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Tp’s, l=1

Thom polynomials of contact singularities of relative dimension 1

(the singularities)

codimension 0

Local algebra C
Thom-Boardman class \Sigma^0
Codimension 0
Thom polynomial in Chern classes 1
Thom polynomial in Schur functions 1
Remarks This contact singularity is open in its Thom-Boardman class.
Source: Implicit function theorem.

codimension 2

Local algebra C[x]/(x^2)
Thom-Boardman class \Sigma^{1,0}
Codimension 2
Thom polynomial in Chern classes c_2
Thom polynomial in Schur functions s_2
Remarks This contact singularity is open in its Thom-Boardman class.
Source: I. Porteous: Simple singularities of maps. In Proc. Liverpool Singularities I, Springer LNM 192 (1971), 268-307

codimension 4

Local algebra C[x]/(x^3)
Thom-Boardman class \Sigma^{1,1,0}
Codimension 4
Thom polynomial in Chern classes 2 c_{4}+ c_{2}^2+ c_{1} c_{3}
Thom polynomial in Schur functions s_{2, 2}+ 2 s_{3, 1}+ 4 s_{4}
Remarks This contact singularity is open in its Thom-Boardman class.
Source: F. Ronga: Le calcul des classes duales aux singularités de Boardman d’ordre deux, Commentarii mathematici Helvetici (1972), Vol. 47, 15-35

codimension 6

Local algebra C[x]/(x^4)
Thom-Boardman class \Sigma^{1,1,1,0}
Codimension 6
Thom polynomial in Chern classes c_{2}^3+ 3 c_{1} c_{2} c_{3}+ 2 c_{1}^2 c_{4}+ c_{3}^2+ 12 c_{6}+ 10 c_{1} c_{5}+ 7 c_{2} c_{4}
Thom polynomial in Schur functions s_{2, 2, 2}+ 5 s_{3, 2, 1}+ 5 s_{3, 3}+ 6 s_{4, 1, 1}+ 19 s_{4, 2}+ 30 s_{5, 1}+ 36 s_{6}
Remarks This contact singularity is open in its Thom-Boardman class.
Source:
Local algebra C[x,y]/(x^2,xy,y^2)
Thom-Boardman class \Sigma^{2,0}
Codimension 6
Thom polynomial in Chern classes c_{3}^2-c_{2} c_{4}
Thom polynomial in Schur functions s_{3,3}
Remarks This contact singularity is open in its Thom-Boardman class.
Source: I. Porteous: Simple singularities of maps. In Proc. Liverpool Singularities I, Springer LNM 192 (1971), 268-307

codimension 7

Local algebra C[x,y]/(x^2,y^2)
Thom-Boardman class \Sigma^{2,0}
Codimension 7
Thom polynomial in Chern classes 2 c_{3} c_{4}-2 c_{2} c_{5}+ c_{1} c_{3}^2-c_{1} c_{2} c_{4}
Thom polynomial in Schur functions s_{3, 3, 1}+ 3 s_{4, 3}
Remarks Source: R. Rimanyi: Thom polynomials, symmetries and incidences of singularities; Inv. Math. 143, 499-521 (2001)

codimension 8

Local algebra C[x]/(x^5)
Thom-Boardman class \Sigma^{1,1,1,1,0}
Codimension 8
Thom polynomial in Chern classes c_{2}^4+ 6 c_{1} c_{2}^2 c_{3}+ 2 c_{1}^2 c_{3}^2+ 4 c_{2} c_{3}^2+ 9 c_{1}^2 c_{2} c_{4}+ 16 c_{2}^2 c_{4}+ 17 c_{1} c_{3} c_{4}+ 11 c_{4}^2+ 6 c_{1}^3 c_{5}+ 53 c_{1} c_{2} c_{5}+ 21 c_{3} c_{5}+ 54 c_{1}^2 c_{6}+ 76 c_{2} c_{6}+ 156 c_{1} c_{7}+ 144 c_{8}
Thom polynomial in Schur functions 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}
Remarks This contact singularity is open in its Thom-Boardman class.
Source: R. Rimanyi: Thom polynomials, symmetries and incidences of singularities; Inv. Math. 143, 499-521 (2001)
Local algebra C[x,y]/(x^2,xy,y^3)
Thom-Boardman class \Sigma^{2,0}
Codimension 8
Thom polynomial in Chern classes 2 c_{2} c_{3}^2-2 c_{2}^2 c_{4}+ 2 c_{1} c_{3} c_{4}-2 c_{4}^2-2 c_{1} c_{2} c_{5}+ 6 c_{3} c_{5}-4 c_{2} c_{6}
Thom polynomial in Schur functions 2 s_{3, 3, 2}+ 4 s_{4, 3, 1}+ 8 s_{5, 3}
Remarks Source: R. Rimanyi: Thom polynomials, symmetries and incidences of singularities; Inv. Math. 143, 499-521 (2001)

