Image Milnor numbers

Image Milnor numbers for quasihomogeneous germs

The image Milnor number $\mu_I(f)$ of a (finite) quasihomogeneous germ $f:(C^m,0) \to (C^{m+1},0)$ with $m \leq 14$ is determined by its weights $(W_1,\ldots,W_m)$ and degrees $(D_1,\ldots,D_{m+1})$ . In this website we present how the weights and degrees determine the image Milnor number, following

[KR] J. Koncki, R. Rimányi: Higher characteristic classes of multisingularity loci, Section 9.

This work builds on
[Oh] T. Ohmoto. Singularities of maps and characteristic classes. In School on Real and Complex Singularities in São Carlos, 2012, number 68 in Adv. Studies in Pure Math., pages 191-265, 2016, Section 6, and
[PPS] I. Pallarés and G. Peñafort-Sanchis: Image Milnor number formulas for weighted-homogeneous map-germs. Results Math., 76(3): Paper No. 152, 23, 2021.

We present the result in three different ways:

1 : [the most economic way] One (reasonable size) polynomial determines the formulas for all m.

2 : [medium way] A list of 15 polynomials (K-polynomials) determine the 15 formulas, recursively.

3 : [direct way, least economic] Absolutely direct: plug in the weights and the degrees, it gives the image Milnor number. The formulas are huge.

1 :

