The multiplicity of a set of homogeneous polynomials over a commutative ring

1998 ◽  
Vol 124 (1) ◽  
pp. 97-105 ◽  
Author(s):  
DAVID KIRBY ◽  
DAVID REES
Keyword(s):  
Commutative Ring ◽  
Polynomial Ring ◽  
Present Note ◽  
Finiteness Condition ◽  
Point Of View ◽  
Koszul Complex ◽  
Homogeneous Polynomials ◽  
Double Complex ◽  
Linear Forms ◽  
Integer Multiple

Some thirty years ago Buchsbaum and Rim [1] extended the notion of multiplicity e(a1, …, an; E) for elements a1, …, an of a commutative ring R with identity and a Noetherian R-module E (≠0) with lengthR (E/[sum ]nt=1aiE) finite to give a multiplicity e((aij); E) associated with E and an m×n matrix (aij) over R satisfying a certain extended finiteness condition. One of their results states that for each of a set of m complexes depending on E, (aij) the Euler–Poincaré characteristic is a certain integer multiple of e((aij); E), at least when R is a local ring.Some of these ideas were taken up in [4] where it is shown that when n[ges ]m−1, each of the complexes K((aij); E; t) with t∈ℤ introduced in [3] also have e((aij); E) as their Euler–Poincaré characteristic. With a slight change in viewpoint (aij) can be replaced by linear forms aj=[sum ]mi=1aijxi (j=1, …, n) of the graded polynomial ring R[x1, …, xm]; the complex K((aij); E; t) then becomes the component of degree t in a certain graded double complexformula herewhere K(a1, …, an; F) is the standard Koszul complex (see [4; section 2]). From this point of view the construction can be extended to allow the homogeneous polynomials a1, …, an to have any (possibly unequal) positive degrees [6]. The main aim of the present note is to extend similarly the results of [4] and to strengthen those results to give information on the vanishing of the multiplicity.

A sequence of complexes generated by a finite set of homogeneous polynomials

1998 ◽  
Vol 124 (1) ◽  
pp. 81-96
Author(s):  
DAVID KIRBY
Keyword(s):  
Present Note ◽  
Infinite Family ◽  
Point Of View ◽  
Koszul Complex ◽  
Homogeneous Polynomials ◽  
Linear Forms ◽  
Algebraic Setting ◽  
Finite Set ◽  
Sensitive Property ◽  
Depth Sensitivity

The usefulness of the Koszul complex in handling in an algebraic setting the two geometric notions of multiplicity and depth first became apparent with the work of Auslander and Buchsbaum [1] following a suggestion of Serre. Regarding the generators a1, …, an of the complex as a 1×n matrix first Eagon and Northcott [4] extended this work to a complex associated with an m×n matrix, then shortly afterwards a different extension was given by Buchsbaum and Rim [2, 3]. These two complexes are two of an infinite family [6] some of which inherit the depth sensitive property of the Koszul complex and all of which under a certain finiteness condition provide the same multiplicity as Euler–Poincaré characteristic [7].These two properties prove useful in geometric applications, see for example Lago and Rodicio [7] for depth sensitivity and Kirby [8] for the characteristic as multiplicity of intersection. During the course of these developments it became clear that it was most appropriate to regard the complexes as being generated by linear forms. From this point of view it is natural to ask if the linearity of the forms is necessary. In the present note we begin a response to this question by extending the work of [6] to complexes associated with forms of arbitrary positive degree. In a sequel we shall similarly extend the results of multiplicity in [7].

Anomalous Behavior of Heterogeneous Materials at Microwaves Frequencies: Introduction to Fractional Derivatives in Electromagnetism

1990 ◽  
Vol 189 ◽  
Author(s):  
F. Heliodore ◽  
D. Cottevieille ◽  
A. Le Mehaute
Keyword(s):  
Electromagnetic Wave ◽  
Physical Meaning ◽  
Fractional Derivatives ◽  
Heterogeneous Media ◽  
Present Note ◽  
Heterogeneous Materials ◽  
Anomalous Behavior ◽  
Point Of View ◽  
Unique Point ◽  
Validity Range

ABSTRACTThe present note introduces new trends in electromagnetic spectroscopy in complex media.When an electromagnetic wave propagates in heterogeneous media, some questions arise about both physical meaning and validity range of the traditional analysis. The aim of our advanced research is related to the generalisation of Maxwell's equations able todescribe both homogeneous and heterogeneous media from an unique point of view.

