Skip to main content

Thom polynomials, relative dimension l=0

Thom polynomials of contact singularities of relative dimension 0

codimension 0

  • Thom polynomial in monomial basis:

    • 1

  • Thom polynomial in Schur basis:

    • s[[0]]

    codimension 1

  • Thom polynomial in monomial basis:

    • c[1]

  • Thom polynomial in Schur basis:

    • s[[1]]

    codimension 2

  • Thom polynomial in monomial basis:

    • c[1]^2 +c[2]

  • Thom polynomial in Schur basis:

    • 2*s[[2]] +s[[1,1]]

    codimension 3

  • Thom polynomial in monomial basis:

    • c[1]^3 +3*c[1]*c[2] +2*c[3]

  • Thom polynomial in Schur basis:

    • 6*s[[3]] +5*s[[2,1]] +s[[1,1,1]]

    codimension 4

  • Thom polynomial in monomial basis:

    • c[1]^4 +6*c[1]^2*c[2] +9*c[1]*c[3] +2*c[2]^2 +6*c[4]

  • Thom polynomial in Schur basis:

    • 24*s[[4]] +10*s[[2,2]] +26*s[[3,1]] +9*s[[2,1,1]] +s[[1,1,1,1]]

  • Thom polynomial in monomial basis:

    • -c[1]*c[3] +c[2]^2

  • Thom polynomial in Schur basis:

    • s[[2,2]]

    codimension 5

  • Thom polynomial in monomial basis:

    • c[1]^5 +10*c[1]^3*c[2] +25*c[1]^2*c[3] +10*c[1]*c[2]^2 +38*c[1]*c[4] +12*c[2]*c[3] +24*c[5]

  • Thom polynomial in Schur basis:

    • 120*s[[5]] +92*s[[3,2]] +154*s[[4,1]] +35*s[[2,2,1]] +71*s[[3,1,1]] +14*s[[2,1,1,1]] +s[[1,1,1,1,1]]

  • Thom polynomial in monomial basis:

    • -2*c[1]^2*c[3] +2*c[1]*c[2]^2 -2*c[1]*c[4] +2*c[2]*c[3]

  • Thom polynomial in Schur basis:

    • 4*s[[3,2]] +2*s[[2,2,1]]

    codimension 6

  • Thom polynomial in monomial basis:

    • c[1]^6 +15*c[1]^4*c[2] +55*c[1]^3*c[3] +30*c[1]^2*c[2]^2 +141*c[1]^2*c[4] +79*c[1]*c[2]*c[3] +5*c[2]^3 +202*c[1]*c[5] +55*c[2]*c[4] +17*c[3]^2 +120*c[6]

  • Thom polynomial in Schur basis:

    • 720*s[[6]] +266*s[[3,3]] +770*s[[4,2]] +1044*s[[5,1]] +70*s[[2,2,2]] +455*s[[3,2,1]] +580*s[[4,1,1]] +84*s[[2,2,1,1]] +155*s[[3,1,1,1]] +20*s[[2,1,1,1,1]] +s[[1,1,1,1,1,1]]

  • Thom polynomial in monomial basis:

    • -2*c[1]^3*c[3] +2*c[1]^2*c[2]^2 -5*c[1]^2*c[4] +2*c[1]*c[2]*c[3] +3*c[2]^3 -6*c[1]*c[5] +9*c[2]*c[4] -3*c[3]^2

  • Thom polynomial in Schur basis:

    • 4*s[[3,3]] +16*s[[4,2]] +5*s[[2,2,2]] +12*s[[3,2,1]] +2*s[[2,2,1,1]]

  • Thom polynomial in monomial basis:

    • -c[1]^3*c[3] +c[1]^2*c[2]^2 -2*c[1]^2*c[4] +3*c[1]*c[2]*c[3] -c[2]^3 -3*c[2]*c[4] +3*c[3]^2

  • Thom polynomial in Schur basis:

    • 6*s[[3,3]] +2*s[[4,2]] +3*s[[3,2,1]] +s[[2,2,1,1]]

    codimension 7

  • Thom polynomial in monomial basis:

    • c[1]^7 +21*c[1]^5*c[2] +105*c[1]^4*c[3] +70*c[1]^3*c[2]^2 +399*c[1]^3*c[4] +301*c[1]^2*c[2]*c[3] +35*c[1]*c[2]^3 +960*c[1]^2*c[5] +467*c[1]*c[2]*c[4] +139*c[1]*c[3]^2 +58*c[2]^2*c[3] +1284*c[1]*c[6] +326*c[2]*c[5] +154*c[3]*c[4] +720*c[7]

  • Thom polynomial in Schur basis:

    • 5040*s[[7]] +3500*s[[4,3]] +6746*s[[5,2]] +8028*s[[6,1]] +1255*s[[3,2,2]] +1891*s[[3,3,1]] +4999*s[[4,2,1]] +5104*s[[5,1,1]] +294*s[[2,2,2,1]] +1456*s[[3,2,1,1]] +1665*s[[4,1,1,1]] +168*s[[2,2,1,1,1]] +295*s[[3,1,1,1,1]] +27*s[[2,1,1,1,1,1]] +s[[1,1,1,1,1,1,1]]

  • Thom polynomial in monomial basis:

    • -2*c[1]^4*c[3] +2*c[1]^3*c[2]^2 -9*c[1]^3*c[4] +9*c[1]*c[2]^3 -26*c[1]^2*c[5] +29*c[1]*c[2]*c[4] -17*c[1]*c[3]^2 +14*c[2]^2*c[3] -24*c[1]*c[6] +34*c[2]*c[5] -10*c[3]*c[4]

  • Thom polynomial in Schur basis:

    • 44*s[[4,3]] +84*s[[5,2]] +47*s[[3,2,2]] +30*s[[3,3,1]] +82*s[[4,2,1]] +13*s[[2,2,2,1]] +24*s[[3,2,1,1]] +2*s[[2,2,1,1,1]]

  • Thom polynomial in monomial basis:

    • -2*c[1]^4*c[3] +2*c[1]^3*c[2]^2 -7*c[1]^3*c[4] +8*c[1]^2*c[2]*c[3] -c[1]*c[2]^3 -6*c[1]^2*c[5] -5*c[1]*c[2]*c[4] +13*c[1]*c[3]^2 -2*c[2]^2*c[3] -10*c[2]*c[5] +10*c[3]*c[4]

  • Thom polynomial in Schur basis:

    • 44*s[[4,3]] +12*s[[5,2]] +9*s[[3,2,2]] +30*s[[3,3,1]] +22*s[[4,2,1]] +3*s[[2,2,2,1]] +12*s[[3,2,1,1]] +2*s[[2,2,1,1,1]]

  • Thom polynomial in monomial basis:

    • -2*c[1]^2*c[2]*c[3] +2*c[1]*c[2]^3 -4*c[1]*c[2]*c[4] +2*c[1]*c[3]^2 +2*c[2]^2*c[3] -2*c[2]*c[5] +2*c[3]*c[4]

  • Thom polynomial in Schur basis:

    • 8*s[[4,3]] +4*s[[5,2]] +6*s[[3,2,2]] +6*s[[3,3,1]] +6*s[[4,2,1]] +2*s[[2,2,2,1]] +2*s[[3,2,1,1]]

    codimension 8

  • Thom polynomial in monomial basis:

    • c[1]^8 +28*c[1]^6*c[2] +182*c[1]^5*c[3] +140*c[1]^4*c[2]^2 +952*c[1]^4*c[4] +868*c[1]^3*c[2]*c[3] +140*c[1]^2*c[2]^3 +3383*c[1]^3*c[5] +2229*c[1]^2*c[2]*c[4] +642*c[1]^2*c[3]^2 +501*c[1]*c[2]^2*c[3] +14*c[2]^4 +7552*c[1]^2*c[6] +3455*c[1]*c[2]*c[5] +1559*c[1]*c[3]*c[4] +364*c[2]^2*c[4] +202*c[2]*c[3]^2 +9468*c[1]*c[7] +2314*c[2]*c[6] +954*c[3]*c[5] +332*c[4]^2 +5040*c[8]

  • Thom polynomial in Schur basis:

    • s[[1,1,1,1,1,1,1,1]] +35*s[[2,1,1,1,1,1,1]] +300*s[[2,2,1,1,1,1]] +840*s[[2,2,2,1,1]] +588*s[[2,2,2,2]] +511*s[[3,1,1,1,1,1]] +3732*s[[3,2,1,1,1]] +6759*s[[3,2,2,1]] +8031*s[[3,3,1,1]] +8412*s[[3,3,2]] +4025*s[[4,1,1,1,1]] +20259*s[[4,2,1,1]] +17484*s[[4,2,2]] +31584*s[[4,3,1]] +12964*s[[4,4]] +18424*s[[5,1,1,1]] +54822*s[[5,2,1]] +39764*s[[5,3]] +48860*s[[6,1,1]] +63808*s[[6,2]] +69264*s[[7,1]] +40320*s[[8]]

  • Thom polynomial in monomial basis:

    • -2*c[1]^5*c[3] +2*c[1]^4*c[2]^2 -14*c[1]^4*c[4] -4*c[1]^3*c[2]*c[3] +18*c[1]^2*c[2]^3 -71*c[1]^3*c[5] +60*c[1]^2*c[2]*c[4] -53*c[1]^2*c[3]^2 +57*c[1]*c[2]^2*c[3] +7*c[2]^4 -154*c[1]^2*c[6] +166*c[1]*c[2]*c[5] -90*c[1]*c[3]*c[4] +74*c[2]^2*c[4] +4*c[2]*c[3]^2 -120*c[1]*c[7] +162*c[2]*c[6] -39*c[3]*c[5] -3*c[4]^2

  • Thom polynomial in Schur basis:

    • 2*s[[2,2,1,1,1,1]] +24*s[[2,2,2,1,1]] +29*s[[2,2,2,2]] +40*s[[3,2,1,1,1]] +194*s[[3,2,2,1]] +108*s[[3,3,1,1]] +194*s[[3,3,2]] +250*s[[4,2,1,1]] +424*s[[4,2,2]] +406*s[[4,3,1]] +124*s[[4,4]] +620*s[[5,2,1]] +412*s[[5,3]] +528*s[[6,2]]

  • Thom polynomial in monomial basis:

    • -2*c[1]^5*c[3] +2*c[1]^4*c[2]^2 -11*c[1]^4*c[4] +8*c[1]^3*c[2]*c[3] +3*c[1]^2*c[2]^3 -26*c[1]^3*c[5] +7*c[1]^2*c[2]*c[4] +13*c[1]^2*c[3]^2 +10*c[1]*c[2]^2*c[3] -4*c[2]^4 -24*c[1]^2*c[6] -2*c[1]*c[2]*c[5] +34*c[1]*c[3]*c[4] -32*c[2]^2*c[4] +24*c[2]*c[3]^2 -40*c[2]*c[6] +60*c[3]*c[5] -20*c[4]^2

  • Thom polynomial in Schur basis:

    • 2*s[[2,2,1,1,1,1]] +9*s[[2,2,2,1,1]] +3*s[[2,2,2,2]] +22*s[[3,2,1,1,1]] +48*s[[3,2,2,1]] +66*s[[3,3,1,1]] +99*s[[3,3,2]] +82*s[[4,2,1,1]] +75*s[[4,2,2]] +232*s[[4,3,1]] +76*s[[4,4]] +122*s[[5,2,1]] +232*s[[5,3]] +60*s[[6,2]]

  • Thom polynomial in monomial basis:

    • -c[1]^5*c[3] +c[1]^4*c[2]^2 -5*c[1]^4*c[4] +6*c[1]^3*c[2]*c[3] -c[1]^2*c[2]^3 -6*c[1]^3*c[5] -9*c[1]^2*c[2]*c[4] +18*c[1]^2*c[3]^2 -5*c[1]*c[2]^2*c[3] +2*c[2]^4 -29*c[1]*c[2]*c[5] +27*c[1]*c[3]*c[4] +13*c[2]^2*c[4] -11*c[2]*c[3]^2 -2*c[2]*c[6] -21*c[3]*c[5] +23*c[4]^2

  • Thom polynomial in Schur basis:

    • s[[2,2,1,1,1,1]] +2*s[[2,2,2,1,1]] +3*s[[2,2,2,2]] +8*s[[3,2,1,1,1]] +15*s[[3,2,2,1]] +36*s[[3,3,1,1]] +21*s[[3,3,2]] +23*s[[4,2,1,1]] +25*s[[4,2,2]] +95*s[[4,3,1]] +86*s[[4,4]] +28*s[[5,2,1]] +62*s[[5,3]] +12*s[[6,2]]