+t^1 * ( -s[[]] )
+t^2 * ( 1/2*s[[1]] )
+t^3 * ( 7/6*s[[2]] -1/3*s[[1,1]] )
+t^4 * ( s[[3]] -5/4*s[[1,2]] +1/4*s[[1,1,1]] )
+t^5 * ( 29/30*s[[4]] -29/30*s[[1,3]] -31/15*s[[2,2]] +13/10*s[[1,1,2]] -1/5*s[[1,1,1,1]] )
+t^6 * ( -4/3*s[[1,1,1,2]] +19/6*s[[1,2,2]] +11/12*s[[1,1,3]] -3*s[[2,3]] -11/12*s[[1,4]] +s[[5]] +1/6*s[[1,1,1,1,1]] )
+t^7 * ( 19/14*s[[1,1,1,1,2]] -30/7*s[[1,1,2,2]] +127/28*s[[2,2,2]] -6/7*s[[1,1,1,3]] +125/28*s[[1,2,3]] +41/84*s[[3,3]] +6/7*s[[1,1,4]] -289/84*s[[2,4]] -43/42*s[[1,5]] +43/42*s[[6]] -1/7*s[[1,1,1,1,1,1]] )
+t^8 * ( -11/8*s[[1,1,1,1,1,2]] +65/12*s[[1,1,1,2,2]] -69/8*s[[1,2,2,2]] +19/24*s[[1,1,1,1,3]] -43/8*s[[1,1,2,3]] +17/2*s[[2,2,3]] +13/24*s[[1,3,3]] -19/24*s[[1,1,1,4]] +91/24*s[[1,2,4]] +3/2*s[[3,4]] +13/12*s[[1,1,5]] -11/2*s[[2,5]] -13/12*s[[1,6]] +s[[7]] +1/8*s[[1,1,1,1,1,1,1]] )
+t^9 * ( -517/90*s[[2,6]] -29/30*s[[1,7]] +29/30*s[[8]] +25/18*s[[1,1,1,1,1,1,2]] -59/9*s[[1,1,1,1,2,2]] +251/18*s[[1,1,2,2,2]] -511/45*s[[2,2,2,2]] -13/18*s[[1,1,1,1,1,3]] +109/18*s[[1,1,1,2,3]] -509/30*s[[1,2,2,3]] -13/30*s[[1,1,3,3]] -209/45*s[[2,3,3]] +13/18*s[[1,1,1,1,4]] -73/15*s[[1,1,2,4]] +923/90*s[[2,2,4]] -209/90*s[[1,3,4]] -61/45*s[[4,4]] -107/90*s[[1,1,1,5]] +302/45*s[[1,2,5]] -29/30*s[[3,5]] +107/90*s[[1,1,6]] -1/9*s[[1,1,1,1,1,1,1,1]] )
+t^10 * ( -17/20*s[[1,8]] -4*s[[3,3,3]] +s[[9]] -13/20*s[[1,1,1,1,1,4]] +103/20*s[[1,1,1,2,4]] -193/20*s[[1,2,2,4]] +19/20*s[[1,1,3,4]] -15*s[[2,3,4]] -59/10*s[[1,4,4]] +27/20*s[[1,1,1,1,5]] -41/5*s[[1,1,2,5]] +28*s[[2,2,5]] +77/20*s[[1,3,5]] -14*s[[4,5]] -27/20*s[[1,1,1,6]] +247/20*s[[1,2,6]] +7*s[[3,6]] +17/20*s[[1,1,7]] -2*s[[2,7]] +1/10*s[[1,1,1,1,1,1,1,1,1]] -7/5*s[[1,1,1,1,1,1,1,2]] +77/10*s[[1,1,1,1,1,2,2]] -41/2*s[[1,1,1,2,2,2]] +251/10*s[[1,2,2,2,2]] +13/20*s[[1,1,1,1,1,1,3]] -13/2*s[[1,1,1,1,2,3]] +477/20*s[[1,1,2,2,3]] -25*s[[2,2,2,3]] +1/5*s[[1,1,1,3,3]] -31/10*s[[1,2,3,3]] )
+t^11 * ( -549/44*s[[1,1,2,6]] +1451/66*s[[1,3,6]] +1305/44*s[[2,2,6]] +45/11*s[[4,6]] -19/33*s[[1,1,1,7]] +713/132*s[[1,2,7]] +1711/66*s[[3,7]] +19/33*s[[1,1,8]] -571/132*s[[2,8]] -71/66*s[[1,9]] +71/66*s[[10]] +19/33*s[[1,1,1,1,1,1,4]] -677/132*s[[1,1,1,1,2,4]] +2315/132*s[[1,1,2,2,4]] -307/11*s[[2,2,2,4]] -41/22*s[[1,1,1,3,4]] +4681/132*s[[1,2,3,4]] -1387/132*s[[3,3,4]] +161/132*s[[1,1,4,4]] +467/66*s[[2,4,4]] -52/33*s[[1,1,1,1,1,5]] +1723/132*s[[1,1,1,2,5]] -1157/33*s[[1,2,2,5]] +161/66*s[[1,1,3,5]] +3617/132*s[[2,3,5]] +131/66*s[[1,4,5]] -2447/132*s[[5,5]] +52/33*s[[1,1,1,1,6]] -1/11*s[[1,1,1,1,1,1,1,1,1,1]] +31/22*s[[1,1,1,1,1,1,1,1,2]] -292/33*s[[1,1,1,1,1,1,2,2]] +1245/44*s[[1,1,1,1,2,2,2]] -509/11*s[[1,1,2,2,2,2]] +2047/66*s[[2,2,2,2,2]] -19/33*s[[1,1,1,1,1,1,1,3]] +295/44*s[[1,1,1,1,1,2,3]] -673/22*s[[1,1,1,2,2,3]] +2078/33*s[[1,2,2,2,3]] +19/132*s[[1,1,1,1,3,3]] +151/66*s[[1,1,2,3,3]] +890/33*s[[2,2,3,3]] +463/132*s[[1,3,3,3]] )
+t^12 * ( -163/12*s[[1,1,3,6]] +235/6*s[[1,4,6]] -119/2*s[[2,3,6]] +35/3*s[[5,6]] +1/24*s[[1,1,1,1,7]] -559/24*s[[1,1,2,7]] -199/6*s[[2,2,7]] -407/12*s[[1,3,7]] +89/3*s[[4,7]] -1/24*s[[1,1,1,8]] -253/8*s[[1,2,8]] -53/6*s[[3,8]] +17/12*s[[1,1,9]] -53/2*s[[2,9]] -17/12*s[[1,10]] +s[[11]] +713/24*s[[1,3,3,4]] -37/24*s[[1,1,1,4,4]] +881/12*s[[1,2,4,4]] -1/3*s[[3,4,4]] +15/8*s[[1,1,1,1,1,1,5]] -433/24*s[[1,1,1,1,2,5]] +87/2*s[[1,1,2,2,5]] -833/6*s[[2,2,2,5]] +45/8*s[[1,1,1,3,5]] -1259/24*s[[1,2,3,5]] +11*s[[3,3,5]] +149/12*s[[1,1,4,5]] +299/2*s[[2,4,5]] +841/24*s[[1,5,5]] -15/8*s[[1,1,1,1,1,6]] +14*s[[1,1,1,2,6]] -2741/24*s[[1,2,2,6]] +1/2*s[[1,1,1,1,1,1,1,1,3]] -20/3*s[[1,1,1,1,1,1,2,3]] +293/8*s[[1,1,1,1,2,2,3]] -595/6*s[[1,1,2,2,2,3]] +461/6*s[[2,2,2,2,3]] -7/12*s[[1,1,1,1,1,3,3]] +2/3*s[[1,1,1,2,3,3]] +33/2*s[[1,2,2,3,3]] -65/24*s[[1,1,3,3,3]] +235/6*s[[2,3,3,3]] -1/2*s[[1,1,1,1,1,1,1,4]] +115/24*s[[1,1,1,1,1,2,4]] -223/12*s[[1,1,1,2,2,4]] +23/3*s[[1,2,2,2,4]] +73/24*s[[1,1,1,1,3,4]] -647/24*s[[1,1,2,3,4]] +296/3*s[[2,2,3,4]] +1/12*s[[1,1,1,1,1,1,1,1,1,1,1]] -17/12*s[[1,1,1,1,1,1,1,1,1,2]] +10*s[[1,1,1,1,1,1,1,2,2]] -112/3*s[[1,1,1,1,1,2,2,2]] +917/12*s[[1,1,1,2,2,2,2]] -923/12*s[[1,2,2,2,2,2]] )
+t^13 * ( -905251/5460*s[[4,8]] +36383/2730*s[[1,2,9]] -6589/2730*s[[1,1,1,9]] -283229/2730*s[[3,9]] +6589/2730*s[[1,1,10]] -5724/455*s[[2,10]] -2039/2730*s[[1,11]] +2039/2730*s[[12]] -2240851/5460*s[[1,2,3,6]] -20471/455*s[[3,3,6]] -299287/5460*s[[1,1,4,6]] -152757/1820*s[[2,4,6]] +87823/455*s[[1,5,6]] +531611/5460*s[[6,6]] +242/273*s[[1,1,1,1,1,7]] -3599/1092*s[[1,1,1,2,7]] -18847/1365*s[[1,2,2,7]] -589447/5460*s[[1,1,3,7]] -1140247/2730*s[[2,3,7]] -1149091/5460*s[[1,4,7]] +93511/2730*s[[5,7]] -242/273*s[[1,1,1,1,8]] +10391/1820*s[[1,1,2,8]] +437/910*s[[2,2,8]] -1176391/5460*s[[1,3,8]] +1289/52*s[[1,1,1,1,1,2,5]] -9017/91*s[[1,1,1,2,2,5]] +46157/273*s[[1,2,2,2,5]] -9109/1092*s[[1,1,1,1,3,5]] -8009/156*s[[1,1,2,3,5]] -388762/1365*s[[2,2,3,5]] -26171/780*s[[1,3,3,5]] +3359/364*s[[1,1,1,4,5]] -1426/65*s[[1,2,4,5]] +210449/5460*s[[3,4,5]] +157163/5460*s[[1,1,5,5]] +1286723/5460*s[[2,5,5]] +88/39*s[[1,1,1,1,1,1,6]] -28037/1092*s[[1,1,1,1,2,6]] +111803/1092*s[[1,1,2,2,6]] -70646/455*s[[2,2,2,6]] -5377/364*s[[1,1,1,3,6]] -456/455*s[[3,3,3,3]] +11/26*s[[1,1,1,1,1,1,1,1,4]] -161/39*s[[1,1,1,1,1,1,2,4]] +863/52*s[[1,1,1,1,2,2,4]] -3765/91*s[[1,1,2,2,2,4]] +36527/546*s[[2,2,2,2,4]] -58/13*s[[1,1,1,1,1,3,4]] +8249/182*s[[1,1,1,2,3,4]] -4003/13*s[[1,2,2,3,4]] -12245/364*s[[1,1,3,3,4]] +7087/65*s[[2,3,3,4]] +488/273*s[[1,1,1,1,4,4]] -1999/78*s[[1,1,2,4,4]] -28891/1365*s[[2,2,4,4]] +95621/2730*s[[1,3,4,4]] +40823/5460*s[[4,4,4]] -88/39*s[[1,1,1,1,1,1,1,5]] +37/26*s[[1,1,1,1,1,1,1,1,1,1,2]] -145/13*s[[1,1,1,1,1,1,1,1,2,2]] +619/13*s[[1,1,1,1,1,1,2,2,2]] -1522/13*s[[1,1,1,1,2,2,2,2]] +4089/26*s[[1,1,2,2,2,2,2]] -8191/91*s[[2,2,2,2,2,2]] -11/26*s[[1,1,1,1,1,1,1,1,1,3]] +83/13*s[[1,1,1,1,1,1,1,2,3]] -538/13*s[[1,1,1,1,1,2,2,3]] +1826/13*s[[1,1,1,2,2,2,3]] -42947/182*s[[1,2,2,2,2,3]] +43/39*s[[1,1,1,1,1,1,3,3]] -147/26*s[[1,1,1,1,2,3,3]] -1416/91*s[[1,1,2,2,3,3]] -35753/273*s[[2,2,2,3,3]] +753/182*s[[1,1,1,3,3,3]] -3551/91*s[[1,2,3,3,3]] -1/13*s[[1,1,1,1,1,1,1,1,1,1,1,1]] )
