Elimination from Homogeneous Polynomials Over a Polynomial Ring

1976 ◽  
Vol 28 (6) ◽  
pp. 1269-1276
Author(s):  
John G. Stevens
Keyword(s):  
Polynomial Ring ◽  
Homogeneous Polynomials ◽  
Fixed Ideal

Let Ω be a field and Γ a parameter. We designate the set of all polynomials homogeneous in (X) = (X1, … , Xn) with coefficients in Ω [Γ] by H Ω Γ[X] and write such polynomials as F, F(X), or F(X, Γ). The degree of a polynomial in H Ω Γ [X] shall mean the degree in (X). Let I = (F1 … , Fr) be a fixed ideal in H Ω Γ [X] generated by F1 … , Fr.

scholarly journals Componentwise linear ideals

1999 ◽  
Vol 153 ◽  
pp. 141-153 ◽  
Author(s):  
Jürgen Herzog ◽  
Takayuki Hibi
Keyword(s):  
Simplicial Complex ◽  
Polynomial Ring ◽  
Betti Numbers ◽  
Monomial Ideals ◽  
Homogeneous Polynomials ◽  
Graded Betti Numbers ◽  
Linear Resolution ◽  
Alexander Dual ◽  
Componentwise Linear Ideals ◽  
The Ideal

AbstractA componentwise linear ideal is a graded ideal I of a polynomial ring such that, for each degree q, the ideal generated by all homogeneous polynomials of degree q belonging to I has a linear resolution. Examples of componentwise linear ideals include stable monomial ideals and Gotzmann ideals. The graded Betti numbers of a componentwise linear ideal can be determined by the graded Betti numbers of its components. Combinatorics on squarefree componentwise linear ideals will be especially studied. It turns out that the Stanley-Reisner ideal IΔ arising from a simplicial complex Δ is componentwise linear if and only if the Alexander dual of Δ is sequentially Cohen-Macaulay. This result generalizes the theorem by Eagon and Reiner which says that the Stanley-Reisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay.

scholarly journals Alternative polarizations of Borel fixed ideals

2012 ◽  
Vol 207 ◽  
pp. 79-93 ◽  
Author(s):  
Kohji Yanagawa
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Simplicial Complexes ◽  
Regular Sequence ◽  
Fixed Ideal

AbstractFor a monomial idealIof a polynomial ringS, apolarizationofIis a square-free monomial idealJof a larger polynomial ringsuch thatS/Iis a quotient of/Jby a (linear) regular sequence. We show that a Borel fixed ideal admits a nonstandard polarization. For example, while the usual polarization sendsours sends it tox1y2y3Using this idea, we recover/refine the results onsquare-free operationin the shifting theory of simplicial complexes. The present paper generalizes a result of Nagel and Reiner, although our approach is very different.

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.

scholarly journals Degrees of Regular Sequences With a Symmetric Group Action

2019 ◽  
Vol 71 (03) ◽  
pp. 557-578
Author(s):  
Federico Galetto ◽  
Anthony Vito Geramita ◽  
David Louis Wehlau
Keyword(s):  
Symmetric Group ◽  
Polynomial Ring ◽  
Group Action ◽  
Homogeneous Polynomials ◽  
Regular Sequences

AbstractWe consider ideals in a polynomial ring that are generated by regular sequences of homogeneous polynomials and are stable under the action of the symmetric group permuting the variables. In previous work, we determined the possible isomorphism types for these ideals. Following up on that work, we now analyze the possible degrees of the elements in such regular sequences. For each case of our classification, we provide some criteria guaranteeing the existence of regular sequences in certain degrees.

Pembentukan Lapangan Faktor dari Suatu Daerah Integral

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Tri Widjajanti ◽  
Dahlia Ramlan ◽  
Rium Hilum
Keyword(s):  
Polynomial Ring ◽  
Integral Domain ◽  
Rational Numbers ◽  
Ring Of Integers ◽  
Binary Operations

<em>Ring of integers under the addition and multiplication as integral domain can be imbedded to the field of rational numbers. In this paper we make&nbsp; a construction such that any integral domain can be&nbsp; a field of quotient. The construction contains three steps. First, we define element of field F from elements of integral domain D. Secondly, we show that the binary operations in fare well-defined. Finally, we prove that </em><em>&nbsp;</em><em>f</em><em> </em><em>:</em><em> </em><em>D </em><em>&reg;</em><em> </em><em>F is an isomorphisma. In this case, the polynomial ring F[x] as the integral domain can be imbedded to the field of quotient.</em>