codimension 9

Local algebra C[x,y],(xy,x^2+y^3)
Thom-Boardman class \Sigma^{2,0}
Codimension 9
Thom polynomial in Chern classes -2 c_1^2 c_2 c_5+ 2 c_1^2 c_3 c_4-2 c_1 c_2^2 c_4+ 2 c_1 c_2 c_3^2-10 c_1 c_2 c_6+ 9 c_1 c_3 c_5+ c_1 c_4^2-5 c_2^2 c_5+ 4 c_2 c_3 c_4+ c_3^3-12 c_2 c_7+ 10 c_3 c_6+ 2 c_4 c_5
Thom polynomial in Schur functions 2 s_{3, 3, 2, 1}+ 3 s_{3, 3, 3}+ 4 s_{4, 3, 1, 1}+ 12 s_{4, 3, 2}+ 12 s_{4, 4, 1}+ 24 s_{5, 3, 1}+ 24 s_{5, 4}+ 32 s_{6, 3}
Remarks Source: unpublished

codimension 10

Local algebra C[x],(x^6)
Thom-Boardman class \Sigma^{1,1,1,1,1,0}
Codimension 10
Thom polynomial in Chern classes 24 c_1^4 c_6+ 38 c_1^3 c_2 c_5+ 12 c_1^3 c_3 c_4+ 25 c_1^2 c_2^2 c_4+ 10 c_1^2 c_2 c_3^2+ 10 c_1 c_2^3 c_3+ c_2^5+ 336 c_1^3 c_7+ 400 c_1^2 c_2 c_6+ 115 c_1^2 c_3 c_5+ 39 c_1^2 c_4^2+ 170 c_1 c_2^2 c_5+ 95 c_1 c_2 c_3 c_4+ 5 c_1 c_3^3+ 30 c_2^3 c_4+ 10 c_2^2 c_3^2+ 1704 c_1^2 c_8+ 1366 c_1 c_2 c_7+ 389 c_1 c_3 c_6+ 233 c_1 c_4 c_5+ 285 c_2^2 c_6+ 136 c_2 c_3 c_5+ 68 c_2 c_4^2+ 19 c_3^2 c_4+ 3696 c_1 c_9+ 1508 c_2 c_8+ 450 c_3 c_7+ 268 c_4 c_6+ 78 c_5^2+ 2880 c_{10}
Thom polynomial in Schur functions s_{2, 2, 2, 2, 2}+ 14 s_{3, 2, 2, 2, 1}+ 35 s_{3, 3, 2, 1, 1}+ 56 s_{3, 3, 2, 2}+ 70 s_{3, 3, 3, 1}+ 71 s_{4, 2, 2, 1, 1}+ 125 s_{4, 2, 2, 2}+ 92 s_{4, 3, 1, 1, 1}+ 573 s_{4, 3, 2, 1}+ 381 s_{4, 3, 3}+ 430 s_{4, 4, 1, 1}+ 783 s_{4, 4, 2}+ 154 s_{5, 2, 1, 1, 1}+ 840 s_{5, 2, 2, 1}+ 1326 s_{5, 3, 1, 1}+ 2232 s_{5, 3, 2}+ 2850 s_{5, 4, 1}+ 