+t^14 * ( -167/35*s[[1,1,1,10]] +44216/105*s[[1,2,10]] +842/3*s[[3,10]] -271/420*s[[1,1,11]] +169*s[[2,11]] +271/420*s[[1,12]] +s[[13]] -161*s[[3,3,7]] -99993/280*s[[1,1,4,7]] -4097/6*s[[2,4,7]] -64257/140*s[[1,5,7]] +818/3*s[[6,7]] +333/140*s[[1,1,1,1,1,8]] +42181/420*s[[1,1,1,2,8]] +110863/140*s[[1,2,2,8]] +74167/280*s[[1,1,3,8]] +579/2*s[[2,3,8]] -537379/840*s[[1,4,8]] -3263/6*s[[5,8]] +167/35*s[[1,1,1,1,9]] +125807/420*s[[1,1,2,9]] +521*s[[2,2,9]] +190549/420*s[[1,3,9]] -659/2*s[[4,9]] +31473/35*s[[1,2,2,2,6]] -2377/168*s[[1,1,1,1,3,6]] +85147/280*s[[1,1,2,3,6]] +693/2*s[[2,2,3,6]] -5602/35*s[[1,3,3,6]] -62683/840*s[[1,1,1,4,6]] -606359/840*s[[1,2,4,6]] -425/2*s[[3,4,6]] -12682/105*s[[1,1,5,6]] -141*s[[2,5,6]] +159779/840*s[[1,6,6]] -333/140*s[[1,1,1,1,1,1,7]] +2923/84*s[[1,1,1,1,2,7]] +39961/105*s[[1,1,2,2,7]] +953/2*s[[2,2,2,7]] +5569/280*s[[1,1,1,3,7]] +92149/140*s[[1,2,3,7]] +1341/2*s[[2,2,2,2,5]] +9887/840*s[[1,1,1,1,1,3,5]] -923/12*s[[1,1,1,2,3,5]] +32509/70*s[[1,2,2,3,5]] -1993/40*s[[1,1,3,3,5]] -369/2*s[[2,3,3,5]] -2377/168*s[[1,1,1,1,4,5]] -6017/30*s[[1,1,2,4,5]] -6421/6*s[[2,2,4,5]] -136039/840*s[[1,3,4,5]] +341/6*s[[4,4,5]] +4363/210*s[[1,1,1,5,5]] -380573/840*s[[1,2,5,5]] +34*s[[3,5,5]] -191/70*s[[1,1,1,1,1,1,1,6]] +15181/420*s[[1,1,1,1,1,2,6]] -8459/84*s[[1,1,1,2,2,6]] +61/10*s[[1,1,1,1,1,1,3,4]] -12373/168*s[[1,1,1,1,2,3,4]] +306*s[[1,1,2,2,3,4]] -3281/6*s[[2,2,2,3,4]] +2621/84*s[[1,1,1,3,3,4]] -2096/5*s[[1,2,3,3,4]] -115/3*s[[3,3,3,4]] -521/280*s[[1,1,1,1,1,4,4]] +75/2*s[[1,1,1,2,4,4]] -39883/70*s[[1,2,2,4,4]] -8447/140*s[[1,1,3,4,4]] -26*s[[2,3,4,4]] +2889/280*s[[1,4,4,4]] +191/70*s[[1,1,1,1,1,1,1,1,5]] -1013/30*s[[1,1,1,1,1,1,2,5]] +3316/21*s[[1,1,1,1,2,2,5]] -8077/42*s[[1,1,2,2,2,5]] +443/10*s[[1,1,1,1,1,1,2,2,3]] -550/3*s[[1,1,1,1,2,2,2,3]] +11419/28*s[[1,1,2,2,2,2,3]] -245*s[[2,2,2,2,2,3]] -59/35*s[[1,1,1,1,1,1,1,3,3]] +193/15*s[[1,1,1,1,1,2,3,3]] -109/21*s[[1,1,1,2,2,3,3]] -1951/21*s[[1,2,2,2,3,3]] -955/168*s[[1,1,1,1,3,3,3]] +134/7*s[[1,1,2,3,3,3]] -1549/6*s[[2,2,3,3,3]] -1801/105*s[[1,3,3,3,3]] -29/84*s[[1,1,1,1,1,1,1,1,1,4]] +219/70*s[[1,1,1,1,1,1,1,2,4]] -158/15*s[[1,1,1,1,1,2,2,4]] +178/7*s[[1,1,1,2,2,2,4]] +8777/84*s[[1,2,2,2,2,4]] +1/14*s[[1,1,1,1,1,1,1,1,1,1,1,1,1]] -10/7*s[[1,1,1,1,1,1,1,1,1,1,1,2]] +517/42*s[[1,1,1,1,1,1,1,1,1,2,2]] -414/7*s[[1,1,1,1,1,1,1,2,2,2]] +849/5*s[[1,1,1,1,1,2,2,2,2]] -856/3*s[[1,1,1,2,2,2,2,2]] +3431/14*s[[1,2,2,2,2,2,2]] +29/84*s[[1,1,1,1,1,1,1,1,1,1,3]] -41/7*s[[1,1,1,1,1,1,1,1,2,3]] )
+t^15 * ( -2797/180*s[[1,2,11]] -248/45*s[[1,1,12]] +7559/6*s[[3,11]] +3187/180*s[[2,12]] -13/6*s[[1,13]] +13/6*s[[14]] +306877/180*s[[1,1,4,8]] +346877/120*s[[2,4,8]] -14787/20*s[[1,5,8]] -54209/60*s[[6,8]] -869/90*s[[1,1,1,1,1,9]] +3241/45*s[[1,1,1,2,9]] -13843/90*s[[1,2,2,9]] +407857/180*s[[1,1,3,9]] +580943/180*s[[2,3,9]] +110759/36*s[[1,4,9]] -1747/6*s[[5,9]] +869/90*s[[1,1,1,1,10]] -1163/15*s[[1,1,2,10]] +3387/20*s[[2,2,10]] +100067/36*s[[1,3,10]] +66439/36*s[[4,10]] +248/45*s[[1,1,1,11]] +937589/360*s[[1,1,2,3,7]] +198796/45*s[[2,2,3,7]] +372973/360*s[[1,3,3,7]] +14977/60*s[[1,1,1,4,7]] +438031/120*s[[1,2,4,7]] -11065/36*s[[3,4,7]] -25097/36*s[[1,1,5,7]] -171817/180*s[[2,5,7]] -25847/20*s[[1,6,7]] -61/12*s[[7,7]] -209/45*s[[1,1,1,1,1,1,8]] +9751/180*s[[1,1,1,1,2,8]] -17789/90*s[[1,1,2,2,8]] +7747/40*s[[2,2,2,8]] +48577/60*s[[1,1,1,3,8]] +1984049/360*s[[1,2,3,8]] +28633/40*s[[3,3,8]] +53717/120*s[[1,1,3,3,6]] +99671/120*s[[2,3,3,6]] -491/45*s[[1,1,1,1,4,6]] +331141/360*s[[1,1,2,4,6]] +314617/360*s[[2,2,4,6]] -25813/90*s[[1,3,4,6]] -17233/180*s[[4,4,6]] -1021/6*s[[1,1,1,5,6]] -187189/60*s[[1,2,5,6]] -8707/20*s[[3,5,6]] -85981/180*s[[1,1,6,6]] -479753/360*s[[2,6,6]] +209/45*s[[1,1,1,1,1,1,1,7]] -2671/60*s[[1,1,1,1,1,2,7]] +3769/30*s[[1,1,1,2,2,7]] -14351/360*s[[1,2,2,2,7]] +28931/180*s[[1,1,1,1,3,7]] +9277/60*s[[1,2,2,4,5]] -1949/15*s[[1,1,3,4,5]] -65849/120*s[[2,3,4,5]] +1061/10*s[[1,4,4,5]] -2857/90*s[[1,1,1,1,5,5]] -20284/45*s[[1,1,2,5,5]] -352873/180*s[[2,2,5,5]] -137329/360*s[[1,3,5,5]] +20039/72*s[[4,5,5]] +33/10*s[[1,1,1,1,1,1,1,1,6]] -4513/90*s[[1,1,1,1,1,1,2,6]] +3467/12*s[[1,1,1,1,2,2,6]] -91141/120*s[[1,1,2,2,2,6]] +294641/360*s[[2,2,2,2,6]] +617/30*s[[1,1,1,1,1,3,6]] +35077/120*s[[1,1,1,2,3,6]] +549563/120*s[[1,2,2,3,6]] -2009/20*s[[3,3,4,4]] +1205/24*s[[1,1,4,4,4]] -717/8*s[[2,4,4,4]] -33/10*s[[1,1,1,1,1,1,1,1,1,5]] +91/2*s[[1,1,1,1,1,1,1,2,5]] -1222/5*s[[1,1,1,1,1,2,2,5]] +5071/8*s[[1,1,1,2,2,2,5]] -9343/12*s[[1,2,2,2,2,5]] -1433/90*s[[1,1,1,1,1,1,3,5]] +3283/24*s[[1,1,1,1,2,3,5]] +12521/20*s[[1,1,2,2,3,5]] +767029/360*s[[2,2,2,3,5]] +7349/120*s[[1,1,1,3,3,5]] +5761/8*s[[1,2,3,3,5]] +647/40*s[[3,3,3,5]] +617/30*s[[1,1,1,1,1,4,5]] -5099/40*s[[1,1,1,2,4,5]] +64/3*s[[1,1,1,1,2,2,2,4]] -47/8*s[[1,1,2,2,2,2,4]] -14069/120*s[[2,2,2,2,2,4]] -119/15*s[[1,1,1,1,1,1,1,3,4]] +1663/15*s[[1,1,1,1,1,2,3,4]] -12425/24*s[[1,1,1,2,2,3,4]] +50591/24*s[[1,2,2,2,3,4]] -253/6*s[[1,1,1,1,3,3,4]] +22933/40*s[[1,1,2,3,3,4]] -87203/120*s[[2,2,3,3,4]] -1871/40*s[[1,3,3,3,4]] +74/45*s[[1,1,1,1,1,1,4,4]] -1043/24*s[[1,1,1,1,2,4,4]] +10391/40*s[[1,1,2,2,4,4]] +4931/180*s[[2,2,2,4,4]] +5311/120*s[[1,1,1,3,4,4]] -28049/60*s[[1,2,3,4,4]] -224/5*s[[1,1,1,1,1,1,1,2,2,3]] +3346/15*s[[1,1,1,1,1,2,2,2,3]] -628*s[[1,1,1,2,2,2,2,3]] +35833/40*s[[1,2,2,2,2,2,3]] +139/60*s[[1,1,1,1,1,1,1,1,3,3]] -112/5*s[[1,1,1,1,1,1,2,3,3]] +46*s[[1,1,1,1,2,2,3,3]] +531/4*s[[1,1,2,2,2,3,3]] +14081/24*s[[2,2,2,2,3,3]] +36/5*s[[1,1,1,1,1,3,3,3]] -269/8*s[[1,1,1,2,3,3,3]] +6707/24*s[[1,2,2,3,3,3]] -461/120*s[[1,1,3,3,3,3]] +1327/120*s[[2,3,3,3,3]] +4/15*s[[1,1,1,1,1,1,1,1,1,1,4]] -107/60*s[[1,1,1,1,1,1,1,1,2,4]] -7/10*s[[1,1,1,1,1,1,2,2,4]] -1/15*s[[1,1,1,1,1,1,1,1,1,1,1,1,1,1]] +43/30*s[[1,1,1,1,1,1,1,1,1,1,1,1,2]] -202/15*s[[1,1,1,1,1,1,1,1,1,1,2,2]] +863/12*s[[1,1,1,1,1,1,1,1,2,2,2]] -3542/15*s[[1,1,1,1,1,1,2,2,2,2]] +7147/15*s[[1,1,1,1,2,2,2,2,2]] -8188/15*s[[1,1,2,2,2,2,2,2]] +32767/120*s[[2,2,2,2,2,2,2]] -4/15*s[[1,1,1,1,1,1,1,1,1,1,1,3]] +61/12*s[[1,1,1,1,1,1,1,1,1,2,3]] )