MODEL OF SOCIAL POTENTIAL OF CHILDHOOD IN THE RUSSIAN REGIONS: THE RESULTS OF EXPERIMENTS

2019 ◽  
pp. 152-162
Author(s):  
A.G. Filipova ◽  
A.V. Vysotskaya
Keyword(s):  
Third Wave ◽  
Russian Regions ◽  
The Third ◽  
Control Factors ◽  
Social Potential ◽  
System Input ◽  
Optimal Values ◽  
Fixed Ideal ◽  
And Control ◽  
Selection Of

The article presents the results of mathematical experiments with the system «Social potential of childhood in the Russian regions». In the structure of system divided into three subsystems – the «Reproduction of children in the region», «Children’s health» and «Education of children», for each defined its target factor (output parameter). The groups of infrastructure factors (education, health, culture and sport, transport), socio-economic, territorial-settlement, demographic and en-vironmental factors are designated as the factors that control the system (input parameters). The aim of the study is to build a model îf «Social potential of childhood in the Russian regions», as well as to conduct experiments to find the optimal ratio of the values of target and control factors. Three waves of experiments were conducted. The first wave is related to the analysis of the dynam-ics of indicators for 6 years. The second – with the selection of optimal values of control factors at fixed ideal values of target factors. The third wave allowed us to calculate the values of the target factors based on the selected optimal values of the control factors of the previous wave.

On Stanley Depths of Certain Monomial Factor Algebras

2015 ◽  
Vol 58 (2) ◽  
pp. 393-401
Author(s):  
Zhongming Tang
Keyword(s):  
Polynomial Ring ◽  
Monomial Ideal ◽  
Primary Decomposition ◽  
Factor Algebra ◽  
Stanley Depth ◽  
Stanley Conjecture ◽  
Stanley Decomposition

AbstractLet S = K[x1 , . . . , xn] be the polynomial ring in n-variables over a ûeld K and I a monomial ideal of S. According to one standard primary decomposition of I, we get a Stanley decomposition of the monomial factor algebra S/I. Using this Stanley decomposition, one can estimate the Stanley depth of S/I. It is proved that sdepthS(S/I) ≤ sizeS(I). When I is squarefree and bigsizeS(I) ≤ 2, the Stanley conjecture holds for S/I, i.e., sdepthS(S/I) ≥ depthS(S/I).

scholarly journals Polynomial ring representations of endomorphisms of exterior powers

2021 ◽  
Author(s):  
Ommolbanin Behzad ◽  
André Contiero ◽  
Letterio Gatto ◽  
Renato Vidal Martins
Keyword(s):  
Lie Algebra ◽  
Polynomial Ring ◽  
Vertex Operators ◽  
Exterior Power ◽  
Infinite Dimensional ◽  
Symmetric Structure ◽  
Rational Polynomials ◽  
Exterior Powers ◽  
Vertex Representation ◽  
Countably Infinite

AbstractAn explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.

scholarly journals The close relation between border and Pommaret marked bases

2021 ◽  
Author(s):  
Cristina Bertone ◽  
Francesca Cioffi
Keyword(s):  
Finite Order ◽  
Polynomial Ring ◽  
Group Actions ◽  
Close Relation ◽  
Open Covering ◽  
Order Ideal ◽  
Lower Dimension ◽  
Hilbert Schemes ◽  
Affine Spaces ◽  
The Ideal

AbstractGiven a finite order ideal $${\mathcal {O}}$$ O in the polynomial ring $$K[x_1,\ldots , x_n]$$ K [ x 1 , … , x n ] over a field K, let $$\partial {\mathcal {O}}$$ ∂ O be the border of $${\mathcal {O}}$$ O and $${\mathcal {P}}_{\mathcal {O}}$$ P O the Pommaret basis of the ideal generated by the terms outside $${\mathcal {O}}$$ O . In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among $$\partial {\mathcal {O}}$$ ∂ O -marked sets (resp. bases) and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked sets (resp. bases). We prove that a $$\partial {\mathcal {O}}$$ ∂ O -marked set B is a marked basis if and only if the $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked set P contained in B is a marked basis and generates the same ideal as B. Using a functorial description of these marked bases, as a byproduct we obtain that the affine schemes respectively parameterizing $$\partial {\mathcal {O}}$$ ∂ O -marked bases and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked bases are isomorphic. We are able to describe this isomorphism as a projection that can be explicitly constructed without the use of Gröbner elimination techniques. In particular, we obtain a straightforward embedding of border schemes in affine spaces of lower dimension. Furthermore, we observe that Pommaret marked schemes give an open covering of Hilbert schemes parameterizing 0-dimensional schemes without any group actions. Several examples are given throughout the paper.

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