Skip to main content

SSM-Tp’s, l=1

SSM-Thom polynomials of contact singularities of relative dimension 1

(Hierarchy of these singularities)

codimension 0

Local algebra C
Thom-Boardman class \Sigma^0
Codimension 0
SSM-Thom polynomial in Chern classes 1+(-c_{1}^6 c_{2}-6 c_{1}^5 c_{3}+5 c_{1}^4 c_{2}^2-15 c_{1}^4 c_{4}+20 c_{1}^3 c_{2} c_{3}-6 c_{1}^2 c_{2}^3-20 c_{1}^3 c_{5}+45 c_{1}^2 c_{2} c_{4}-15 c_{1}^2 c_{3}^2-12 c_{1} c_{2}^2 c_{3}+c_{2}^4-15 c_{1}^2 c_{6}+35 c_{1} c_{2} c_{5}-15 c_{1} c_{3} c_{4}-15 c_{2}^2 c_{4}+9 c_{2} c_{3}^2-6 c_{1} c_{7}+14 c_{2} c_{6}-21 c_{3} c_{5}+12 c_{4}^2-c_{8}) T^8+(c_{1}^5 c_{2}+5 c_{1}^4 c_{3}-4 c_{1}^3 c_{2}^2+10 c_{1}^3 c_{4}-12 c_{1}^2 c_{2} c_{3}+3 c_{1} c_{2}^3+10 c_{1}^2 c_{5}-20 c_{1} c_{2} c_{4}+8 c_{1} c_{3}^2+3 c_{2}^2 c_{3}+5 c_{1} c_{6}-8 c_{2} c_{5}+4 c_{3} c_{4}+c_{7}) T^7+(-c_{1}^4 c_{2}-4 c_{1}^3 c_{3}+3 c_{1}^2 c_{2}^2-6 c_{1}^2 c_{4}+6 c_{1} c_{2} c_{3}-c_{2}^3-4 c_{1} c_{5}+6 c_{2} c_{4}-3 c_{3}^2-c_{6}) T^6+(c_{1}^3 c_{2}+3 c_{1}^2 c_{3}-2 c_{1} c_{2}^2+3 c_{1} c_{4}-2 c_{2} c_{3}+c_{5}) T^5+(-c_{1}^2 c_{2}-2 c_{1} c_{3}+c_{2}^2-c_{4}) T^4+(c_{1} c_{2}+c_{3}) T^3-T^2 c_{2}
SSM-Thom polynomial in Schur functions s_{0}-T^2 s_{2}+ T^3 (2 s_{3}+ s_{2, 1})+T^4 (-3 s_{4}-3 s_{3, 1}- s_{2, 1, 1})+T^5 (4 s_{5}+6 s_{4, 1}+4 s_{3, 1, 1}+ s_{2, 1, 1, 1})+T^6 (-5 s_{6}+ s_{3, 3}-10 s_{5, 1}-10 s_{4, 1, 1}-5 s_{3, 1, 1, 1}- s_{2, 1, 1, 1, 1})+T^7 (6 s_{7}-3 s_{4, 3}+15 s_{6, 1}-2 s_{3, 3, 1}+20 s_{5, 1, 1}+15 s_{4, 1, 1, 1}+6 s_{3, 1, 1, 1, 1}+ s_{2, 1, 1, 1, 1, 1})+T^8 (- s_{2, 1, 1, 1, 1, 1, 1}-7 s_{3, 1, 1, 1, 1, 1}+3 s_{3, 3, 1, 1}+ s_{3, 3, 2}-21 s_{4, 1, 1, 1, 1}+7 s_{4, 3, 1}+3 s_{4, 4}-35 s_{5, 1, 1, 1}+6 s_{5, 3}-35 s_{6, 1, 1}-21 s_{7, 1}-7 s_{8})