2 :
With K_m-polynomials, see [KR]

s[[]]/1! (
1
)

s[[]]/2! (
-s[[]] -c[1] )

s[[]]/3! (
2*c[1]^2 +3*c[1]*s[[]] +s[[]]^2 -7*c[2] )

s[[]]/4! (
-6*c[1]^3 -11*c[1]^2*s[[]] -6*c[1]*s[[]]^2 -s[[]]^3 +30*c[1]*c[2] +28*c[2]*s[[]] -24*c[3] )

s[[]]/5! (
24*c[1]^4 +50*c[1]^3*s[[]] +35*c[1]^2*s[[]]^2 +10*c[1]*s[[]]^3 +s[[]]^4 -156*c[1]^2*c[2] -220*c[1]*c[2]*s[[]] -70*c[2]*s[[]]^2 +116*c[1]*c[3] +248*c[2]^2 +120*c[3]*s[[]] -116*c[4] )

s[[]]/6! (
-120*c[1]^5 -274*c[1]^4*s[[]] -225*c[1]^3*s[[]]^2 -85*c[1]^2*s[[]]^3 -15*c[1]*s[[]]^4 -s[[]]^5 +960*c[1]^3*c[2] +1666*c[1]^2*c[2]*s[[]] +870*c[1]*c[2]*s[[]]^2 +140*c[2]*s[[]]^3 -660*c[1]^2*c[3] -2280*c[1]*c[2]^2 -1056*c[1]*c[3]*s[[]] -1978*c[2]^2*s[[]] -360*c[3]*s[[]]^2 +660*c[1]*c[4] +2160*c[2]*c[3] +696*c[4]*s[[]] -720*c[5] )

s[[]]/7! (
720*c[1]^6 +1764*c[1]^5*s[[]] +1624*c[1]^4*s[[]]^2 +735*c[1]^3*s[[]]^3 +175*c[1]^2*s[[]]^4 +21*c[1]*s[[]]^5 +s[[]]^6 -6840*c[1]^4*c[2] -13566*c[1]^3*c[2]*s[[]] -9121*c[1]^2*c[2]*s[[]]^2 -2520*c[1]*c[2]*s[[]]^3 -245*c[2]*s[[]]^4 +4320*c[1]^3*c[3] +21600*c[1]^2*c[2]^2 +8736*c[1]^2*c[3]*s[[]] +28518*c[1]*c[2]^2*s[[]] +4956*c[1]*c[3]*s[[]]^2 +8638*c[2]^2*s[[]]^2 +840*c[3]*s[[]]^3 -4320*c[1]^2*c[4] -22500*c[1]*c[2]*c[3] -7056*c[1]*c[4]*s[[]] -22860*c[2]^3 -21000*c[2]*c[3]*s[[]] -2436*c[4]*s[[]]^2 +5160*c[1]*c[5] +17340*c[2]*c[4] -2460*c[3]^2 +5040*c[5]*s[[]] -5160*c[6] )

s[[]]/8! (
-5040*c[1]^7 -13068*c[1]^6*s[[]] -13132*c[1]^5*s[[]]^2 -6769*c[1]^4*s[[]]^3 -1960*c[1]^3*s[[]]^4 -322*c[1]^2*s[[]]^5 -28*c[1]*s[[]]^6 -s[[]]^7 +55440*c[1]^5*c[2] +121080*c[1]^4*c[2]*s[[]] +95788*c[1]^3*c[2]*s[[]]^2 +35056*c[1]^2*c[2]*s[[]]^3 +6020*c[1]*c[2]*s[[]]^4 +392*c[2]*s[[]]^5 -31920*c[1]^4*c[3] -218400*c[1]^3*c[2]^2 -76112*c[1]^3*c[3]*s[[]] -357068*c[1]^2*c[2]^2*s[[]] -56448*c[1]^2*c[3]*s[[]]^2 -178024*c[1]*c[2]^2*s[[]]^2 -16576*c[1]*c[3]*s[[]]^3 -27608*c[2]^2*s[[]]^3 -1680*c[3]*s[[]]^4 +31920*c[1]^3*c[4] +216720*c[1]^2*c[2]*c[3] +66032*c[1]^2*c[4]*s[[]] +347760*c[1]*c[2]^3 +336352*c[1]*c[2]*c[3]*s[[]] +37968*c[1]*c[4]*s[[]]^2 +280096*c[2]^3*s[[]] +107520*c[2]*c[3]*s[[]]^2 +6496*c[4]*s[[]]^3 -43680*c[1]^2*c[5] -152880*c[1]*c[2]*c[4] -21840*c[1]*c[3]^2 -61440*c[1]*c[5]*s[[]] -342720*c[2]^2*c[3] -184192*c[2]*c[4]*s[[]] -480*c[3]^2*s[[]] -20160*c[5]*s[[]]^2 +43680*c[1]*c[6] +221760*c[2]*c[5] -60480*c[3]*c[4] +41280*c[6]*s[[]] -40320*c[7] )

s[[]]/9! (
40320*c[1]^8 +109584*c[1]^7*s[[]] +118124*c[1]^6*s[[]]^2 +67284*c[1]^5*s[[]]^3 +22449*c[1]^4*s[[]]^4 +4536*c[1]^3*s[[]]^5 +546*c[1]^2*s[[]]^6 +36*c[1]*s[[]]^7 +s[[]]^8 -504000*c[1]^6*c[2] -1185696*c[1]^5*c[2]*s[[]] -1054488*c[1]^4*c[2]*s[[]]^2 -463176*c[1]^3*c[2]*s[[]]^3 -107436*c[1]^2*c[2]*s[[]]^4 -12600*c[1]*c[2]*s[[]]^5 -588*c[2]*s[[]]^6 +262080*c[1]^5*c[3] +2378880*c[1]^4*c[2]^2 +713952*c[1]^4*c[3]*s[[]] +4467888*c[1]^3*c[2]^2*s[[]] +633816*c[1]^3*c[3]*s[[]]^2 +2876676*c[1]^2*c[2]^2*s[[]]^2 +248976*c[1]^2*c[3]*s[[]]^3 +767088*c[1]*c[2]^2*s[[]]^3 +44856*c[1]*c[3]*s[[]]^4 +72408*c[2]^2*s[[]]^4 +3024*c[3]*s[[]]^5 -262080*c[1]^4*c[4] -2197440*c[1]^3*c[2]*c[3] -641376*c[1]^3*c[4]*s[[]] -5060160*c[1]^2*c[2]^3 -4421664*c[1]^2*c[2]*c[3]*s[[]] -482616*c[1]^2*c[4]*s[[]]^2 -6230880*c[1]*c[2]^3*s[[]] -2428848*c[1]*c[2]*c[3]*s[[]]^2 -143136*c[1]*c[4]*s[[]]^3 -1793944*c[2]^3*s[[]]^2 -393120*c[2]*c[3]*s[[]]^3 -14616*c[4]*s[[]]^4 +431424*c[1]^3*c[5] +1766016*c[1]^2*c[2]*c[4] +157248*c[1]^2*c[3]^2 +699840*c[1]^2*c[5]*s[[]] +6156864*c[1]*c[2]^2*c[3] +2826720*c[1]*c[2]*c[4]*s[[]] +458784*c[1]*c[3]^2*s[[]] +367200*c[1]*c[5]*s[[]]^2 +4120704*c[2]^4 +5104512*c[2]^2*c[3]*s[[]] +1033488*c[2]*c[4]*s[[]]^2 +92880*c[3]^2*s[[]]^2 +60480*c[5]*s[[]]^3 -431424*c[1]^2*c[6] -2435328*c[1]*c[2]*c[5] +842688*c[1]*c[3]*c[4] -578880*c[1]*c[6]*s[[]] -3721536*c[2]^2*c[4] +1685376*c[2]*c[3]^2 -2419200*c[2]*c[5]*s[[]] +193536*c[3]*c[4]*s[[]] -185760*c[6]*s[[]]^2 +350784*c[1]*c[7] +2084544*c[2]*c[6] +350784*c[3]*c[5] +491904*c[4]^2 +362880*c[7]*s[[]] -350784*c[8] )