1432 s_{5, 5}+ 120 s_{6, 1, 1, 1, 1}+ 2036 s_{6, 2, 1, 1}+ 2885 s_{6, 2, 2}+ 6146 s_{6, 3, 1}+ 4880 s_{6, 4}+ 1680 s_{7, 1, 1, 1}+ 9254 s_{7, 2, 1}+ 9524 s_{7, 3}+ 8520 s_{8, 1, 1}+ 15196 s_{8, 2}+ 18480 s_{9, 1}+ 14400 s_{10}
Remarks Source: unpublished
Local algebra C[x,y],(x^2,xy,y^4)
Thom-Boardman class \Sigma^{2,0}
Codimension 10
Thom polynomial in Chern classes -6 c_1^2 c_2 c_6+ 9 c_1^2 c_3 c_5-3 c_1^2 c_4^2-5 c_1 c_2^2 c_5+ 2 c_1 c_2 c_3 c_4+ 3 c_1 c_3^3-2 c_2^3 c_4+ 2 c_2^2 c_3^2-30 c_1 c_2 c_7+ 43 c_1 c_3 c_6-13 c_1 c_4 c_5-13 c_2^2 c_6+ 13 c_2 c_3 c_5-9 c_2 c_4^2+ 9 c_3^2 c_4-36 c_2 c_8+ 50 c_3 c_7-12 c_4 c_6-2 c_5^2
Thom polynomial in Schur functions 104 s_{7, 3}+ 2 s_{3, 3, 2, 2}+ 5 s_{3, 3, 3, 1}+ 12 s_{4, 3, 2, 1}+ 24 s_{4, 3, 3}+ 4 s_{4, 4, 1, 1}+ 12 s_{4, 4, 2}+ 16 s_{5, 3, 1, 1}+ 50 s_{5, 3, 2}+ 28 s_{5, 4, 1}+ 8 s_{5, 5}+ 84 s_{6, 3, 1}+ 40 s_{6, 4}
Remarks Source: unpublished
Local algebra C[x,y],(x^3,xy,y^3)
Thom-Boardman class \Sigma^{2,0}
Codimension 10
Thom polynomial in Chern classes -3 c_1^2 c_3 c_5+ 3 c_1^2 c_4^2-2 c_1 c_2^2 c_5+ 3 c_1 c_2 c_3 c_4-c_1 c_3^3-c_2^3 c_4+ c_2^2 c_3^2-7 c_1 c_3 c_6+ 7 c_1 c_4 c_5-4 c_2^2 c_6+ 8 c_2 c_3 c_5-2 c_2 c_4^2-2 c_3^2 c_4-2 c_3 c_7-9 c_4 c_6+ 11 c_5^2
Thom polynomial in Schur functions 7 s_{5, 3, 2}+ 20 s_{5, 4, 1}+ 28 s_{5, 5}+ 6 s_{6, 3, 1}+ 16 s_{6, 4}+ 4 s_{7, 3}+ s_{3, 3, 2, 2}+ 3 s_{4, 3, 2, 1}+ 6 s_{4, 4, 1, 1}+ 3 s_{4, 4, 2}+ 2 s_{5, 3, 1, 1}
Remarks Source: unpublished