codimension 2

Local algebra C[x]/(x^2)
Thom-Boardman class \Sigma^{1,0}
Codimension 2
SSM-Thom polynomial in Chern classes (c_{1}^6 c_{2}+c_{1}^5 c_{3}-10 c_{1}^4 c_{2}^2-29 c_{1}^4 c_{4}-42 c_{1}^3 c_{2} c_{3}+18 c_{1}^2 c_{2}^3-139 c_{1}^3 c_{5}-35 c_{1}^2 c_{2} c_{4}-17 c_{1}^2 c_{3}^2+54 c_{1} c_{2}^2 c_{3}-4 c_{2}^4-281 c_{1}^2 c_{6}+57 c_{1} c_{2} c_{5}-55 c_{1} c_{3} c_{4}+53 c_{2}^2 c_{4}-7 c_{2} c_{3}^2-279 c_{1} c_{7}+72 c_{2} c_{6}-24 c_{3} c_{5}-18 c_{4}^2-113 c_{8}) T^8+(-c_{1}^5 c_{2}-c_{1}^4 c_{3}+8 c_{1}^3 c_{2}^2+19 c_{1}^3 c_{4}+27 c_{1}^2 c_{2} c_{3}-9 c_{1} c_{2}^3+71 c_{1}^2 c_{5}+22 c_{1} c_{2} c_{4}+5 c_{1} c_{3}^2-15 c_{2}^2 c_{3}+99 c_{1} c_{6}-4 c_{2} c_{5}+10 c_{3} c_{4}+51 c_{7}) T^7+(c_{1}^4 c_{2}+c_{1}^3 c_{3}-6 c_{1}^2 c_{2}^2-11 c_{1}^2 c_{4}-15 c_{1} c_{2} c_{3}+3 c_{2}^3-29 c_{1} c_{5}-10 c_{2} c_{4}-21 c_{6}) T^6+(-c_{1}^3 c_{2}-c_{1}^2 c_{3}+4 c_{1} c_{2}^2+5 c_{1} c_{4}+6 c_{2} c_{3}+7 c_{5}) T^5+(c_{1}^2 c_{2}+c_{1} c_{3}-2 c_{2}^2-c_{4}) T^4+(-c_{1} c_{2}-c_{3}) T^3+T^2 c_{2}
SSM-Thom polynomial in Schur functions T^2 s_{2}+T^3 (-2 s_{3}- s_{2, 1})+T^4 (- s_{4}- s_{2, 2}+ s_{3, 1}+ s_{2, 1, 1})+T^5 (20 s_{5}+10 s_{3, 2}+14 s_{4, 1}+2 s_{2, 2, 1}- s_{2, 1, 1, 1})+T^6 (-87 s_{6}-20 s_{3, 3}-57 s_{4, 2}-96 s_{5, 1}- s_{2, 2, 2}-23 s_{3, 2, 1}-32 s_{4, 1, 1}-3 s_{2, 2, 1, 1}- s_{3, 1, 1, 1}+ s_{2, 1, 1, 1, 1})+T^7 (282 s_{7}+111 s_{4, 3}+246 s_{5, 2}+405 s_{6, 1}+13 s_{3, 2, 2}+51 s_{3, 3, 1}+149 s_{4, 2, 1}+226 s_{5, 1, 1}+2 s_{2, 2, 2, 1}+39 s_{3, 2, 1, 1}+55 s_{4, 1, 1, 1}+4 s_{2, 2, 1, 1, 1}+2 s_{3, 1, 1, 1, 1}- s_{2, 1, 1, 1, 1, 1})+T^8 ( s_{2, 1, 1, 1, 1, 1, 1}-5 s_{2, 2, 1, 1, 1, 1}-3 s_{2, 2, 2, 1, 1}- s_{2, 2, 2, 2}-3 s_{3, 1, 1, 1, 1, 1}-58 s_{3, 2, 1, 1, 1}-29 s_{3, 2, 2, 1}-95 s_{3, 3, 1, 1}-30 s_{3, 3, 2}-83 s_{4, 1, 1, 1, 1}-283 s_{4, 2, 1, 1}-94 s_{4, 2, 2}-318 s_{4, 3, 1}-130 s_{4, 4}-423 s_{5, 1, 1, 1}-722 s_{5, 2, 1}-425 s_{5, 3}-1049 s_{6, 1, 1}-897 s_{6, 2}-1389 s_{7, 1}-797 s_{8})