s[[]]/10! (
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s[[]]/11! (
3628800*c[1]^10 +10628640*c[1]^9*s[[]] +12753576*c[1]^8*s[[]]^2 +8409500*c[1]^7*s[[]]^3 +3416930*c[1]^6*s[[]]^4 +902055*c[1]^5*s[[]]^5 +157773*c[1]^4*s[[]]^6 +18150*c[1]^3*s[[]]^7 +1320*c[1]^2*s[[]]^8 +55*c[1]*s[[]]^9 +s[[]]^10 -56246400*c[1]^8*c[2] -147684240*c[1]^7*c[2]*s[[]] -154472868*c[1]^6*c[2]*s[[]]^2 -85675260*c[1]^5*c[2]*s[[]]^3 -27913215*c[1]^4*c[2]*s[[]]^4 -5520900*c[1]^3*c[2]*s[[]]^5 -651882*c[1]^2*c[2]*s[[]]^6 -42240*c[1]*c[2]*s[[]]^7 -1155*c[2]*s[[]]^8 +22982400*c[1]^7*c[3] +353203200*c[1]^6*c[2]^2 +78899040*c[1]^6*c[3]*s[[]] +793005840*c[1]^5*c[2]^2*s[[]] +89797488*c[1]^5*c[3]*s[[]]^2 +678915600*c[1]^4*c[2]^2*s[[]]^2 +49412880*c[1]^4*c[3]*s[[]]^3 +288792240*c[1]^3*c[2]^2*s[[]]^3 +14851320*c[1]^3*c[3]*s[[]]^4 +65164440*c[1]^2*c[2]^2*s[[]]^4 +2494800*c[1]^2*c[3]*s[[]]^5 +7461300*c[1]*c[2]^2*s[[]]^5 +219912*c[1]*c[3]*s[[]]^6 +340956*c[2]^2*s[[]]^6 +7920*c[3]*s[[]]^7 -22982400*c[1]^6*c[4] -267624000*c[1]^5*c[2]*c[3] -73196640*c[1]^5*c[4]*s[[]] -1129464000*c[1]^4*c[2]^3 -753160320*c[1]^4*c[2]*c[3]*s[[]] -75826608*c[1]^4*c[4]*s[[]]^2 -1997035920*c[1]^3*c[2]^3*s[[]] -663432792*c[1]^3*c[2]*c[3]*s[[]]^2 -36550800*c[1]^3*c[4]*s[[]]^3 -1227336396*c[1]^2*c[2]^3*s[[]]^2 -255443760*c[1]^2*c[2]*c[3]*s[[]]^3 -9030120*c[1]^2*c[4]*s[[]]^4 -315298720*c[1]*c[2]^3*s[[]]^3 -44865480*c[1]*c[2]*c[3]*s[[]]^4 -1108800*c[1]*c[4]*s[[]]^5 -28866640*c[2]^3*s[[]]^4 -2938320*c[2]*c[3]*s[[]]^5 -53592*c[4]*s[[]]^6 +62899200*c[1]^5*c[5] +204724800*c[1]^4*c[2]*c[4] -5745600*c[1]^4*c[3]^2 +110230560*c[1]^4*c[5]*s[[]] +1221091200*c[1]^3*c[2]^2*c[3] +566137440*c[1]^3*c[2]*c[4]*s[[]] +88553520*c[1]^3*c[3]^2*s[[]] +80712720*c[1]^3*c[5]*s[[]]^2 +1847059200*c[1]^2*c[2]^4 +2401359840*c[1]^2*c[2]^2*c[3]*s[[]] +432034152*c[1]^2*c[2]*c[4]*s[[]]^2 +91715976*c[1]^2*c[3]^2*s[[]]^2 +29066400*c[1]^2*c[5]*s[[]]^3 +2117760480*c[1]*c[2]^4*s[[]] +1267583856*c[1]*c[2]^2*c[3]*s[[]]^2 +127527840*c[1]*c[2]*c[4]*s[[]]^3 +28005120*c[1]*c[3]^2*s[[]]^3 +5029200*c[1]*c[5]*s[[]]^4 +579251904*c[2]^4*s[[]]^2 +194816160*c[2]^2*c[3]*s[[]]^3 +13225080*c[2]*c[4]*s[[]]^4 +2514600*c[3]^2*s[[]]^4 +332640*c[5]*s[[]]^5 -62899200*c[1]^4*c[6] -521035200*c[1]^3*c[2]*c[5] +74390400*c[1]^3*c[3]*c[4] -102247200*c[1]^3*c[6]*s[[]] -700056000*c[1]^2*c[2]^2*c[4] -91324800*c[1]^2*c[2]*c[3]^2 -687867840*c[1]^2*c[2]*c[5]*s[[]] -728640*c[1]^2*c[3]*c[4]*s[[]] -64080720*c[1]^2*c[6]*s[[]]^2 -2513548800*c[1]*c[2]^3*c[3] -1135933920*c[1]*c[2]^2*c[4]*s[[]] -325876320*c[1]*c[2]*c[3]^2*s[[]] -364573440*c[1]*c[2]*c[5]*s[[]]^2 -16898112*c[1]*c[3]*c[4]*s[[]]^2 -17424000*c[1]*c[6]*s[[]]^3 -1238025600*c[2]^5 -1822296960*c[2]^3*c[3]*s[[]] -470911056*c[2]^2*c[4]*s[[]]^2 -27609120*c[2]*c[3]^2*s[[]]^2 -59875200*c[2]*c[5]*s[[]]^3 -9313920*c[3]*c[4]*s[[]]^3 -1702800*c[6]*s[[]]^4 +22982400*c[1]^3*c[7] +498052800*c[1]^2*c[2]*c[6] -97372800*c[1]^2*c[3]*c[5] -48686400*c[1]^2*c[4]^2 +66528000*c[1]^2*c[7]*s[[]] +1399507200*c[1]*c[2]^2*c[5] -1415534400*c[1]*c[2]*c[3]*c[4] +709156800*c[1]*c[2]*c[6]*s[[]] -140011200*c[1]*c[3]^3 +252426240*c[1]*c[3]*c[5]*s[[]] -173083680*c[1]*c[4]^2*s[[]] +39251520*c[1]*c[7]*s[[]]^2 +1114041600*c[2]^3*c[4] -1076544000*c[2]^2*c[3]^2 +1456297920*c[2]^2*c[5]*s[[]] -415514880*c[2]*c[3]*c[4]*s[[]] +162328320*c[2]*c[6]*s[[]]^2 -179150400*c[3]^3*s[[]] +59209920*c[3]*c[5]*s[[]]^2 +45704736*c[4]^2*s[[]]^2 +6652800*c[7]*s[[]]^3 -22982400*c[1]^2*c[8] -215611200*c[1]*c[2]*c[7] -877564800*c[1]*c[3]*c[6] -79228800*c[1]*c[4]*c[5] -53222400*c[1]*c[8]*s[[]] -1183896000*c[2]^2*c[6] -1093780800*c[2]*c[3]*c[5] -282441600*c[2]*c[4]^2 -126403200*c[2]*c[7]*s[[]] +419428800*c[3]^2*c[4] +238550400*c[3]*c[6]*s[[]] -597421440*c[4]*c[5]*s[[]] -19293120*c[8]*s[[]]^2 +42940800*c[1]*c[9] +172670400*c[2]*c[8] -1034812800*c[3]*c[7] -163296000*c[4]*c[6] +739972800*c[5]^2 +39916800*c[9]*s[[]] -42940800*c[10] )

s[[]]/12! (
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s[[]]/13! (
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s[[]]/14! (
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s[[]]/15! (
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-1070262454060800*c[2]^4*c[6] -2786178230035200*c[2]^3*c[3]*c[5] -35823012825600*c[2]^3*c[4]^2 +662370278169600*c[2]^3*c[7]*s[[]] +950276065939200*c[2]^2*c[3]^2*c[4] +490101351667200*c[2]^2*c[3]*c[6]*s[[]] -1738895039481600*c[2]^2*c[4]*c[5]*s[[]] -10386171250560*c[2]^2*c[8]*s[[]]^2 -14460699052800*c[2]*c[3]^4 -288088352697600*c[2]*c[3]^2*c[5]*s[[]] -22351843987200*c[2]*c[3]*c[4]^2*s[[]] +304576621977600*c[2]*c[3]*c[7]*s[[]]^2 +47741819662080*c[2]*c[4]*c[6]*s[[]]^2 -190322373427200*c[2]*c[5]^2*s[[]]^2 -6538371840000*c[2]*c[9]*s[[]]^3 -32288093510400*c[3]^3*c[4]*s[[]] +38554707628800*c[3]^2*c[6]*s[[]]^2 -44033315800320*c[3]*c[4]*c[5]*s[[]]^2 -2557229875200*c[3]*c[8]*s[[]]^3 -6798977894016*c[4]^3*s[[]]^2 +5833680652800*c[4]*c[7]*s[[]]^3 +1873295424000*c[5]*c[6]*s[[]]^3 -58614192000*c[10]*s[[]]^4 -7206738739200*c[1]^3*c[11] +101388352665600*c[1]^2*c[2]*c[10] -2963023026163200*c[1]^2*c[3]*c[9] -2229417705715200*c[1]^2*c[4]*c[8] +911630655936000*c[1]^2*c[5]*c[7] +624639721305600*c[1]^2*c[6]^2 +80472268800*c[1]^2*c[11]*s[[]] +201134847513600*c[1]*c[2]^2*c[9] -7206916728211200*c[1]*c[2]*c[3]*c[8] -4773349259078400*c[1]*c[2]*c[4]*c[7] +4079704287859200*c[1]*c[2]*c[5]*c[6] +562813039756800*c[1]*c[2]*c[10]*s[[]] -1354797866822400*c[1]*c[3]^2*c[7] +375055538457600*c[1]*c[3]*c[4]*c[6] +498837814675200*c[1]*c[3]*c[5]^2 +663780691123200*c[1]*c[3]*c[9]*s[[]] -138744250444800*c[1]*c[4]^2*c[5] -725930166044160*c[1]*c[4]*c[8]*s[[]] -619987577932800*c[1]*c[5]*c[7]*s[[]] +186526536134400*c[1]*c[6]^2*s[[]] +1142179315200*c[1]*c[11]*s[[]]^2 -253263833222400*c[2]^3*c[8] -5776898525798400*c[2]^2*c[3]*c[7] -1142823851769600*c[2]^2*c[4]*c[6] +2563572095884800*c[2]^2*c[5]^2 +724429805299200*c[2]^2*c[9]*s[[]] -1086143432774400*c[2]*c[3]^2*c[6] +717575412153600*c[2]*c[3]*c[4]*c[5] +401496978355200*c[2]*c[3]*c[8]*s[[]] +117200315232000*c[2]*c[4]^3 -931156517491200*c[2]*c[4]*c[7]*s[[]] -187305672153600*c[2]*c[5]*c[6]*s[[]] +9866612851200*c[2]*c[10]*s[[]]^2 -21151632902400*c[3]^3*c[5] +131355890265600*c[3]^2*c[4]^2 -245074310208000*c[3]^2*c[7]*s[[]] -294087190924800*c[3]*c[4]*c[6]*s[[]] +69966522662400*c[3]*c[5]^2*s[[]] +69141246451200*c[3]*c[9]*s[[]]^2 +93789311616000*c[4]^2*c[5]*s[[]] +109626118398720*c[4]*c[8]*s[[]]^2 -21088284940800*c[5]*c[7]*s[[]]^2 -62975286950400*c[6]^2*s[[]]^2 +217945728000*c[11]*s[[]]^3 +7206738739200*c[1]^2*c[12] +20319806707200*c[1]*c[2]*c[11] -3634862527296000*c[1]*c[3]*c[10] -4023241814592000*c[1]*c[4]*c[9] +966829043980800*c[1]*c[5]*c[8] +1689972969484800*c[1]*c[6]*c[7] +355419187200*c[1]*c[12]*s[[]] -221454654220800*c[2]^2*c[10] -4220468168716800*c[2]*c[3]*c[9] -3780018014572800*c[2]*c[4]*c[8] +1248226038259200*c[2]*c[5]*c[7] +1742668614086400*c[2]*c[6]^2 +219471348096000*c[2]*c[11]*s[[]] -936066004473600*c[3]^2*c[8] +401928246720000*c[3]*c[4]*c[7] +569296036108800*c[3]*c[5]*c[6] +365613865344000*c[3]*c[10]*s[[]] +125195291020800*c[4]^2*c[6] -363951203616000*c[4]*c[5]^2 -432142789478400*c[4]*c[9]*s[[]] -712420995686400*c[5]*c[8]*s[[]] +355220401536000*c[6]*c[7]*s[[]] -488342131200*c[12]*s[[]]^2 +2833294464000*c[1]*c[13] -23153101171200*c[2]*c[12] -1647451757952000*c[3]*c[11] -2413349370432000*c[4]*c[10] +380751186816000*c[5]*c[9] +1181461996915200*c[6]*c[8] +6647344704000*c[7]^2 +1307674368000*c[13]*s[[]] -2833294464000*c[14] )

