The implicitization problem for φ{symbol} : Pn (P1)n + 1

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We develop in this paper methods for studying the implicitization problem for a rational map φ{symbol} : Pn (P1)n + 1 defining a hypersurface in (P1)n + 1, based on computing the determinant of a graded strand of a Koszul complex. We show that the classical study of Macaulay resultants and Koszul co...

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Autor principal: Botbol, N.
Formato: Artículo publishedVersion
Lenguaje:Inglés
Publicado: 2009
Materias:
Approximation complex
Elimination theory
Implicitization
Koszul complex
Rational map
Syzygy
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00218693_v322_n11_p3878_Botbol
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling paperaa:paper_00218693_v322_n11_p3878_Botbol2023-06-12T16:42:26Z The implicitization problem for φ{symbol} : Pn (P1)n + 1 J. Algebra 2009;322(11):3878-3895 Botbol, N. Approximation complex Elimination theory Implicitization Koszul complex Rational map Syzygy We develop in this paper methods for studying the implicitization problem for a rational map φ{symbol} : Pn (P1)n + 1 defining a hypersurface in (P1)n + 1, based on computing the determinant of a graded strand of a Koszul complex. We show that the classical study of Macaulay resultants and Koszul complexes coincides, in this case, with the approach of approximation complexes and we study and give a geometric interpretation for the acyclicity conditions. Under suitable hypotheses, these techniques enable us to obtain the implicit equation, up to a power, and up to some extra factor. We give algebraic and geometric conditions for determining when the computed equation defines the scheme theoretic image of φ{symbol}, and, what are the extra varieties that appear. We also give some applications to the problem of computing sparse discriminants. © 2009 Elsevier Inc. All rights reserved. 2009 info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion application/pdf eng info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00218693_v322_n11_p3878_Botbol
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
language Inglés
orig_language_str_mv eng
topic Approximation complex
Elimination theory
Implicitization
Koszul complex
Rational map
Syzygy
spellingShingle Approximation complex
Elimination theory
Implicitization
Koszul complex
Rational map
Syzygy
Botbol, N.
The implicitization problem for φ{symbol} : Pn (P1)n + 1
topic_facet Approximation complex
Elimination theory
Implicitization
Koszul complex
Rational map
Syzygy
description We develop in this paper methods for studying the implicitization problem for a rational map φ{symbol} : Pn (P1)n + 1 defining a hypersurface in (P1)n + 1, based on computing the determinant of a graded strand of a Koszul complex. We show that the classical study of Macaulay resultants and Koszul complexes coincides, in this case, with the approach of approximation complexes and we study and give a geometric interpretation for the acyclicity conditions. Under suitable hypotheses, these techniques enable us to obtain the implicit equation, up to a power, and up to some extra factor. We give algebraic and geometric conditions for determining when the computed equation defines the scheme theoretic image of φ{symbol}, and, what are the extra varieties that appear. We also give some applications to the problem of computing sparse discriminants. © 2009 Elsevier Inc. All rights reserved.
format Artículo
Artículo
publishedVersion
author Botbol, N.
author_facet Botbol, N.
author_sort Botbol, N.
title The implicitization problem for φ{symbol} : Pn (P1)n + 1
title_short The implicitization problem for φ{symbol} : Pn (P1)n + 1
title_full The implicitization problem for φ{symbol} : Pn (P1)n + 1
title_fullStr The implicitization problem for φ{symbol} : Pn (P1)n + 1
title_full_unstemmed The implicitization problem for φ{symbol} : Pn (P1)n + 1
title_sort implicitization problem for φ{symbol} : pn (p1)n + 1
publishDate 2009
url http://hdl.handle.net/20.500.12110/paper_00218693_v322_n11_p3878_Botbol
work_keys_str_mv AT botboln theimplicitizationproblemforphsymbolpnp1n1
AT botboln implicitizationproblemforphsymbolpnp1n1
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