  • Thom polynomial in monomial basis:

    • -2*c[1]^3*c[2]*c[3] +2*c[1]^2*c[2]^3 -7*c[1]^2*c[2]*c[4] +2*c[1]^2*c[3]^2 +4*c[1]*c[2]^2*c[3] +c[2]^4 -9*c[1]*c[2]*c[5] +5*c[1]*c[3]*c[4] +2*c[2]^2*c[4] +2*c[2]*c[3]^2 -4*c[2]*c[6] +3*c[3]*c[5] +c[4]^2

  • Thom polynomial in Schur basis:

    • 2*s[[2,2,2,1,1]] +3*s[[2,2,2,2]] +2*s[[3,2,1,1,1]] +15*s[[3,2,2,1]] +12*s[[3,3,1,1]] +21*s[[3,3,2]] +12*s[[4,2,1,1]] +25*s[[4,2,2]] +40*s[[4,3,1]] +20*s[[4,4]] +22*s[[5,2,1]] +32*s[[5,3]] +12*s[[6,2]]

    codimension 9 — beyond Mather’s nice dimensions

  • Thom polynomial in monomial basis:

    • c[1]^9 +36*c[1]^7*c[2] +294*c[1]^6*c[3] +252*c[1]^5*c[2]^2 +2016*c[1]^5*c[4] +2100*c[1]^4*c[2]*c[3] +420*c[1]^3*c[2]^3 +9831*c[1]^4*c[5] +7869*c[1]^3*c[2]*c[4] +2202*c[1]^3*c[3]^2 +2421*c[1]^2*c[2]^2*c[3] +126*c[1]*c[2]^4 +32244*c[1]^3*c[6] +20123*c[1]^2*c[2]*c[5] +8727*c[1]^2*c[3]*c[4] +3856*c[1]*c[2]^2*c[4] +2074*c[1]*c[2]*c[3]^2 +260*c[2]^3*c[3] +67076*c[1]^2*c[7] +29746*c[1]*c[2]*c[6] +11726*c[1]*c[3]*c[5] +4016*c[1]*c[4]^2 +2940*c[2]^2*c[5] +2400*c[2]*c[3]*c[4] +220*c[3]^3 +79344*c[1]*c[8] +18920*c[2]*c[7] +7300*c[3]*c[6] +4020*c[4]*c[5] +40320*c[9]

  • Thom polynomial in Schur basis:

    • 495*s[[2,2,1,1,1,1,1]] +1980*s[[2,2,2,1,1,1]] +2772*s[[2,2,2,2,1]] +826*s[[3,1,1,1,1,1,1]] +8295*s[[3,2,1,1,1,1]] +24003*s[[3,2,2,1,1]] +16919*s[[3,2,2,2]] +26115*s[[3,3,1,1,1]] +58346*s[[3,3,2,1]] +23836*s[[3,3,3]] +8624*s[[4,1,1,1,1,1]] +64032*s[[4,2,1,1,1]] +115766*s[[4,2,2,1]] +165354*s[[4,3,1,1]] +174584*s[[4,3,2]] +149236*s[[4,4,1]] +54649*s[[5,1,1,1,1]] +271449*s[[5,2,1,1]] +232290*s[[5,2,2]] +437690*s[[5,3,1]] +218520*s[[5,4]] +214676*s[[6,1,1,1]] +627826*s[[6,2,1]] +455220*s[[6,3]] +509004*s[[7,1,1]] +655824*s[[7,2]] +663696*s[[8,1]] +362880*s[[9]] +s[[1,1,1,1,1,1,1,1,1]] +44*s[[2,1,1,1,1,1,1,1]]

  • Thom polynomial in monomial basis:

