## Algebraic Approach of Residues and Applications (1996)

Venue: | The Mathematics of Numerical Analysis, volume 32 of Lectures in Applied Math |

Citations: | 28 - 11 self |

### BibTeX

@INPROCEEDINGS{Cardinal96algebraicapproach,

author = {J.P. Cardinal and B. Mourrain},

title = {Algebraic Approach of Residues and Applications},

booktitle = {The Mathematics of Numerical Analysis, volume 32 of Lectures in Applied Math},

year = {1996},

pages = {18921--0},

publisher = {AMS}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper, we give an overview of the theory of residues and its applications. We adopt the algebraic point of view and show how this theory can lead to effective computations. For this purpose, we analyze the properties of Bezoutians, objects which underly many problems related to polynomial systems. We recall the main properties of a Gorenstein Algebra, where we can naturally speak of residues. We show how we can construct a residue, in the case of complete intersection, and we give the algebraic counterparts of some theorems of the analytical theory of residues. We also give some direct applications of this approach to the computations of Chow Forms and show how the information concerning multiple roots can be recovered from the local residues in a very simple way. Finally, we propose a new method for solving zero-dimensional systems of n equations P1 = \Delta \Delta \Delta = Pn = 0 in n variables x1 , ..., xn . This method involves elementary manipulations on the matrices...

### Citations

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Citation Context ... : +un i n ), the roots being counted with multiplicity. It is usually de��ned as the condition on the coordinates of an hyperplane u : (u 0 ; : : : ; un ) such that this hyperplane intersects V (=-=see [HP52]). I-=-t can also be de��ned, up to a scalar, by det(u 0 Id d + u 1 [x 1 ] + \Delta \Delta \Delta + un [x n ]), where [x l ] is the matrix of multiplication by x l in the quotient B associated to V . Pro... |

104 |
and J.Harris, Principles of algebraic geometry, Wiley-Interscience
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Citation Context ...es of these objects in the analytical theory of residues. The determinant that appears in the transformation law is similar to \Phi(1; P 1 ; : : : ; Pn ). For instance, the determinant det(A) used in =-=[GH78]-=-[p 657] is exactly \Phi(1; f 1 ; : : : ; fn )(z; 0) when g i = x i . Using methods from complex analysis, people have relied on these objects to obtain representation formulae. We refer, for instance,... |

104 |
Commutative Algebra
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Citation Context ...ls P 1 ; : : : ; Pn in n variables and assume that the quotient B of R by I is a K -vector space of dimension d. More precisely, we assume that P = (P 1 ; : : : ; Pn ) is a regular sequence of R (see =-=[Mat80]-=-). We will see how to construct a residue for B. 3.1. Construction of a residue. Let B = B\Omega K B and \Delta P = \Phi(1; P 1 ; : : : ; Pn ). We will also note \Phi(P ) = \Phi(P; P 1 ; : : : ; Pn ),... |

92 |
The eliminant of three quantics in two independent variables
- Dixon
- 1994
(Show Context)
Citation Context ...see [CE93]) are the analog of Sylvester's resultants, whereas the resultants we will deduce from \Phi will correspond to Bezout's resultants. We also ��nd the presence of \Phi in the work of A. Di=-=xon [Dix08]. Fo-=-r 3 polynomialssP 0 ; P 1 ; P 2 in two variables, of the same degree, he took some coeOEcients in �� of \Phi(P 0 ; P 1 ; P 2 ), added some multiples of the initial polynomials and took the vectors... |

84 |
An elimination algorithm for computation of all zeros of a system of multivariate polynomial equations
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Citation Context ...t in @ l . This polynomial (or residue) t characterizes completely the singular point i. Other work has been done for the description of the endomorphism of multiplication by variables, (see [M#l93], =-=[AS88]-=-). Also, see [MMM95] for a similar approach to this one but with a dioeerent language. As we have seen, the eigenspaces of these endomorphisms are also the pseudo-eigenspaces of ([\Phi(x l )]; [\Phi(1... |

80 |
On some formulae in elimination
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Citation Context ... initial polynomials and took the determinant of the corresponding (m+n) \Theta (m + n) matrix. As we will see, this comparison can be carried further in several variables. "Macaulay" result=-=ants (see [Mac02] or [VdW48]) o-=-r "sparse" resultants (see [CE93]) are the analog of Sylvester's resultants, whereas the resultants we will deduce from \Phi will correspond to Bezout's resultants. We also ��nd the pres... |

