Resultant over the residual of a complete intersection
In this article, we study the residual resultant which is the necessary and sufficient condition for a polynomial system F to have a solution in the residual of a variety, defined here by a complete intersection G. We show that it corresponds to an irreducible divisor and give an explicit formula fo...
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| Veröffentlicht in: | Journal of pure and applied algebra Jg. 164; H. 1; S. 35 - 57 |
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Elsevier B.V
24.10.2001
Elsevier |
| Schriftenreihe: | Effective methods in algebraic geometry (Bath, 2000) |
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| ISSN: | 0022-4049, 1873-1376 |
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| Abstract | In this article, we study the residual resultant which is the necessary and sufficient condition for a polynomial system
F to have a solution in the residual of a variety, defined here by a complete intersection
G. We show that it corresponds to an irreducible divisor and give an explicit formula for its degree in the coefficients of each polynomial. Using the resolution of the ideal
(F
:
G)
and computing its regularity, we give a method for computing the residual resultant using a matrix which involves a Macaulay and a Bezout part. In particular, we show that this resultant is the gcd of all the maximal minors of this matrix. We illustrate our approach for the residual of points and end by some explicit examples. |
|---|---|
| AbstractList | In this article, we study the residual resultant which is the necessary and sufficient condition for a polynomial system
F to have a solution in the residual of a variety, defined here by a complete intersection
G. We show that it corresponds to an irreducible divisor and give an explicit formula for its degree in the coefficients of each polynomial. Using the resolution of the ideal
(F
:
G)
and computing its regularity, we give a method for computing the residual resultant using a matrix which involves a Macaulay and a Bezout part. In particular, we show that this resultant is the gcd of all the maximal minors of this matrix. We illustrate our approach for the residual of points and end by some explicit examples. In this article, we study the residual resultant which is the necessary and sufficient condition for a polynomial system F to have a solution in the residual of a variety, defined here by a complete intersection G. We show that it corresponds to an irreducible divisor and give an explicit formula for its degree in the coefficients of each polynomial. Using the resolution of the ideal (F:G) and computing its regularity, we give a method for computing the residual resultant using a matrix which involves a Macaulay and a Bezout part. In particular, we show that this resultant is the gcd of all the maximal minors of this matrix. We illustrate our approach for the residual of points and end by some explicit examples. |
| Author | Busé, L. Elkadi, M. Mourrain, B. |
| Author_xml | – sequence: 1 givenname: L. surname: Busé fullname: Busé, L. email: lbuse@math.unice.fr organization: UNSA, UMR 6621, Parc Valrose, BP 71, 06108 Nice Cedex 02, France – sequence: 2 givenname: M. surname: Elkadi fullname: Elkadi, M. email: elkadi@math.unice.fr organization: UNSA, UMR 6621, Parc Valrose, BP 71, 06108 Nice Cedex 02, France – sequence: 3 givenname: B. surname: Mourrain fullname: Mourrain, B. email: mourrain@sophia.inria.fr organization: SAGA, INRIA, BP 93, 06902 Sophia Antipolis, France |
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| Cites_doi | 10.1112/plms/s1-35.1.3 10.1006/aima.1996.1609 10.1115/1.2836473 10.1006/jsco.1999.0304 10.1007/BF01389151 10.1016/0001-8708(91)90031-2 10.1006/jsco.1998.0266 10.1007/978-0-8176-4771-1 10.1016/0021-8693(90)90050-X |
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Symbolic Comput. doi: 10.1006/jsco.1998.0266 – ident: 10.1016/S0022-4049(00)00144-4_BIB13 doi: 10.1007/978-0-8176-4771-1 – volume: 128 start-page: 214 year: 1990 ident: 10.1016/S0022-4049(00)00144-4_BIB2 article-title: The resolution of the generic residual intersection of a complete intersection publication-title: J. Algebra doi: 10.1016/0021-8693(90)90050-X – ident: 10.1016/S0022-4049(00)00144-4_BIB5 – volume: 390 start-page: 1 year: 1998 ident: 10.1016/S0022-4049(00)00144-4_BIB16 article-title: Residual intersections publication-title: J. Reine Angew. Math. – ident: 10.1016/S0022-4049(00)00144-4_BIB6 – year: 1992 ident: 10.1016/S0022-4049(00)00144-4_BIB7 – year: 1977 ident: 10.1016/S0022-4049(00)00144-4_BIB15 |
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F to have a solution in the residual... In this article, we study the residual resultant which is the necessary and sufficient condition for a polynomial system F to have a solution in the residual... |
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| Title | Resultant over the residual of a complete intersection |
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