The implicitization problem for φ{symbol} : Pn (P1)n + 1
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...
Guardado en:
Autor principal: | |
---|---|
Formato: | JOUR |
Materias: | |
Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_00218693_v322_n11_p3878_Botbol |
Aporte de: |
id |
todo:paper_00218693_v322_n11_p3878_Botbol |
---|---|
record_format |
dspace |
spelling |
todo:paper_00218693_v322_n11_p3878_Botbol2023-10-03T14:21:27Z The implicitization problem for φ{symbol} : Pn (P1)n + 1 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. JOUR 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) |
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 |
JOUR |
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 |
url |
http://hdl.handle.net/20.500.12110/paper_00218693_v322_n11_p3878_Botbol |
work_keys_str_mv |
AT botboln theimplicitizationproblemforphsymbolpnp1n1 AT botboln implicitizationproblemforphsymbolpnp1n1 |
_version_ |
1782026783252545536 |