codimension 4

Local algebra C[x]/(x^3)
Thom-Boardman class \Sigma^{1,1,0}
Codimension 4
SSM-Thom polynomial in Chern classes (5 c_{1}^5 c_{3}+5 c_{1}^4 c_{2}^2+32 c_{1}^4 c_{4}+4 c_{1}^3 c_{2} c_{3}-18 c_{1}^2 c_{2}^3+33 c_{1}^3 c_{5}-120 c_{1}^2 c_{2} c_{4}+c_{1}^2 c_{3}^2-66 c_{1} c_{2}^2 c_{3}+6 c_{2}^4-206 c_{1}^2 c_{6}-350 c_{1} c_{2} c_{5}-56 c_{1} c_{3} c_{4}-51 c_{2}^2 c_{4}-15 c_{2} c_{3}^2-611 c_{1} c_{7}-306 c_{2} c_{6}-64 c_{3} c_{5}-29 c_{4}^2-486 c_{8}) T^8+(-4 c_{1}^4 c_{3}-4 c_{1}^3 c_{2}^2-23 c_{1}^3 c_{4}-6 c_{1}^2 c_{2} c_{3}+9 c_{1} c_{2}^3-33 c_{1}^2 c_{5}+38 c_{1} c_{2} c_{4}-2 c_{1} c_{3}^2+21 c_{2}^2 c_{3}+22 c_{1} c_{6}+62 c_{2} c_{5}+8 c_{3} c_{4}+56 c_{7}) T^7+(3 c_{1}^3 c_{3}+3 c_{1}^2 c_{2}^2+15 c_{1}^2 c_{4}+6 c_{1} c_{2} c_{3}-3 c_{2}^3+23 c_{1} c_{5}-2 c_{2} c_{4}+c_{3}^2+10 c_{6}) T^6+(-2 c_{1}^2 c_{3}-2 c_{1} c_{2}^2-8 c_{1} c_{4}-4 c_{2} c_{3}-8 c_{5}) T^5+(c_{1} c_{3}+c_{2}^2+2 c_{4}) T^4
SSM-Thom polynomial in Schur functions T^4 (4 s_{4}+ s_{2, 2}+2 s_{3, 1})+T^5 (-24 s_{5}-10 s_{3, 2}-20 s_{4, 1}-2 s_{2, 2, 1}-4 s_{3, 1, 1})+T^6 (56 s_{6}+13 s_{3, 3}+38 s_{4, 2}+76 s_{5, 1}+18 s_{3, 2, 1}+36 s_{4, 1, 1}+3 s_{2, 2, 1, 1}+6 s_{3, 1, 1, 1})+T^7 (144 s_{7}+39 s_{4, 3}+72 s_{5, 2}+48 s_{6, 1}+14 s_{3, 2, 2}-2 s_{3, 3, 1}-14 s_{4, 2, 1}-84 s_{5, 1, 1}+ s_{2, 2, 2, 1}-24 s_{3, 2, 1, 1}-52 s_{4, 1, 1, 1}-4 s_{2, 2, 1, 1, 1}-8 s_{3, 1, 1, 1, 1})+T^8 (5 s_{2, 2, 1, 1, 1, 1}-3 s_{2, 2, 2, 1, 1}-2 s_{2, 2, 2, 2}+10 s_{3, 1, 1, 1, 1, 1}+28 s_{3, 2, 1, 1, 1}-58 s_{3, 2, 2, 1}-41 s_{3, 3, 1, 1}-144 s_{3, 3, 2}+68 s_{4, 1, 1, 1, 1}-95 s_{4, 2, 1, 1}-295 s_{4, 2, 2}-573 s_{4, 3, 1}-361 s_{4, 4}+26 s_{5, 1, 1, 1}-1028 s_{5, 2, 1}-1210 s_{5, 3}-908 s_{6, 1, 1}-2105 s_{6, 2}-2682 s_{7, 1}-2292 s_{8})

codimension 6

Local algebra C[x]/(x^4)
Thom-Boardman class \Sigma^{1,1,1,0}
Codimension 6
SSM-Thom polynomial in Chern classes (12 c_{1}^4 c_{4}+18 c_{1}^3 c_{2} c_{3}+6 c_{1}^2 c_{2}^3+120 c_{1}^3 c_{5}+104 c_{1}^2 c_{2} c_{4}+26 c_{1}^2 c_{3}^2+18 c_{1} c_{2}^2 c_{3}-4 c_{2}^4+448 c_{1}^2 c_{6}+208 c_{1} c_{2} c_{5}+106 c_{1} c_{3} c_{4}-5 c_{2}^2 c_{4}+11 c_{2} c_{3}^2+740 c_{1} c_{7}+146 c_{2} c_{6}+83 c_{3} c_{5}+27 c_{4}^2+456 c_{8}) T^8+(-6 c_{1}^3 c_{4}-9 c_{1}^2 c_{2} c_{3}-3 c_{1} c_{2}^3-48 c_{1}^2 c_{5}-42 c_{1} c_{2} c_{4}-9 c_{1} c_{3}^2-9 c_{2}^2 c_{3}-126 c_{1} c_{6}-51 c_{2} c_{5}-21 c_{3} c_{4}-108 c_{7}) T^7+(2 c_{1}^2 c_{4}+3 c_{1} c_{2} c_{3}+c_{2}^3+10 c_{1} c_{5}+7 c_{2} c_{4}+c_{3}^2+12 c_{6}) T^6
SSM-Thom polynomial in Schur functions T^6 (36 s_{6}+5 s_{3, 3}+19 s_{4, 2}+30 s_{5, 1}+ s_{2, 2, 2}+5 s_{3, 2, 1}+6 s_{4, 1, 1})+T^7 (-432 s_{7}-144 s_{4, 3}-318 s_{5, 2}-468 s_{6, 1}-27 s_{3, 2, 2}-45 s_{3, 3, 1}-135 s_{4, 2, 1}-162 s_{5, 1, 1}-3 s_{2, 2, 2, 1}-15 s_{3, 2, 1, 1}-18 s_{4, 1, 1, 1})+T^8 (6 s_{2, 2, 2, 1, 1}+2 s_{2, 2, 2, 2}+30 s_{3, 2, 1, 1, 1}+78 s_{3, 2, 2, 1}+120 s_{3, 3, 1, 1}+151 s_{3, 3, 2}+36 s_{4, 1, 1, 1, 1}+352 s_{4, 2, 1, 1}+334 s_{4, 2, 2}+773 s_{4, 3, 1}+409 s_{4, 4}+408 s_{5, 1, 1, 1}+1540 s_{5, 2, 1}+1383 s_{5, 3}+1776 s_{6, 1, 1}+2542 s_{6, 2}+3468 s_{7, 1}+2520 s_{8})
Local algebra C[x,y]/(x^2,xy,y^2)
Thom-Boardman class \Sigma^{2,0}
Codimension 6
SSM-Thom polynomial in Chern classes (-6 c_{1}^2 c_{2} c_{4}+6 c_{1}^2 c_{3}^2-12 c_{1} c_{2} c_{5}+12 c_{1} c_{3} c_{4}+2 c_{2}^2 c_{4}-2 c_{2} c_{3}^2-8 c_{2} c_{6}+5 c_{3} c_{5}+3 c_{4}^2) T^8+(3 c_{1} c_{2} c_{4}-3 c_{1} c_{3}^2+3 c_{2} c_{5}-3 c_{3} c_{4}) T^7+(-c_{2} c_{4}+c_{3}^2) T^6
SSM-Thom polynomial in Schur functions T^6 s_{3, 3}+T^7 (-6 s_{4, 3}-3 s_{3, 3, 1})+T^8 (6 s_{3, 3, 1, 1}+4 s_{3, 3, 2}+22 s_{4, 3, 1}+21 s_{4, 4}+24 s_{5, 3})