75 | An efficient algorithm for the sparse mixed resultant
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Citation Context ... the corresponding (m+n) \Theta (m + n) matrix. As we will see, this comparison can be carried further in several variables. "Macaulay" resultants (see [Mac02] or [VdW48]) or "sparse&qu=-=ot; resultants (see [CE93]) ar-=-e the analog of Sylvester's resultants, whereas the resultants we will deduce from \Phi will correspond to Bezout's resultants. We also ��nd the presence of \Phi in the work of A. Dixon [Dix08]. F... |

56 |
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Citation Context ...rresponds to the right rank. There are 22 (approximate) real roots. 5.2. Numerical accuracy and degrees of intersection. This problem comes from Computational Geometry; more precisely, from the paper =-=[BMS94]-=-. It consists of giving a bound on the numerical accuracy required to ensure an answer. We will see on an example that this question is closely connected to the degree of a Chow form. Problem: Given 3... |

41 |
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Citation Context ...oblem. For Macaulay resultants, this applies to cases where the polynomials are not dense polynomials. For sparse resultants, this applies when the polynomials are not generic for their supports (see =-=[Ber75]-=- for non-genericity in this case). We consider two matrices M 0 ; M 1 where the columns are indexed by monomials. Each row of these matrices represent a polynomial. Assume that for a polynomial f in R... |

23 |
Multivariate Bezoutians, Kronecker symbol and Eisenbud & Levine formula
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- 1996
(Show Context)
Citation Context ...approach of residues, but in more algorithmic and eoeective language, we mention the thesis of the ��rst author [Car93]. We also mention some applications in real algebraic geometry, such as the w=-=ork [BCRS94]-=-, connecting the topological degree of a real map to signatures of quadratic forms associated to the residue. Our motivation here is to show that all the information concerning the structure associate... |

11 | Grobner duality and multiplicities in polynomial system solving
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Citation Context ...ynomial (or residue) t characterizes completely the singular point i. Other work has been done for the description of the endomorphism of multiplication by variables, (see [M#l93], [AS88]). Also, see =-=[MMM95]-=- for a similar approach to this one but with a dioeerent language. As we have seen, the eigenspaces of these endomorphisms are also the pseudo-eigenspaces of ([\Phi(x l )]; [\Phi(1)]). Example: For th... |

5 |
Multdimensional analogues of newtons formulas for systems of nonlinear algebraic equations and some of their applications. Trans. from Sib
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Citation Context ...residues has been proposed in [CDS94], in a special case where the polynomials P 1 ; : : : ; Pn are a AEat deformation of (x d1 1 ; : : : ; x dn n ). Some applications to Newton sums also exist (c.f. =-=[AK81]-=-). ALGEBRAIC APPROACH OF RESIDUES AND APPLICATIONS 3 As we will see, most of the theorems of this analytical approach of residues (transformation laws, Euler-Jacoby formula, : : : ) can be proved in a... |

5 |
G#om#trie des points #pais
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(Show Context)
Citation Context ...generalized eigenspaces of the two matrices [\Phi(1)]; [\Phi(x l )] in any basis. For this purpose, we need to introduce some tools relative to D-modules. For more information, we refer the reader to =-=[Ems78]. In-=- this section, we assume that the ��eld K is of characteristic 0. For any root i of fP 1 = \Delta \Delta \Delta = Pn = 0g, the linear form @ 0 i : p 7! p(i) (2 R) vanishes on I and induces a linea... |

4 |
elimination and localization
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- 1970
(Show Context)
Citation Context ...erating the construction of a new point �� with two points x and ��: �� = ( \Phi(x 1 ) \Phi(1) ; : : : ; \Phi(x n ) \Phi(1) ) This method is also described in [OR70]. In the same domain, a=-=nother work [Hou70] als-=-o used the Bezoutians to localize the root of a polynomial in one variable. The proposed method is not far from the Schur-Cohn method. Of course we ��nd traces of these objects in the analytical t... |