    • -2*c[1]^6*c[3] +2*c[1]^5*c[2]^2 -20*c[1]^5*c[4] -10*c[1]^4*c[2]*c[3] +30*c[1]^3*c[2]^3 -155*c[1]^4*c[5] +100*c[1]^3*c[2]*c[4] -125*c[1]^3*c[3]^2 +145*c[1]^2*c[2]^2*c[3] +35*c[1]*c[2]^4 -580*c[1]^3*c[6] +480*c[1]^2*c[2]*c[5] -400*c[1]^2*c[3]*c[4] +440*c[1]*c[2]^2*c[4] -10*c[1]*c[2]*c[3]^2 +70*c[2]^3*c[3] -1044*c[1]^2*c[7] +1084*c[1]*c[2]*c[6] -471*c[1]*c[3]*c[5] -73*c[1]*c[4]^2 +426*c[2]^2*c[5] +106*c[2]*c[3]*c[4] -28*c[3]^3 -720*c[1]*c[8] +936*c[2]*c[7] -196*c[3]*c[6] -20*c[4]*c[5]

  • Thom polynomial in Schur basis:

    • 2*s[[2,2,1,1,1,1,1]] +38*s[[2,2,2,1,1,1]] +105*s[[2,2,2,2,1]] +60*s[[3,2,1,1,1,1]] +520*s[[3,2,2,1,1]] +565*s[[3,2,2,2]] +280*s[[3,3,1,1,1]] +1210*s[[3,3,2,1]] +464*s[[3,3,3]] +590*s[[4,2,1,1,1]] +2410*s[[4,2,2,1]] +1910*s[[4,3,1,1]] +3080*s[[4,3,2]] +1580*s[[4,4,1]] +2580*s[[5,2,1,1]] +4004*s[[5,2,2]] +4660*s[[5,3,1]] +1880*s[[5,4]] +5168*s[[6,2,1]] +3856*s[[6,3]] +3840*s[[7,2]]

  • Thom polynomial in monomial basis:

    • -2*c[1]^6*c[3] +2*c[1]^5*c[2]^2 -16*c[1]^5*c[4] +6*c[1]^4*c[2]*c[3] +10*c[1]^3*c[2]^3 -71*c[1]^4*c[5] +32*c[1]^3*c[2]*c[4] -5*c[1]^3*c[3]^2 +53*c[1]^2*c[2]^2*c[3] -9*c[1]*c[2]^4 -154*c[1]^3*c[6] +90*c[1]^2*c[2]*c[5] +32*c[1]^2*c[3]*c[4] -74*c[1]*c[2]^2*c[4] +126*c[1]*c[2]*c[3]^2 -20*c[2]^3*c[3] -120*c[1]^2*c[7] -64*c[1]*c[2]*c[6] +351*c[1]*c[3]*c[5] -131*c[1]*c[4]^2 -124*c[2]^2*c[5] +46*c[2]*c[3]*c[4] +42*c[3]^3 -204*c[2]*c[7] +294*c[3]*c[6] -90*c[4]*c[5]

  • Thom polynomial in Schur basis:

    • 2*s[[2,2,1,1,1,1,1]] +18*s[[2,2,2,1,1,1]] +21*s[[2,2,2,2,1]] +36*s[[3,2,1,1,1,1]] +164*s[[3,2,2,1,1]] +99*s[[3,2,2,2]] +144*s[[3,3,1,1,1]] +474*s[[3,3,2,1]] +312*s[[3,3,3]] +230*s[[4,2,1,1,1]] +534*s[[4,2,2,1]] +854*s[[4,3,1,1]] +1192*s[[4,3,2]] +716*s[[4,4,1]] +660*s[[5,2,1,1]] +652*s[[5,2,2]] +1940*s[[5,3,1]] +856*s[[5,4]] +848*s[[6,2,1]] +1552*s[[6,3]] +384*s[[7,2]]

  • Thom polynomial in monomial basis:

    • -2*c[1]^6*c[3] +2*c[1]^5*c[2]^2 -14*c[1]^5*c[4] +14*c[1]^4*c[2]*c[3] -32*c[1]^4*c[5] -24*c[1]^3*c[2]*c[4] +58*c[1]^3*c[3]^2 -6*c[1]^2*c[2]^2*c[3] +4*c[1]*c[2]^4 -24*c[1]^3*c[6] -136*c[1]^2*c[2]*c[5] +158*c[1]^2*c[3]*c[4] +16*c[1]*c[2]^2*c[4] -22*c[1]*c[2]*c[3]^2 +8*c[2]^3*c[3] -134*c[1]*c[2]*c[6] -30*c[1]*c[3]*c[5] +152*c[1]*c[4]^2 +24*c[2]^2*c[5] +2*c[2]*c[3]*c[4] -14*c[3]^3 -12*c[2]*c[7] -98*c[3]*c[6] +110*c[4]*c[5]

