id summary reporter owner description type status priority milestone component version resolution keywords cc 474 insufficient develloped from hnexpansion gorzel pfister "The following germ has 4 branches at the origin, with 167 = mu = 2*delta -r + 1 = 2*85 -4+1 {{{ LIB ""hnoether.lib""; ring rds = 0,(y,z),ds; poly f = -y^7-7*y^6*z^2-7*y^6*z^3-21*y^5*z^4+8*y^6*z^4-42*y^5*z^5-35*y^4*z^6+27*y^5*z^6-105*y^4*z^7-35*y^3*z^8+48*y^5*z^7+15*y^4*z^8-140*y^3*z^9-21*y^2*z^10-25*y^5*z^8+205*y^4*z^9-50*y^3*z^10-105*y^2*z^11-7*y*z^12-5*y^4*z^10+340*y^3*z^11-90*y^2*z^12-42*y*z^13-z^14-125*y^4*z^11+195*y^3*z^12+270*y^2*z^13-57*y*z^14-7*z^15+38*y^4*z^12-340*y^3*z^13+365*y^2*z^14+100*y*z^15-13*z^16-98*y^3*z^14-291*y^2*z^15+250*y*z^16+13*z^17+152*y^3*z^15-402*y^2*z^16-62*y*z^17+60*z^18-28*y^3*z^16+206*y^2*z^17-365*y*z^18+14*z^19+144*y^2*z^18+4*y*z^19-99*z^20-84*y^2*z^19+247*y*z^20-51*z^21+8*y^2*z^20-16*y*z^21+83*z^22-68*y*z^22+43*z^23+16*y*z^23-38*z^24-12*z^25+y*z^25+8*z^26+z^27+z^28; vdim(std(jacob(f))); // 167 }}} {{{ with Maple > singularities(f,y,z); {[[0, 0, 1], 7, 85, 4], [[1, 0, 0], 21, 267, 2]} }}} But hnexpansion can not compute it from the given equation {{{ > displayInvariants(f); //** The two HNE's are identical! //** You have either tried to compare a branch with itself, //** or the two branches have been developed separately. // In the latter case use `extdevelop' to extend the HNE's until they differ. //** The two HNE's are identical! --- invariants of branch number 1 : --- characteristic exponents : 1 generators of semigroup : 1 Puiseux pairs : degree of the conductor : 0 delta invariant : 0 sequence of multiplicities: 1 --- invariants of branch number 2 : --- characteristic exponents : 1 generators of semigroup : 1 Puiseux pairs : degree of the conductor : 0 delta invariant : 0 sequence of multiplicities: 1 --- invariants of branch number 3 : --- characteristic exponents : 3,13 generators of semigroup : 3,13 Puiseux pairs : (13,3) degree of the conductor : 24 delta invariant : 12 sequence of multiplicities: 3,3,3,3,1,1,1 // <-- One of them, either this --- invariants of branch number 4 : --- characteristic exponents : 3,13 generators of semigroup : 3,13 Puiseux pairs : (13,3) degree of the conductor : 24 delta invariant : 12 sequence of multiplicities: 3,3,3,3,1,1,1 // <-- or this must be wrong !?? -------------- contact numbers : -------------- branch | 4 3 2 -------+----------------- 1 | 4, 4, 5 2 | 4, 4 3 | -1 // <-- negative contact number ?!? -------------- intersection multiplicities : -------------- branch | 4 3 2 -------+----------------- 1 | 12, 12, 5 2 | 12, 12 3 | -1 // <-- negative intersection number ?!? -------------- delta invariant of the curve : 76 }}} The curve is reducible with a non-linear polynomial, defining a smooth branch at the origin. {{{ > factorize(f,1); _[1]=y+z^2+z^3 _[2]=-y^6-6*y^5*z^2-6*y^5*z^3-15*y^4*z^4+8*y^5*z^4-30*y^4*z^5-20*y^3*z^6+25*y^4*z^6-60*y^3*z^7-15*y^2*z^8+40*y^4*z^7+20*y^3*z^8-60*y^2*z^9-6*y*z^10-25*y^4*z^8+140*y^3*z^9-10*y^2*z^10-30*y*z^11-z^12-20*y^3*z^10+180*y^2*z^11-20*y*z^12-6*z^13-100*y^3*z^11+75*y^2*z^12+100*y*z^13-7*z^14+38*y^3*z^12-220*y^2*z^13+110*y*z^14+20*z^15-36*y^2*z^14-146*y*z^15+40*z^16+114*y^2*z^15-146*y*z^16-26*z^17-28*y^2*z^16+128*y*z^17-73*z^18+58*y*z^18+22*z^19-56*y*z^19+61*z^20+8*y*z^20-18*z^21-20*z^22+8*z^23+z^25 > poly g = _[2]; > displayInvariants(g); --- invariants of branch number 1 : --- characteristic exponents : 3,13 generators of semigroup : 3,13 Puiseux pairs : (13,3) degree of the conductor : 24 delta invariant : 12 sequence of multiplicities: 3,3,3,3,1,1,1 --- invariants of branch number 2 : --- characteristic exponents : 2,9 generators of semigroup : 2,9 Puiseux pairs : (9,2) degree of the conductor : 8 delta invariant : 4 sequence of multiplicities: 2,2,2,2,1,1 // <--- compare with the multsequence above --- invariants of branch number 3 : --- characteristic exponents : 1 generators of semigroup : 1 Puiseux pairs : degree of the conductor : 0 delta invariant : 0 sequence of multiplicities: 1 -------------- contact numbers : -------------- branch | 3 2 -------+----------- 1 | 4, 4 2 | 4 -------------- intersection multiplicities : -------------- branch | 3 2 -------+----------- 1 | 12, 24 2 | 8 -------------- delta invariant of the curve : 60 }}} for the second component, obviously {{{ > poly h =y+z^2+z^3; > displayInvariants(h); --- invariants of branch number 1 : --- characteristic exponents : 1 generators of semigroup : 1 Puiseux pairs : degree of the conductor : 0 delta invariant : 0 sequence of multiplicities: 1 -------------- delta invariant of the curve : 0 }}} " bug new minor 3-2-0 and higher singular-libs 3-1-6 hnexpansion