3 |
Sur la complexit# du th#or#me des z#ros
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Citation Context ... and P. Solerno [SS95], uses the algebraic approach developed in [SS75], [Kun86] in order to ��nd representation formula, when the ideal I = (P 1 ; : : : ; Pn ) is radical (no multiplicity). The w=-=ork [FGS93]-=-, following the previous approach, assuming the multiplication tables in the quotient R=I are known, also constructs representation formulae, for polynomials in the ideal I . Closely related to the al... |

3 | Systems of algebraic equations solved by means of endomorphisms - Mller - 1993 |

2 |
Eoeective B#zout identities
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(Show Context)
Citation Context ...ly \Phi(1; f 1 ; : : : ; fn )(z; 0) when g i = x i . Using methods from complex analysis, people have relied on these objects to obtain representation formulae. We refer, for instance, to [BGVY93] or =-=[BY91]-=-. An algorithmic approach to compute residues has been proposed in [CDS94], in a special case where the polynomials P 1 ; : : : ; Pn are a AEat deformation of (x d1 1 ; : : : ; x dn n ). Some applicat... |

2 |
Dualit# et algorithmes it#ratifs pour la r#solution de syst#mes polynomiaux
- Cardinal
- 1993
(Show Context)
Citation Context ...resentation formulae, for polynomials in the ideal I . Closely related to the algebraic approach of residues, but in more algorithmic and eoeective language, we mention the thesis of the ��rst aut=-=hor [Car93]-=-. We also mention some applications in real algebraic geometry, such as the work [BCRS94], connecting the topological degree of a real map to signatures of quadratic forms associated to the residue. O... |

1 |
Residue currents and bezout identities, Prog
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- 1993
(Show Context)
Citation Context ...57] is exactly \Phi(1; f 1 ; : : : ; fn )(z; 0) when g i = x i . Using methods from complex analysis, people have relied on these objects to obtain representation formulae. We refer, for instance, to =-=[BGVY93]-=- or [BY91]. An algorithmic approach to compute residues has been proposed in [CDS94], in a special case where the polynomials P 1 ; : : : ; Pn are a AEat deformation of (x d1 1 ; : : : ; x dn n ). Som... |

1 |
Computing multidimensional residues, Eoeective Methods
- Cattani, Dickenstein, et al.
- 1994
(Show Context)
Citation Context ...omplex analysis, people have relied on these objects to obtain representation formulae. We refer, for instance, to [BGVY93] or [BY91]. An algorithmic approach to compute residues has been proposed in =-=[CDS94]-=-, in a special case where the polynomials P 1 ; : : : ; Pn are a AEat deformation of (x d1 1 ; : : : ; x dn n ). Some applications to Newton sums also exist (c.f. [AK81]). ALGEBRAIC APPROACH OF RESIDU... |

1 |
Le formalisme du r#sultant, Adv
- Jouanolou
- 1991
(Show Context)
Citation Context ...he algebraic approach of residues are settled. The ��rst part of this paper borrows heavily on their works. These objects are useful in elimination theory, as we can see in the work of J.P. Jouano=-=lou [Jou91]-=-. The function \Phi appears in his work under the name of Morley form. The resultant of n homogeneous polynomials in n variables is constructed via this polynomial \Phi. In another direction, the work... |

1 |
K#hkler dioeerentials, Advanced lectures
- Kunz
- 1986
(Show Context)
Citation Context ...ansformation laws, Euler-Jacoby formula, : : : ) can be proved in an algebraic setting. The main contributions in this direction are certainly the works of G. Scheja and U. Storch [SS75], and E. Kunz =-=[Kun86], wh-=-ere the foundations of the algebraic approach of residues are settled. The ��rst part of this paper borrows heavily on their works. These objects are useful in elimination theory, as we can see in... |

1 |
R#solution des syst#mes d'#quations alg#briques, Theo
- Lazard
- 1981
(Show Context)
Citation Context ...d of two matrices M 0 ; M 1 we have several matrices M 1 ; : : : ; M s connected by the same relations. Each row of M i is f i times the corresponding row of M 0 . A similar procedure can be found in =-=[Laz81] where i-=-t is used in the case where P = (P 1 ; : : : ; Pn ) de��nes a complete intersection with no point at in��nity. Here, we use this compression process for any complete intersection. 4.2. Computa... |