codimension 11

Local algebra C[x,y]/(xy,x^2+y^4)
Thom-Boardman class \Sigma^{2,0}
Codimension 11
Thom polynomial in Chern classes -6 c_1^3 c_2 c_6+ 9 c_1^3 c_3 c_5-3 c_1^3 c_4^2-5 c_1^2 c_2^2 c_5+ 2 c_1^2 c_2 c_3 c_4+ 3 c_1^2 c_3^3-2 c_1 c_2^3 c_4+ 2 c_1 c_2^2 c_3^2-54 c_1^2 c_2 c_7+ 67 c_1^2 c_3 c_6-13 c_1^2 c_4 c_5-33 c_1 c_2^2 c_6+ 19 c_1 c_2 c_3 c_5-7 c_1 c_2 c_4^2+ 21 c_1 c_3^2 c_4-6 c_2^3 c_5+ 4 c_2^2 c_3 c_4+ 2 c_2 c_3^3-156 c_1 c_2 c_8+ 162 c_1 c_3 c_7+ 24 c_1 c_4 c_6-30 c_1 c_5^2-52 c_2^2 c_7+ 28 c_2 c_3 c_6-12 c_2 c_4 c_5+ 16 c_3^2 c_5+ 20 c_3 c_4^2-144 c_2 c_9+ 128 c_3 c_8+ 52 c_4 c_7-36 c_5 c_6
Thom polynomial in Schur functions 242 s_{5, 4, 2}+ 100 s_{5, 5, 1}+ 164 s_{6, 3, 1, 1}+ 334 s_{6, 3, 2}+ 432 s_{6, 4, 1}+ 160 s_{6, 5}+ 524 s_{7, 3, 1}+ 512 s_{7, 4}+ 520 s_{8, 3}+ 2 s_{3, 3, 2, 2, 1}+ 5 s_{3, 3, 3, 1, 1}+ 9 s_{3, 3, 3, 2}+ 12 s_{4, 3, 2, 1, 1}+ 22 s_{4, 3, 2, 2}+ 63 s_{4, 3, 3, 1}+ 4 s_{4, 4, 1, 1, 1}+ 56 s_{4, 4, 2, 1}+ 96 s_{4, 4, 3}+ 16 s_{5, 3, 1, 1, 1}+ 126 s_{5, 3, 2, 1}+ 172 s_{5, 3, 3}+ 96 s_{5, 4, 1, 1}
Remarks Source: unpublished
Local algebra C[x,y]/(xy,x^3+y^3)
Thom-Boardman class \Sigma^{2,0}
Codimension 11
Thom polynomial in Chern classes -3 c_1^3 c_3 c_5+ 3 c_1^3 c_4^2-2 c_1^2 c_2^2 c_5+ 3 c_1^2 c_2 c_3 c_4-c_1^2 c_3^3-c_1 c_2^3 c_4+ c_1 c_2^2 c_3^2-17 c_1^2 c_3 c_6+ 17 c_1^2 c_4 c_5-10 c_1 c_2^2 c_6+ 9 c_1 c_2 c_3 c_5+ 5 c_1 c_2 c_4^2-4 c_1 c_3^2 c_4-3 c_2^3 c_5+ 2 c_2^2 c_3 c_4+ c_2 c_3^3-26 c_1 c_3 c_7-6 c_1 c_4 c_6+ 32 c_1 c_5^2-12 c_2^2 c_7+ 9 c_2 c_3 c_6+ 7 c_2 c_4 c_5+ 3 c_3^2 c_5-7 c_3 c_4^2-8 c_3 c_8-32 c_4 c_7+ 40 c_5 c_6
Thom polynomial in Schur functions 3 s_{4, 3, 2, 1, 1}+ 8 s_{4, 3, 2, 2}+ 6 s_{4, 3, 3, 1}+ 6 s_{4, 4, 1, 1, 1}+ 24 s_{4, 4, 2, 1}+ 6 s_{4, 4, 3}+ 2 s_{5, 3, 1, 1, 1}+ 24 s_{5, 3, 2, 1}+ 14 s_{5, 3, 3}+ 56 s_{5, 4, 1, 1}+ 70 s_{5, 4, 2}+ 116 s_{5, 5, 1}+ 16 s_{6, 3, 1, 1}+ 41 s_{6, 3, 2}+ 134 s_{6, 4, 1}+ 188 s_{6, 5}+ 34 s_{7, 3, 1}+ 92 s_{7, 4}+ 20 s_{8, 3}+ s_{3, 3, 2, 2, 1}+ 2 s_{3, 3, 3, 2}
Remarks Source: unpublished
Local algebra C[x,y]/(x^2,xy^2,y^3)
Thom-Boardman class \Sigma^{2,1,0}
Codimension 11
Thom polynomial in Chern classes -2 c_1^2 c_3 c_6+ 2 c_1^2 c_4 c_5-4 c_1 c_2 c_3 c_5+ 2 c_1 c_2 c_4^2+ 2 c_1 c_3^2 c_4-2 c_2^2 c_3 c_4+ 2 c_2 c_3^3-6 c_1 c_3 c_7+ 6 c_1 c_4 c_6-6 c_2 c_3 c_6+ 2 c_2 c_4 c_5+ 6 c_3^2 c_5-2 c_3 c_4^2-4 c_3 c_8+ 2 c_4 c_7+ 2 c_5 c_6
Thom polynomial in Schur functions 2 s_{3, 3, 3, 2}+ 2 s_{4, 3, 2, 2}+ 6 s_{4, 3, 3, 1}+ 6 s_{4, 4, 2, 1}+ 6 s_{4, 4, 3}+ 6 s_{5, 3, 2, 1}+ 14 s_{5, 3, 3}+ 8 s_{5, 4, 1, 1}+ 20 s_{5, 4, 2}+ 16 s_{5, 5, 1}+ 4 s_{6, 3, 1, 1}+ 14 s_{6, 3, 2}+ 28 s_{6, 4, 1}+ 24 s_{6, 5}+ 12 s_{7, 3, 1}+ 24 s_{7, 4}+ 8 s_{8, 3}
Remarks This contact singularity is open in its Thom-Boardman class.
Source: L. M. Fehér and B. Kőműves: On second order Thom–Boardman singularities, Fund. Math. 191 (2006) 249–264