codimension 7

Local algebra C[x,y]/(xy,x^2+y^2)
Thom-Boardman class \Sigma^{2,0}
Codimension 7
SSM-Thom polynomial in Chern classes (3 c_{1}^2 c_{2} c_{4}-3 c_{1}^2 c_{3}^2+11 c_{1} c_{2} c_{5}-11 c_{1} c_{3} c_{4}+2 c_{2}^2 c_{4}-2 c_{2} c_{3}^2+10 c_{2} c_{6}-6 c_{3} c_{5}-4 c_{4}^2) T^8+(-c_{1} c_{2} c_{4}+c_{1} c_{3}^2-2 c_{2} c_{5}+2 c_{3} c_{4}) T^7
SSM-Thom polynomial in Schur functions T^7 (3 s_{4, 3}+ s_{3, 3, 1})+T^8 (-3 s_{3, 3, 1, 1}-5 s_{3, 3, 2}-19 s_{4, 3, 1}-18 s_{4, 4}-26 s_{5, 3})

codimension 8

Local algebra C[x]/(x^5)
Thom-Boardman class \Sigma^{1,1,1,1,0}
Codimension 8
SSM-Thom polynomial in Chern classes (6 c_{1}^3 c_{5}+9 c_{1}^2 c_{2} c_{4}+2 c_{1}^2 c_{3}^2+6 c_{1} c_{2}^2 c_{3}+c_{2}^4+54 c_{1}^2 c_{6}+53 c_{1} c_{2} c_{5}+17 c_{1} c_{3} c_{4}+16 c_{2}^2 c_{4}+4 c_{2} c_{3}^2+156 c_{1} c_{7}+76 c_{2} c_{6}+21 c_{3} c_{5}+11 c_{4}^2+144 c_{8}) T^8
SSM-Thom polynomial in Schur functions T^8 ( s_{2, 2, 2, 2}+9 s_{3, 2, 2, 1}+10 s_{3, 3, 1, 1}+21 s_{3, 3, 2}+26 s_{4, 2, 1, 1}+55 s_{4, 2, 2}+104 s_{4, 3, 1}+76 s_{4, 4}+24 s_{5, 1, 1, 1}+210 s_{5, 2, 1}+240 s_{5, 3}+216 s_{6, 1, 1}+460 s_{6, 2}+624 s_{7, 1}+576 s_{8})
Local algebra C[x,y]/(x^2,xy,y^3)
Thom-Boardman class \Sigma^{2,0}
Codimension 8
SSM-Thom polynomial in Chern classes (-2 c_{1} c_{2} c_{5}+2 c_{1} c_{3} c_{4}-2 c_{2}^2 c_{4}+2 c_{2} c_{3}^2-4 c_{2} c_{6}+6 c_{3} c_{5}-2 c_{4}^2) T^8
SSM-Thom polynomial in Schur functions T^8 (2 s_{3, 3, 2}+4 s_{4, 3, 1}+8 s_{5, 3})