3 :
Let
w[i]=i’th elementary symmetric polynomial of the weights W[i] (i=1,…,m).
d[i]=i’th elementary symmetric polynomial of the degrees D[i] (i=1,…,m+1).
Then the functions below give the image Milnor numbers.

1/(2*w[1]^2)*
( -d[1]*w[1] +w[1]^2 +d[2] )

1/(6*w[2]^3)*
( -d[1]^2*w[2]^2 +6*d[1]*w[1]*w[2]^2 -5*w[1]^2*w[2]^2 -d[2]*w[2]^2 -3*d[3]*w[1]*w[2] +w[2]^3 +d[3]^2 )

1/(24*w[3]^4)*
( -2*d[1]^3*w[3]^3 +2*d[1]*w[1]^2*w[3]^3 -d[1]^2*d[4]*w[3]^2 +10*d[1]*d[2]*w[3]^3 +18*d[1]*d[4]*w[1]*w[3]^2 -22*d[1]*w[2]*w[3]^3 -6*d[2]*w[1]*w[3]^3 -17*d[4]*w[1]^2*w[3]^2 +18*w[1]*w[2]*w[3]^3 +2*d[1]*d[4]^2*w[3] -16*d[2]*d[4]*w[3]^2 -6*d[4]^2*w[1]*w[3] +28*d[4]*w[2]*w[3]^2 +d[4]^3 )

1/(120*w[4]^5)*
( -6*d[1]^4*w[4]^4 +130*d[1]^2*w[1]^2*w[4]^4 -240*d[1]*w[1]^3*w[4]^4 +116*w[1]^4*w[4]^4 -5*d[1]^3*d[5]*w[4]^3 +34*d[1]^2*d[2]*w[4]^4 +40*d[1]^2*d[5]*w[1]*w[4]^3 +10*d[1]^2*w[2]*w[4]^4 -270*d[1]*d[2]*w[1]*w[4]^4 -85*d[1]*d[5]*w[1]^2*w[4]^3 +210*d[1]*w[1]*w[2]*w[4]^4 +240*d[2]*w[1]^2*w[4]^4 +50*d[5]*w[1]^3*w[4]^3 -224*w[1]^2*w[2]*w[4]^4 +5*d[1]^2*d[5]^2*w[4]^2 -20*d[1]*d[2]*d[5]*w[4]^3 -64*d[1]*d[3]*w[4]^4 +30*d[1]*d[5]^2*w[1]*w[4]^2 -40*d[1]*d[5]*w[2]*w[4]^3 +120*d[1]*w[3]*w[4]^4 +108*d[2]^2*w[4]^4 +40*d[2]*d[5]*w[1]*w[4]^3 -240*d[2]*w[2]*w[4]^4 +60*d[3]*w[1]*w[4]^4 -35*d[5]^2*w[1]^2*w[4]^2 +20*d[5]*w[1]*w[2]*w[4]^3 -116*w[1]*w[3]*w[4]^4 +132*w[2]^2*w[4]^4 +5*d[1]*d[5]^3*w[4] -50*d[2]*d[5]^2*w[4]^2 +60*d[3]*d[5]*w[4]^3 +4*d[4]*w[4]^4 -10*d[5]^3*w[1]*w[4] +70*d[5]^2*w[2]*w[4]^2 -120*d[5]*w[3]*w[4]^3 -4*w[4]^5 +d[5]^4 )

1/(720*w[5]^6)*
( -24*d[1]^5*w[5]^5 +60*d[1]^3*w[1]^2*w[5]^5 -36*d[1]*w[1]^4*w[5]^5 -26*d[1]^4*d[6]*w[5]^4 +156*d[1]^3*d[2]*w[5]^5 +150*d[1]^3*d[6]*w[1]*w[5]^4 +60*d[1]^3*w[2]*w[5]^5 -180*d[1]^2*d[2]*w[1]*w[5]^5 +700*d[1]^2*d[6]*w[1]^2*w[5]^4 -960*d[1]^2*w[1]*w[2]*w[5]^5 -1770*d[1]*d[6]*w[1]^3*w[5]^4 +1584*d[1]*w[1]^2*w[2]*w[5]^5 +946*d[6]*w[1]^4*w[5]^4 -660*w[1]^3*w[2]*w[5]^5 +15*d[1]^3*d[6]^2*w[5]^3 -16*d[1]^2*d[2]*d[6]*w[5]^4 -276*d[1]^2*d[3]*w[5]^5 +195*d[1]^2*d[6]^2*w[1]*w[5]^3 +10*d[1]^2*d[6]*w[2]*w[5]^4 -108*d[1]*d[2]^2*w[5]^5 -2100*d[1]*d[2]*d[6]*w[1]*w[5]^4 +720*d[1]*d[2]*w[2]*w[5]^5 +1140*d[1]*d[3]*w[1]*w[5]^5 -495*d[1]*d[6]^2*w[1]^2*w[5]^3 +2280*d[1]*d[6]*w[1]*w[2]*w[5]^4 -684*d[1]*w[1]*w[3]*w[5]^5 -612*d[1]*w[2]^2*w[5]^5 +60*d[2]^2*w[1]*w[5]^5 +2140*d[2]*d[6]*w[1]^2*w[5]^4 -600*d[2]*w[1]*w[2]*w[5]^5 -840*d[3]*w[1]^2*w[5]^5 +285*d[6]^2*w[1]^3*w[5]^3 -2314*d[6]*w[1]^2*w[2]*w[5]^4 +660*w[1]^2*w[3]*w[5]^5 +540*w[1]*w[2]^2*w[5]^5 +25*d[1]^2*d[6]^3*w[5]^2 -270*d[1]*d[2]*d[6]^2*w[5]^3 -24*d[1]*d[3]*d[6]*w[5]^4 +396*d[1]*d[4]*w[5]^5 +30*d[1]*d[6]^3*w[1]*w[5]^2 +90*d[1]*d[6]^2*w[2]*w[5]^3 +360*d[1]*d[6]*w[3]*w[5]^4 -756*d[1]*w[4]*w[5]^5 +1138*d[2]^2*d[6]*w[5]^4 -600*d[2]*d[3]*w[5]^5 +330*d[2]*d[6]^2*w[1]*w[5]^3 -2420*d[2]*d[6]*w[2]*w[5]^4 +720*d[2]*w[3]*w[5]^5 +600*d[3]*w[2]*w[5]^5 -420*d[4]*w[1]*w[5]^5 -55*d[6]^3*w[1]^2*w[5]^2 -150*d[6]^2*w[1]*w[2]*w[5]^3 -336*d[6]*w[1]*w[3]*w[5]^4 +1282*d[6]*w[2]^2*w[5]^4 +780*w[1]*w[4]*w[5]^5 -720*w[2]*w[3]*w[5]^5 +9*d[1]*d[6]^4*w[5] -110*d[2]*d[6]^3*w[5]^2 +240*d[3]*d[6]^2*w[5]^3 -336*d[4]*d[6]*w[5]^4 -15*d[6]^4*w[1]*w[5] +140*d[6]^3*w[2]*w[5]^2 -360*d[6]^2*w[3]*w[5]^3 +696*d[6]*w[4]*w[5]^4 +d[6]^5 )

