Applications to Commutative Algebra and Harmonic Analysis
Identifieur interne : 000603 ( France/Analysis ); précédent : 000602; suivant : 000604Applications to Commutative Algebra and Harmonic Analysis
Auteurs : Carlos A. Berenstein [États-Unis] ; Alekos Vidras [Japon] ; Roger Gay [France] ; Alain Yger [France]Source :
- Progress in Mathematics ; 1993.
Abstract
Abstract: Let F be a subfield of ℂ, and P 1,…,P m be elements of F[X 1,…, X n ]. If Q ∈ F[X 1 ,…,X n ,] vanishes on V = {P 1 =…= P m, = 0}, some power of Q lies in the ideal I generated by P 1 ,…, P m . This classical result, which is known as the global algebraic Nullstellensatz, can be reduced to the following special case, where the polynomials have no common zeroes in ℂn. In that case, solving the Nullstellensatz corresponds to the problem of finding m polynomials A 1,…, A m , in F[X 1,…, X n ] such that 5.1 $$ 1 = {A_1}{P_1} \ldots + {A_m}{P_m}.$$ This polynomial equation (5.1) j called the (algebraic) Bezout equation.
Url:
DOI: 10.1007/978-3-0348-8560-7_5
Affiliations:
- France, Japon, États-Unis
- Aquitaine, Maryland, Nouvelle-Aquitaine, Région du Kansai
- College Park (Maryland), Kyoto, Talence (
- Université de Kyoto, Université du Maryland
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Berenstein</name><affiliation wicri:level="4"><country xml:lang="fr">États-Unis</country><wicri:regionArea>Mathematics Department & Institute of Systems Research, University of Maryland, College Park, 20742, MD</wicri:regionArea><placeName><region type="state">Maryland</region><settlement type="city">College Park (Maryland)</settlement></placeName><orgName type="university">Université du Maryland</orgName></affiliation></author><author><name sortKey="Vidras, Alekos" sort="Vidras, Alekos" uniqKey="Vidras A" first="Alekos" last="Vidras">Alekos Vidras</name><affiliation wicri:level="4"><country xml:lang="fr">Japon</country><wicri:regionArea>Research Institute for Mathematical Sciences, Kyoto University, 606, Kyoto</wicri:regionArea><orgName type="university">Université de Kyoto</orgName><placeName><settlement type="city">Kyoto</settlement><region type="prefecture">Région du Kansai</region></placeName></affiliation></author><author><name sortKey="Gay, Roger" sort="Gay, Roger" uniqKey="Gay R" first="Roger" last="Gay">Roger Gay</name><affiliation wicri:level="3"><country xml:lang="fr">France</country><wicri:regionArea>Centre de Recherche en Mathématiques, Université de Bordeaux I, 33405, Talence (Cedex)</wicri:regionArea><placeName><region type="region" nuts="2">Nouvelle-Aquitaine</region><region type="old region" nuts="2">Aquitaine</region><settlement type="city">Talence (</settlement></placeName></affiliation></author><author><name sortKey="Yger, Alain" sort="Yger, Alain" uniqKey="Yger A" first="Alain" last="Yger">Alain Yger</name><affiliation wicri:level="1"><country xml:lang="fr">France</country><wicri:regionArea>Centre de Recherche en Mathématiques, Université de Bordeaux I, 33405, Talence (Cedex)</wicri:regionArea><wicri:noRegion>33405, Talence (Cedex)</wicri:noRegion><wicri:noRegion>Talence (Cedex)</wicri:noRegion></affiliation></author></analytic><monogr></monogr><series><title level="s">Progress in Mathematics</title><imprint><date>1993</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: Let F be a subfield of ℂ, and P 1,…,P m be elements of F[X 1,…, X n ]. If Q ∈ F[X 1 ,…,X n ,] vanishes on V = {P 1 =…= P m, = 0}, some power of Q lies in the ideal I generated by P 1 ,…, P m . This classical result, which is known as the global algebraic Nullstellensatz, can be reduced to the following special case, where the polynomials have no common zeroes in ℂn. In that case, solving the Nullstellensatz corresponds to the problem of finding m polynomials A 1,…, A m , in F[X 1,…, X n ] such that 5.1 $$ 1 = {A_1}{P_1} \ldots + {A_m}{P_m}.$$ This polynomial equation (5.1) j called the (algebraic) Bezout equation.</div></front></TEI><affiliations><list><country><li>France</li><li>Japon</li><li>États-Unis</li></country><region><li>Aquitaine</li><li>Maryland</li><li>Nouvelle-Aquitaine</li><li>Région du Kansai</li></region><settlement><li>College Park (Maryland)</li><li>Kyoto</li><li>Talence (</li></settlement><orgName><li>Université de Kyoto</li><li>Université du Maryland</li></orgName></list><tree><country name="États-Unis"><region name="Maryland"><name sortKey="Berenstein, Carlos A" sort="Berenstein, Carlos A" uniqKey="Berenstein C" first="Carlos A." last="Berenstein">Carlos A. 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