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
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 http://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00218693_v322_n11_p3878_Botbol_oai
Aporte de:	Repositorio Digital de la Universidad de Buenos Aires (UBA) de Universidad de Buenos Aires

id	I28-R145-paper_00218693_v322_n11_p3878_Botbol_oai
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spelling	I28-R145-paper_00218693_v322_n11_p3878_Botbol_oai2020-10-19 Botbol, N. 2009 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. application/pdf http://hdl.handle.net/20.500.12110/paper_00218693_v322_n11_p3878_Botbol info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar J. Algebra 2009;322(11):3878-3895 Approximation complex Elimination theory Implicitization Koszul complex Rational map Syzygy The implicitization problem for φ{symbol} : Pn (P1)n + 1 info:eu-repo/semantics/article info:ar-repo/semantics/artículo info:eu-repo/semantics/publishedVersion http://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00218693_v322_n11_p3878_Botbol_oai
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-145
collection	Repositorio Digital de la Universidad de Buenos Aires (UBA)
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 http://repositoriouba.sisbi.uba.ar/gsdl/cgi-bin/library.cgi?a=d&c=artiaex&d=paper_00218693_v322_n11_p3878_Botbol_oai
work_keys_str_mv	AT botboln theimplicitizationproblemforphsymbolpnp1n1 AT botboln implicitizationproblemforphsymbolpnp1n1
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The implicitization problem for φ{symbol} : Pn (P1)n + 1

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