Changeset 906458 in git for Singular/LIB/phindex.lib
- Timestamp:
- Apr 7, 2009, 6:18:06 PM (15 years ago)
- Branches:
- (u'spielwiese', '17f1d200f27c5bd38f5dfc6e8a0879242279d1d8')
- Children:
- 5d98f437864469b655868be585350eeb57da2863
- Parents:
- 2ae96e40fc5453bcb155aec76d376d79dd549cbe
- File:
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- 1 edited
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Singular/LIB/phindex.lib (modified) (8 diffs)
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Singular/LIB/phindex.lib
r2ae96e r906458 1 1 ////////////////////////////////////////////////////////////////////////////// 2 version="$Id: phindex.lib,v 1. 3 2008-10-09 09:31:57 SingularExp $";2 version="$Id: phindex.lib,v 1.4 2009-04-07 16:18:05 seelisch Exp $"; 3 3 category=" "; 4 4 info=" … … 10 10 germ found by Eisenbud-Levine in 1997. This result was also proved by 11 11 Khimshiashvili. If the isolated singularity is non algebraically 12 isolated and the vector field has areduced complex zeroes of12 isolated and the vector field has similar reduced complex zeroes of 13 13 codimension 1, we use a formula as the Eisenbud-Levine found by Victor 14 14 Castellanos, in both cases is necessary to use a local order (ds,...). … … 30 30 "USAGE: signatureL(M[,r]); M symmetric matrix, r int (optional). 31 31 RETURN: the signature of M of type int or if r is given and !=0 then 32 intvec with (signature, n o of +, noof -) is returned.33 THEORY: Given the matrix M we construct the quadratic form associated, after32 intvec with (signature, nr. of +, nr. of -) is returned. 33 THEORY: Given the matrix M, we construct the quadratic form associated. Afterwards 34 34 we use the method of Lagrange to compute the signature. The law of 35 inertia for a real quadratic form A(x,x) say that in a35 inertia for a real quadratic form A(x,x) says that in a 36 36 representation of A(x,x) as a sum of independent squares 37 37 A(x,x)=sum_{i=1}^r a_iX_i^2. … … 98 98 "USAGE: signatureLqf(h); h quadratic form (poly type). 99 99 RETURN: the signature of h of type int or if r is given and !=0 then 100 intvec with (signature, n o of +, noof -) is returned.100 intvec with (signature, nr. of +, nr. of -) is returned. 101 101 THEORY: To compute the signature we use the method of Lagrange. The law of 102 inertia for a real quadratic form h(x,x) say that in a102 inertia for a real quadratic form h(x,x) says that in a 103 103 representation of h(x,x) as a sum of independent squares 104 h(x,x)=sum_{i=1}^r a_iX_i^ 2 105 the number of positive and the number of negative squares are 104 h(x,x)=sum_{i=1}^r a_i*X_i^2 the number of positive and the number of negative squares are 106 105 independent of the choice of representation. The signature -s- of 107 106 h(x,x) is the difference between the number -pi- of positive squares … … 229 228 NOTE: the isolated singularity must be algebraically isolated. 230 229 THEORY: The Poincare-Hopf index of a real vector field X at the isolated 231 singularity 0 is the degree of map 232 (X/|X|) : S_epsilon ---> S, 230 singularity 0 is the degree of the map (X/|X|) : S_epsilon ---> S, 233 231 where S is the unit sphere, and the spheres are oriented as 234 232 (n-1)-spheres in R^n. The degree depends only on the germ, X, of X … … 245 243 composition of the product in the algebra Qx with a linear 246 244 functional map 247 . L 248 <,> : (Qx)x(Qx)----->Qx----->R 245 <,> : (Qx)x(Qx) ---(.)--> Qx ---(L)--> R 249 246 with L(Jo)>0, where Jo is the residue class of the Jacobian 250 247 determinant in Qx. Here, we use a natural linear functional defined 251 as follows. Suppose that E={E_1,..E_r} is a basis of Qx, then Jo is252 writingas253 Jo=a_1E_{j1}+...+a_kE_{jk}, js\in {1...r}, s=1..k, k<=r .248 as follows. Suppose that E={E_1,..E_r} is a basis of Qx, then Jo can 249 be written as 250 Jo=a_1E_{j1}+...+a_kE_{jk}, js\in {1...r}, s=1..k, k<=r, 254 251 where a_s are constant. The linear functional L:Qx--->R is defined as 255 252 L(E_{j1})=(a_1)/|a_1|=sign of a_1, … … 355 352 at 0, with reduced complex zeros of codimension 1. 356 353 THEORY: Suppose that 0 is an algebraically isolated singularity of the real 357 analytic vector field X, geometrically its meanthat the354 analytic vector field X, geometrically this corresponds to the fact that the 358 355 complexified vector field has positive dimension singular locus, 359 algebraically this mean that the local ring 360 Qx=R[[x1..xn]]/Ix 356 algebraically this mean that the local ring Qx=R[[x1..xn]]/Ix 361 357 where R[[x1,..,xn]] is the ring of germs at 0 of real-valued analytic 362 358 functions on R^n, and Ix is the ideal generated by the components 363 359 of X is infinite dimensional as real vector space. In the case that 364 X has a reduced hypersurface as complex zero es we have the next.360 X has a reduced hypersurface as complex zeros we have the next. 365 361 There exist a real analytic function f:R^n-->R, and a real analytic 366 362 vector field Y s. t. X=fY. The function f does not change of sign … … 371 367 bilinear form <,> obtained by composition of the product in the 372 368 algebra Mx with a linear functional map 373 . L 374 <,> : (Mx)x(Mx)----->Mx----->R 369 <,> : (Mx)x(Mx) ---(.)--> Mx ---(L)--> R 375 370 with L(Jp)>0, where Jp is the residue class of the Jacobian 376 371 determinant of X, JX, over f^n, JX/(f^n) in Mx. Here, we use a 377 372 natural linear functional defined as follows. Suppose that 378 373 E={E_1,..E_r} is a basis of Mx, then Jp is writing as 379 Jp=a_1E_{j1}+...+a_kE_{jk}, js\in {1...r}, s=1..k, k<=r .374 Jp=a_1E_{j1}+...+a_kE_{jk}, js\in {1...r}, s=1..k, k<=r, 380 375 where a_s are constant. The linear functional L:M--->R is defined as 381 376 L(E_{j1})=(a_1)/|a_1|=sign of a_1,
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