the same properties. The preceding results offer refined versions of well known theorems. In fact, from (2.1) we obtain (cf. (2.3)) a more complete form of Dubreil’s inequalities (cf. [2, 11); moreover (3.3) and (4.1) improve, in the particular hypotheses under consideration, a classic result of Macaulay (cf. [lo, 143) and its reduced version (cf. [6, 11, 131). Finally, particular cases of our estimates in (2.1) can be found in [7, 8, 5, 121 (for points “in generic position” in the projective plane), in [ 1 ] (for “Hilbert-function-completeintersections”), and in [4] (for monomial ideals of height 2).

On the Ideals of a Noetherian Ring

We construct various examples of Noetherian rings with peculiar ideal structure. For example, there exists a Noetherian domain $R$ with a minimal, nonzero ideal $I$, such that $R/I$ is a commutative polynomial ring in $n$ variables, and a Noetherian domain $S$ with a (second layer) clique that is not locally finite. The key step in the construction of these rings is to idealize at a right ideal $I$ in a Noetherian domain $T$ such that $T/I$ is not Artinian.

On the ideals of a Noetherian ring

We construct various examples of Noetherian rings with peculiar ideal structure. For example, there exists a Noetherian domain R R with a minimal, nonzero ideal I I , such that R / I R/I is a commutative polynomial ring in n n variables, and a Noetherian domain S S with a (second layer) clique that is not locally finite. The key step in the construction of these rings is to idealize at a right ideal I I in a Noetherian domain T T such that T / I T/I is not Artinian.