Amenability and Translation Experiments

1983 ◽  
Vol 35 (1) ◽  
pp. 49-58 ◽  
Author(s):  
Alan L. T. Paterson
Keyword(s):  
Present Note ◽  
Locally Compact Group ◽  
Finiteness Condition ◽  
Probability Measures ◽  
Multiplier Theorem ◽  
Bounded Linear ◽  
Discrete Semigroups ◽  
Transition Operators ◽  
Semigroup Version ◽  
Similar Issue

In [11] it is shown that the deficiency of a translation experiment with respect to another on a σ-finite, amenable, locally compact group can be calculated in terms of probability measures on the group. This interesting result, brought to the writer's notice by [1], does not seem to be as wellknown in the theory of amenable groups as it should be. The present note presents a simple proof of the result, removing the σ-finiteness condition and repairing a gap in Torgersen's argument. The main novelty is the use of Wendel's multiplier theorem to replace Torgersen's approach which is based on disintegration of a bounded linear operator from L1(G) into C(G)* for G σ-finite (cf. [5], VI.8.6). The writer claims no particular competence in mathematical statistics, but hopes that the discussion of the above result from the “harmonic analysis” perspective may prove illuminating.We then investigate a similar issue for discrete semigroups. A set of transition operators, which reduce to multipliers in the group case, is introduced, and a semigroup version of Torgersen's theorem is established.

Complete Decomposability in the Exterior Algebra of a Free Module

1980 ◽  
Vol 32 (1) ◽  
pp. 27-33 ◽  
Author(s):  
M. Boratynski ◽  
E. D. Davis ◽  
A. V. Geramita
Keyword(s):  
Commutative Ring ◽  
Present Note ◽  
Free Module ◽  
Exterior Algebra ◽  
Standard Bases ◽  
Classical Criterion ◽  
Generally True

Recall the classical criterion for the complete decomposability of exterior vectors: the completely decomposable vectors in ∧pRn, R a field, are precisely the “Plücker vectors,” i.e. those whose coordinates (relative to the standard bases for ∧pRn) satisfy the Plücker equations. For R an arbitrary commutative ring, completely decomposable exterior vectors are still Plücker vectors, but the converse is not generally true. Rings for which the converse is true (for all 1 ≤ p ≤ n) are called Towber rings. Noetherian Towber rings are regular and, in fact, are characterized by the property that every Plücker vector in ∧2R4 is completely decomposable. (See [10] for these two results as well as for the above mentioned facts.) The present note develops a new characterization of Towber rings, combining it with results of Kleiner [9] and Estes-Matijevic [5] in (1) below.

scholarly journals Unique factorization in polynomial rings with zero divisors

2020 ◽  
pp. 2150113 ◽  
Author(s):  
D. D. Anderson ◽  
Ranthony A. C. Edmonds
Keyword(s):  
Commutative Ring ◽  
Polynomial Ring ◽  
Commutative Rings ◽  
Polynomial Rings ◽  
Unique Factorization ◽  
Factorization Property ◽  
Zero Divisors ◽  
Unique Factorization Domain

Given a certain factorization property of a ring [Formula: see text], we can ask if this property extends to the polynomial ring over [Formula: see text] or vice versa. For example, it is well known that [Formula: see text] is a unique factorization domain if and only if [Formula: see text] is a unique factorization domain. If [Formula: see text] is not a domain, this is no longer true. In this paper, we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization.

