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A.1 Milnor and TjurinaThe Milnor number, resp. the Tjurina number, of a power
series f in
![]() respectively ![]() where jacob(f) is the ideal generated by the partials
of f . tjurina(f) is finite, if and only if f has an
isolated singularity. The same holds for milnor(f) if
K has characteristic 0.
SINGULAR displays -1 if the dimension is infinite.
SINGULAR cannot compute with infinite power series. But it can
work in
We shall show in the example below how to realize the following:
option(prot); ring r1 = 32003,(x,y,z),ds; r1; → // characteristic : 32003 → // number of vars : 3 → // block 1 : ordering ds → // : names x y z → // block 2 : ordering C int a,b,c,t=11,5,3,0; poly f = x^a+y^b+z^(3*c)+x^(c+2)*y^(c-1)+x^(c-1)*y^(c-1)*z3+ x^(c-2)*y^c*(y^2+t*x)^2; f; → y5+x5y2+x2y2z3+xy7+z9+x11 ideal i=jacob(f); i; → i[1]=5x4y2+2xy2z3+y7+11x10 → i[2]=5y4+2x5y+2x2yz3+7xy6 → i[3]=3x2y2z2+9z8 ideal j=std(i); → [1023:2]7(2)s8s10s11s12s(3)s13(4)s(5)s14(6)s(7)15--.s(6)-16.-.s(5)17.s(7)\ s--s18(6).--19-..sH(24)20(3)...21....22....23.--24- → product criterion:10 chain criterion:69 "The Milnor number of f(11,5,3) for t=0 is", vdim(j); → The Milnor number of f(11,5,3) for t=0 is 250 j=i+f; // overwrite j j=std(j); → [1023:2]7(3)s8(2)s10s11(3)ss12(4)s(5)s13(6)s(8)s14(9).s(10).15--sH(23)(8)\ ...16......17.......sH(21)(9)sH(20)16(10).17...........18.......19..----.\ .sH(19) → product criterion:10 chain criterion:53 vdim(j); // compute the Tjurina number for t=0 → 195 t=1; f=x^a+y^b+z^(3*c)+x^(c+2)*y^(c-1)+x^(c-1)*y^(c-1)*z3 +x^(c-2)*y^c*(y^2+t*x)^2; ideal i1=jacob(f); ideal j1=std(i1); → [1023:2]7(2)s8s10s11s12s13(3)ss(4)s14(5)s(6)s15(7).....s(8)16.s...s(9)..1\ 7............s18(10).....s(11)..-.19.......sH(24)(10).....20...........21\ ..........22.............................23..............................\ .24.----------.25.26 → product criterion:11 chain criterion:83 "The Milnor number of f(11,5,3) for t=1:",vdim(j1); → The Milnor number of f(11,5,3) for t=1: 248 vdim(std(j1+f)); // compute the Tjurina number for t=1 → [1023:2]7(16)s8(15)s10s11ss(16)-12.s-s13s(17)s(18)s(19)-s(18).-14-s(17)-s\ (16)ss(17)s15(18)..-s...--.16....-.......s(16).sH(23)s(18)...17..........\ 18.........sH(20)17(17)....................18..........19..---....-.-....\ .....20.-----...s17(9).........18..............19..-.......20.-......21..\ .......sH(19)16(5).....18......19.----- → product criterion:15 chain criterion:174 → 195 option(noprot); |
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