1/(5040*w[6]^7)*
( -120*d[1]^6*w[6]^6 +420*d[1]^4*w[1]^2*w[6]^6 +10080*d[1]^3*w[1]^3*w[6]^6 -35700*d[1]^2*w[1]^4*w[6]^6 +40320*d[1]*w[1]^5*w[6]^6 -15000*w[1]^6*w[6]^6 -154*d[1]^5*d[7]*w[6]^5 +888*d[1]^4*d[2]*w[6]^6 +756*d[1]^4*d[7]*w[1]*w[6]^5 +420*d[1]^4*w[2]*w[6]^6 -1260*d[1]^3*d[2]*w[1]*w[6]^6 +2870*d[1]^3*d[7]*w[1]^2*w[6]^5 +420*d[1]^3*w[1]*w[2]*w[6]^6 -32760*d[1]^2*d[2]*w[1]^2*w[6]^6 -9660*d[1]^2*d[7]*w[1]^3*w[6]^5 +36960*d[1]^2*w[1]^2*w[2]*w[6]^6 +75600*d[1]*d[2]*w[1]^3*w[6]^6 +8624*d[1]*d[7]*w[1]^4*w[6]^5 -87360*d[1]*w[1]^3*w[2]*w[6]^6 -42588*d[2]*w[1]^4*w[6]^6 -2436*d[7]*w[1]^5*w[6]^5 +49680*w[1]^4*w[2]*w[6]^6 +49*d[1]^4*d[7]^2*w[6]^4 +196*d[1]^3*d[2]*d[7]*w[6]^5 -1560*d[1]^3*d[3]*w[6]^6 +1260*d[1]^3*d[7]^2*w[1]*w[6]^4 +560*d[1]^3*d[7]*w[2]*w[6]^5 -912*d[1]^2*d[2]^2*w[6]^6 -10374*d[1]^2*d[2]*d[7]*w[1]*w[6]^5 -2520*d[1]^2*d[2]*w[2]*w[6]^6 +840*d[1]^2*d[3]*w[1]*w[6]^6 +910*d[1]^2*d[7]^2*w[1]^2*w[6]^4 -630*d[1]^2*d[7]*w[1]*w[2]*w[6]^5 +10500*d[1]^2*w[1]*w[3]*w[6]^6 -3780*d[1]^2*w[2]^2*w[6]^6 +36120*d[1]*d[2]^2*w[1]*w[6]^6 +16100*d[1]*d[2]*d[7]*w[1]^2*w[6]^5 -71400*d[1]*d[2]*w[1]*w[2]*w[6]^6 -9240*d[1]*d[3]*w[1]^2*w[6]^6 -5880*d[1]*d[7]^2*w[1]^3*w[6]^4 +3304*d[1]*d[7]*w[1]^2*w[2]*w[6]^5 -8400*d[1]*w[1]^2*w[3]*w[6]^6 +50400*d[1]*w[1]*w[2]^2*w[6]^6 -40320*d[2]^2*w[1]^2*w[6]^6 -6090*d[2]*d[7]*w[1]^3*w[6]^5 +84672*d[2]*w[1]^2*w[2]*w[6]^6 +10080*d[3]*w[1]^3*w[6]^6 +3661*d[7]^2*w[1]^4*w[6]^4 -3066*d[7]*w[1]^3*w[2]*w[6]^5 -2220*w[1]^3*w[3]*w[6]^6 -52260*w[1]^2*w[2]^2*w[6]^6 +140*d[1]^3*d[7]^3*w[6]^3 -1561*d[1]^2*d[2]*d[7]^2*w[6]^4 -966*d[1]^2*d[3]*d[7]*w[6]^5 +1980*d[1]^2*d[4]*w[6]^6 +525*d[1]^2*d[7]^3*w[1]*w[6]^3 +595*d[1]^2*d[7]^2*w[2]*w[6]^4 +840*d[1]^2*d[7]*w[3]*w[6]^5 -420*d[1]^2*w[4]*w[6]^6 +5432*d[1]*d[2]^2*d[7]*w[6]^5 +3792*d[1]*d[2]*d[3]*w[6]^6 -7560*d[1]*d[2]*d[7]^2*w[1]*w[6]^4 -1960*d[1]*d[2]*d[7]*w[2]*w[6]^5 -10080*d[1]*d[2]*w[3]*w[6]^6 +11844*d[1]*d[3]*d[7]*w[1]*w[6]^5 +10920*d[1]*d[3]*w[2]*w[6]^6 -10920*d[1]*d[4]*w[1]*w[6]^6 -1610*d[1]*d[7]^3*w[1]^2*w[6]^3 +10080*d[1]*d[7]^2*w[1]*w[2]*w[6]^4 -12264*d[1]*d[7]*w[1]*w[3]*w[6]^5 -3472*d[1]*d[7]*w[2]^2*w[6]^5 +8400*d[1]*w[1]*w[4]*w[6]^6 -5040*d[1]*w[2]*w[3]*w[6]^6 -12444*d[2]^3*w[6]^6 -3318*d[2]^2*d[7]*w[1]*w[6]^5 +35280*d[2]^2*w[2]*w[6]^6 +8820*d[2]*d[3]*w[1]*w[6]^6 +9205*d[2]*d[7]^2*w[1]^2*w[6]^4 -2100*d[2]*d[7]*w[1]*w[2]*w[6]^5 -2772*d[2]*w[1]*w[3]*w[6]^6 -33516*d[2]*w[2]^2*w[6]^6 -10710*d[3]*d[7]*w[1]^2*w[6]^5 -23940*d[3]*w[1]*w[2]*w[6]^6 +8820*d[4]*w[1]^2*w[6]^6 +945*d[7]^3*w[1]^3*w[6]^3 -10759*d[7]^2*w[1]^2*w[2]*w[6]^4 +11256*d[7]*w[1]^2*w[3]*w[6]^5 +5418*d[7]*w[1]*w[2]^2*w[6]^5 -7860*w[1]^2*w[4]*w[6]^6 +18300*w[1]*w[2]*w[3]*w[6]^6 +10680*w[2]^3*w[6]^6 +70*d[1]^2*d[7]^4*w[6]^2 -1120*d[1]*d[2]*d[7]^3*w[6]^3 +1176*d[1]*d[3]*d[7]^2*w[6]^4 +336*d[1]*d[4]*d[7]*w[6]^5 -2400*d[1]*d[5]*w[6]^6 +700*d[1]*d[7]^3*w[2]*w[6]^3 -2856*d[1]*d[7]*w[4]*w[6]^5 +5040*d[1]*w[5]*w[6]^6 +5698*d[2]^2*d[7]^2*w[6]^4 -10080*d[2]*d[3]*d[7]*w[6]^5 +6588*d[2]*d[4]*w[6]^6 +1260*d[2]*d[7]^3*w[1]*w[6]^3 -11900*d[2]*d[7]^2*w[2]*w[6]^4 +10920*d[2]*d[7]*w[3]*w[6]^5 -7308*d[2]*w[4]*w[6]^6 -7500*d[3]^2*w[6]^6 -1260*d[3]*d[7]^2*w[1]*w[6]^4 +10080*d[3]*d[7]*w[2]*w[6]^5 +15120*d[3]*w[3]*w[6]^6 -504*d[4]*d[7]*w[1]*w[6]^5 -6300*d[4]*w[2]*w[6]^6 +2520*d[5]*w[1]*w[6]^6 -70*d[7]^4*w[1]^2*w[6]^2 -840*d[7]^3*w[1]*w[2]*w[6]^3 +84*d[7]^2*w[1]*w[3]*w[6]^4 +6202*d[7]^2*w[2]^2*w[6]^4 +3024*d[7]*w[1]*w[4]*w[6]^5 -10920*d[7]*w[2]*w[3]*w[6]^5 -5160*w[1]*w[5]*w[6]^6 +7020*w[2]*w[4]*w[6]^6 -7620*w[3]^2*w[6]^6 +14*d[1]*d[7]^5*w[6] -203*d[2]*d[7]^4*w[6]^2 +630*d[3]*d[7]^3*w[6]^3 -1596*d[4]*d[7]^2*w[6]^4 +2520*d[5]*d[7]*w[6]^5 -120*d[6]*w[6]^6 -21*d[7]^5*w[1]*w[6] +245*d[7]^4*w[2]*w[6]^2 -840*d[7]^3*w[3]*w[6]^3 +2436*d[7]^2*w[4]*w[6]^4 -5040*d[7]*w[5]*w[6]^5 +120*w[6]^7 +d[7]^6 )