Interpolation in the Automorphism Group of a Polynomial Ring

2020 ◽  
Vol 27 (03) ◽  
pp. 587-598
Author(s):  
M’hammed El Kahoui ◽  
Najoua Essamaoui ◽  
Mustapha Ouali
Keyword(s):  
Automorphism Group ◽  
Commutative Ring ◽  
Polynomial Ring ◽  
Proper Ideal ◽  
Polynomial Rings ◽  
Group Homomorphism ◽  
Volume Preserving

Let R be a commutative ring with unity and SAn(R) be the group of volume-preserving automorphisms of the polynomial R-algebra R[n]. Given a proper ideal 𝔞 of R, we address in this paper the question of whether the canonical group homomorphism SAn(R) → SAn(R/𝔞) is surjective. As an application, we retrieve and generalize, in a unified way, several known results on residual coordinates in polynomial rings.

scholarly journals Radicals Of Polynomial Rings

1956 ◽  
Vol 8 ◽  
pp. 355-361 ◽  
Author(s):  
S. A. Amitsur
Keyword(s):  
Polynomial Ring ◽  
Jacobson Radical ◽  
Present Note ◽  
Polynomial Rings ◽  
Lower Radical

Introduction. Let R be a ring and let R[x] be the ring of all polynomials in a commutative indeterminate x over R. Let J(R) denote the Jacobson radical (5) of the ring R and let L(R) be the lower radical (4) of R. The main object of the present note is to determine the radicals J(R[x]) and L(R[x]). The Jacobson radical J(R[x]) is shown to be a polynomial ring N[x] over a nil ideal N of R and the lower radical L(R[x]) is the polynomial ring L(R)[x].

On the Prime Ideals in a Commutative Ring

2000 ◽  
Vol 43 (3) ◽  
pp. 312-319 ◽  
Author(s):  
David E. Dobbs
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Polynomial Ring ◽  
Sufficient Conditions ◽  
Prime Ideals ◽  
Preceding Result ◽  
Finite Commutative Ring ◽  
The Axiom Of Choice ◽  
Necessary And Sufficient ◽  
Positive Integers

AbstractIf n and m are positive integers, necessary and sufficient conditions are given for the existence of a finite commutative ring R with exactly n elements and exactly m prime ideals. Next, assuming the Axiom of Choice, it is proved that if R is a commutative ring and T is a commutative R-algebra which is generated by a set I, then each chain of prime ideals of T lying over the same prime ideal of R has at most 2|I| elements. A polynomial ring example shows that the preceding result is best-possible.

scholarly journals A Characterization of separable polynomials over a skew polynomial ring

1985 ◽  
Vol 38 (2) ◽  
pp. 275-280
Author(s):  
George Szeto
Keyword(s):  
Commutative Ring ◽  
Finite Order ◽  
Polynomial Ring ◽  
Subject Classification ◽  
Skew Polynomial Ring ◽  
Skew Polynomial

AbstractThe characterization of a separable polynomial over an indecomposable commutative ring (with no idempotents but 0 and 1) in terms of the discriminant proved by G. J. Janusz is generalized to a skew polynomial ring R [ X, ρ] over a not necessarily commutative ring R where ρ is an automorphism of R with a finite order. 1980 Mathematics subject classification (Amer. Math. Soc.): 16 A 05.

scholarly journals Identifiability of homogeneous polynomials and Cremona transformations

2019 ◽  
Vol 2019 (757) ◽  
pp. 279-308 ◽  
Author(s):  
Francesco Galuppi ◽  
Massimiliano Mella
Keyword(s):  
Special Class ◽  
Homogeneous Polynomial ◽  
Projective Spaces ◽  
Homogeneous Polynomials ◽  
Additive Decomposition ◽  
Linear Forms ◽  
Cremona Transformations ◽  
General Polynomial ◽  
Powers Of Linear Forms

AbstractA homogeneous polynomial of degree d in {n+1} variables is identifiable if it admits a unique additive decomposition in powers of linear forms. Identifiability is expected to be very rare. In this paper we conclude a work started more than a century ago and we describe all values of d and n for which a general polynomial of degree d in {n+1} variables is identifiable. This is done by classifying a special class of Cremona transformations of projective spaces.

Sign in / Sign up

Export Citation Format

Share Document