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Good rings and homogeneous polynomials

hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorFRESNEL, J.
hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorMATIGNON, Michel
dc.date.accessioned2024-04-04T02:49:17Z
dc.date.available2024-04-04T02:49:17Z
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/191809
dc.description.abstractEnIn 2011, Khurana, Lam and Wang define the following property. (*)A commutative unital ring A satisfies the property "power stable range one" if for all a, b ∈ A with aA + bA = A there are an integer N = N (a, b) ≥ 1 and λ = λ(a, b) ∈ A such that b N + λa ∈ A × , the unit group of A. In 2019, Berman and Erman consider rings with the following property (**) A commutative unital ring A has enough homogeneous polynomials if for any k ≥ 1 and set S := {p 1 , p 2 , ..., p k } , of primitive points in A n and any n ≥ 2, there exists an homogeneous polynomial P (X 1 , X 2 , ..., X n) ∈ A[X 1 , X 2 , ..., X n ]) with deg P ≥ 1 and P (p i) ∈ A × for 1 ≤ i ≤ k. We show in this article that the two properties (*) and (**) are equivalent and we shall call a commutative unital ring with these properties a good ring. When A is a commutative unital ring of pictorsion as defined by Gabber, Lorenzini and Liu in 2015, we show that A is a good ring. Using a Dedekind domain we built by Goldman in 1963,we show that the converse is false.
dc.language.isoen
dc.title.enGood rings and homogeneous polynomials
dc.typeDocument de travail - Pré-publication
dc.subject.halMathématiques [math]/Algèbre commutative [math.AC]
dc.subject.halMathématiques [math]/Géométrie algébrique [math.AG]
dc.identifier.arxiv1907.05655
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
hal.identifierhal-02173007
hal.version1
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-02173007v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=FRESNEL,%20J.&MATIGNON,%20Michel&rft.genre=preprint


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