1/(40320*w[7]^8)*
( -720*d[1]^7*w[7]^7 +3360*d[1]^5*w[1]^2*w[7]^7 -5040*d[1]^3*w[1]^4*w[7]^7 +2400*d[1]*w[1]^6*w[7]^7 -1044*d[1]^6*d[8]*w[7]^6 +6000*d[1]^5*d[2]*w[7]^7 +4704*d[1]^5*d[8]*w[1]*w[7]^6 +3360*d[1]^5*w[2]*w[7]^7 -10080*d[1]^4*d[2]*w[1]*w[7]^7 +18760*d[1]^4*d[8]*w[1]^2*w[7]^6 +20160*d[1]^4*w[1]*w[2]*w[7]^7 -40320*d[1]^3*d[2]*w[1]^2*w[7]^7 +58800*d[1]^3*d[8]*w[1]^3*w[7]^6 -141120*d[1]^3*w[1]^2*w[2]*w[7]^7 +80640*d[1]^2*d[2]*w[1]^3*w[7]^7 -338996*d[1]^2*d[8]*w[1]^4*w[7]^6 +336000*d[1]^2*w[1]^3*w[2]*w[7]^7 -35280*d[1]*d[2]*w[1]^4*w[7]^7 +410256*d[1]*d[8]*w[1]^5*w[7]^6 -336960*d[1]*w[1]^4*w[2]*w[7]^7 -152480*d[8]*w[1]^6*w[7]^6 +117600*w[1]^5*w[2]*w[7]^7 +140*d[1]^5*d[8]^2*w[7]^5 +2792*d[1]^4*d[2]*d[8]*w[7]^6 -10704*d[1]^4*d[3]*w[7]^7 +9128*d[1]^4*d[8]^2*w[1]*w[7]^5 +6328*d[1]^4*d[8]*w[2]*w[7]^6 -8160*d[1]^3*d[2]^2*w[7]^7 -73416*d[1]^3*d[2]*d[8]*w[1]*w[7]^6 -40320*d[1]^3*d[2]*w[2]*w[7]^7 -10080*d[1]^3*d[3]*w[1]*w[7]^7 +14840*d[1]^3*d[8]^2*w[1]^2*w[7]^5 +12600*d[1]^3*d[8]*w[1]*w[2]*w[7]^6 -15120*d[1]^3*w[1]*w[3]*w[7]^7 +55440*d[1]^3*w[2]^2*w[7]^7 +90720*d[1]^2*d[2]^2*w[1]*w[7]^7 -287000*d[1]^2*d[2]*d[8]*w[1]^2*w[7]^6 +102480*d[1]^2*d[2]*w[1]*w[2]*w[7]^7 +199920*d[1]^2*d[3]*w[1]^2*w[7]^7 -83440*d[1]^2*d[8]^2*w[1]^3*w[7]^5 +432824*d[1]^2*d[8]*w[1]^2*w[2]*w[7]^6 -13440*d[1]^2*w[1]^2*w[3]*w[7]^7 -322560*d[1]^2*w[1]*w[2]^2*w[7]^7 -80640*d[1]*d[2]^2*w[1]^2*w[7]^7 +824040*d[1]*d[2]*d[8]*w[1]^3*w[7]^6 -221760*d[1]*d[2]*w[1]^2*w[2]*w[7]^7 -403200*d[1]*d[3]*w[1]^3*w[7]^7 +85820*d[1]*d[8]^2*w[1]^4*w[7]^5 -974904*d[1]*d[8]*w[1]^3*w[2]*w[7]^6 +147120*d[1]*w[1]^3*w[3]*w[7]^7 +537360*d[1]*w[1]^2*w[2]^2*w[7]^7 -467376*d[2]*d[8]*w[1]^4*w[7]^6 +152880*d[2]*w[1]^3*w[2]*w[7]^7 +223104*d[3]*w[1]^4*w[7]^7 -26488*d[8]^2*w[1]^5*w[7]^5 +524112*d[8]*w[1]^4*w[2]*w[7]^6 -117600*w[1]^4*w[3]*w[7]^7 -265440*w[1]^3*w[2]^2*w[7]^7 +889*d[1]^4*d[8]^3*w[7]^4 -10220*d[1]^3*d[2]*d[8]^2*w[7]^5 -10576*d[1]^3*d[3]*d[8]*w[7]^6 +12720*d[1]^3*d[4]*w[7]^7 +5460*d[1]^3*d[8]^3*w[1]*w[7]^4 +6020*d[1]^3*d[8]^2*w[2]*w[7]^5 +3360*d[1]^3*d[8]*w[3]*w[7]^6 -5040*d[1]^3*w[4]*w[7]^7 +35628*d[1]^2*d[2]^2*d[8]*w[7]^6 +52656*d[1]^2*d[2]*d[3]*w[7]^7 -82292*d[1]^2*d[2]*d[8]^2*w[1]*w[7]^5 -45752*d[1]^2*d[2]*d[8]*w[2]*w[7]^6 +20160*d[1]^2*d[2]*w[3]*w[7]^7 +65856*d[1]^2*d[3]*d[8]*w[1]*w[7]^6 -62160*d[1]^2*d[3]*w[2]*w[7]^7 +30240*d[1]^2*d[4]*w[1]*w[7]^7 -4410*d[1]^2*d[8]^3*w[1]^2*w[7]^4 +38780*d[1]^2*d[8]^2*w[1]*w[2]*w[7]^5 +51856*d[1]^2*d[8]*w[1]*w[3]*w[7]^6 -27412*d[1]^2*d[8]*w[2]^2*w[7]^6 -188160*d[1]^2*w[1]*w[4]*w[7]^7 +120960*d[1]^2*w[2]*w[3]*w[7]^7 -37200*d[1]*d[2]^3*w[7]^7 +428568*d[1]*d[2]^2*d[8]*w[1]*w[7]^6 +40320*d[1]*d[2]^2*w[2]*w[7]^7 -344400*d[1]*d[2]*d[3]*w[1]*w[7]^7 +159740*d[1]*d[2]*d[8]^2*w[1]^2*w[7]^5 -895440*d[1]*d[2]*d[8]*w[1]*w[2]*w[7]^6 +55440*d[1]*d[2]*w[1]*w[3]*w[7]^7 +65520*d[1]*d[2]*w[2]^2*w[7]^7 -158480*d[1]*d[3]*d[8]*w[1]^2*w[7]^6 +556080*d[1]*d[3]*w[1]*w[2]*w[7]^7 -42000*d[1]*d[4]*w[1]^2*w[7]^7 -11340*d[1]*d[8]^3*w[1]^3*w[7]^4 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1/(362880*w[8]^9)*
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1/(39916800*w[10]^11)*
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1/(479001600*w[11]^12)*
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1/(6227020800*w[12]^13)*
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+75881237848416000*w[14]^14*d[2]^3*d[3]*w[1]^2*w[3] +87476713388064000*w[14]^14*d[2]^3*d[3]*w[1]*w[2]^2 +32580750467865600*w[14]^14*d[2]^3*d[4]*w[1]^2*w[2] -237290404006656000*w[14]^14*d[1]*d[2]^3*w[1]^3*w[4] -210667368143232000*w[14]^14*d[1]*d[2]^3*w[1]*w[2]^3 -101639062901376000*w[14]^14*d[1]*d[2]^2*d[5]*w[1]^4 -190654509453408000*w[14]^14*d[2]^3*d[3]*w[1]^3*w[2] +130415149325260800*w[14]^14*d[1]^2*d[2]^2*w[1]^2*w[6] +87996023571456000*w[14]^14*d[1]^2*d[2]*d[7]*w[1]^3 +472095759687552000*w[14]^14*d[1]*d[2]^3*w[1]^4*w[3] +982958273256960000*w[14]^14*d[1]*d[2]^3*w[1]^3*w[2]^2 +124444831230720000*w[14]^14*d[1]*d[2]^2*d[4]*w[1]^5 -367369013992780800*w[14]^14*d[1]^2*d[2]^2*w[1]^3*w[5] -181321168905676800*w[14]^14*d[1]^2*d[2]^2*w[1]^2*w[3]^2 -161900155851033600*w[14]^14*d[1]^2*d[2]*d[6]*w[1]^4 -24935301507916800*w[14]^14*d[1]^2*d[2]^2*w[1]*w[7] -30384358804800000*w[14]^14*d[1]^2*d[2]*d[8]*w[1]^2 -25402041624960000*w[14]^14*d[1]*d[2]^3*w[1]*w[6] 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-27029799014400*d[15]^2*w[14]^12*d[1]^4*d[6]*w[2] )

l=1 Master Function

K_1

K_2

K_3

K_4

K_5

K_6

K_7

K_8

K_9

K_{10}

K_{11}

K_{12}

K_{13}

K_{14}

K_{15}

m=1 Image Milnor function

m=2 Image Milnor function

m=3 Image Milnor function

m=4 Image Milnor function

m=5 Image Milnor function

m=6 Image Milnor function

m=7 Image Milnor function

m=8 Image Milnor function

m=9 Image Milnor function

m=10 Image Milnor function

m=11 Image Milnor function

m=12 Image Milnor function

m=13 Image Milnor function

m=14 Image Milnor function