  • Thom polynomial in Schur basis:

    • 2*s[[2,2,1,1,1,1,1]] +8*s[[2,2,2,1,1,1]] +14*s[[2,2,2,2,1]] +24*s[[3,2,1,1,1,1]] +78*s[[3,2,2,1,1]] +66*s[[3,2,2,2]] +136*s[[3,3,1,1,1]] +270*s[[3,3,2,1]] +84*s[[3,3,3]] +110*s[[4,2,1,1,1]] +250*s[[4,2,2,1]] +638*s[[4,3,1,1]] +612*s[[4,3,2]] +860*s[[4,4,1]] +240*s[[5,2,1,1]] +276*s[[5,2,2]] +1036*s[[5,3,1]] +1016*s[[5,4]] +248*s[[6,2,1]] +544*s[[6,3]] +96*s[[7,2]]

  • Thom polynomial in monomial basis:

    • -2*c[1]^4*c[2]*c[3] +2*c[1]^3*c[2]^3 -10*c[1]^3*c[2]*c[4] +2*c[1]^3*c[3]^2 +8*c[1]^2*c[2]^2*c[3] -18*c[1]^2*c[2]*c[5] +8*c[1]^2*c[3]*c[4] -2*c[1]*c[2]^2*c[4] +12*c[1]*c[2]*c[3]^2 -14*c[1]*c[2]*c[6] +4*c[1]*c[3]*c[5] +6*c[1]*c[4]^2 -12*c[2]^2*c[5] +22*c[2]*c[3]*c[4] -6*c[3]^3 -4*c[2]*c[7] -6*c[3]*c[6] +10*c[4]*c[5]

  • Thom polynomial in Schur basis:

    • 2*s[[2,2,2,1,1,1]] +4*s[[2,2,2,2,1]] +2*s[[3,2,1,1,1,1]] +20*s[[3,2,2,1,1]] +18*s[[3,2,2,2]] +16*s[[3,3,1,1,1]] +66*s[[3,3,2,1]] +30*s[[3,3,3]] +16*s[[4,2,1,1,1]] +64*s[[4,2,2,1]] +94*s[[4,3,1,1]] +146*s[[4,3,2]] +118*s[[4,4,1]] +46*s[[5,2,1,1]] +70*s[[5,2,2]] +174*s[[5,3,1]] +128*s[[5,4]] +56*s[[6,2,1]] +100*s[[6,3]] +24*s[[7,2]]

  • Thom polynomial in monomial basis:

    • -c[1]^3*c[2]*c[4] -4*c[1]^2*c[2]^2*c[3] +5*c[1]*c[2]^4 -11*c[1]^2*c[2]*c[5] +c[1]^2*c[3]*c[4] +12*c[1]*c[2]^2*c[4] -10*c[1]*c[2]*c[3]^2 +8*c[2]^3*c[3] -22*c[1]*c[2]*c[6] +18*c[1]*c[3]*c[5] -8*c[1]*c[4]^2 +21*c[2]^2*c[5] -16*c[2]*c[3]*c[4] +7*c[3]^3 -12*c[2]*c[7] +22*c[3]*c[6] -10*c[4]*c[5]

  • Thom polynomial in Schur basis:

    • 5*s[[2,2,2,2,1]] +11*s[[3,2,2,1,1]] +24*s[[3,2,2,2]] +6*s[[3,3,1,1,1]] +30*s[[3,3,2,1]] +12*s[[3,3,3]] +6*s[[4,2,1,1,1]] +60*s[[4,2,2,1]] +42*s[[4,3,1,1]] +68*s[[4,3,2]] +30*s[[4,4,1]] +36*s[[5,2,1,1]] +85*s[[5,2,2]] +96*s[[5,3,1]] +36*s[[5,4]] +66*s[[6,2,1]] +72*s[[6,3]] +36*s[[7,2]]

  • Thom polynomial in monomial basis:

    • -c[1]*c[3]*c[5] +c[1]*c[4]^2 +c[2]^2*c[5] -2*c[2]*c[3]*c[4] +c[3]^3

  • Thom polynomial in Schur basis:

    • s[[3,3,3]]