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KTp’s, l=1

K-theoretic Thom polynomials of contact singularities of relative dimension 1, in Grothendieck expansions

(Hierarchy of these singularities)

codimension 0

Local algebra C
Thom-Boardman class \Sigma^0
Codimension 0
K-Thom polynomial in Grothendieck expansion g_{0}=1
Remarks This contact singularity is open in its Thom-Boardman class.

codimension 2

Local algebra C[x]/(x^2)
Thom-Boardman class \Sigma^{1,0}
Codimension 2
K-Thom polynomial in Grothendieck expansion g_{2}
Remarks This contact singularity is open in its Thom-Boardman class.

codimension 4

Local algebra C[x]/(x^3)
Thom-Boardman class \Sigma^{1,1,0}
Codimension 4
K-Thom polynomial in Grothendieck expansion g_{2, 2}+ 2g_{3, 1}+ 4g_{4}
– ( 2g_{3, 2}+ 5g_{4, 1}+ 4g_{5} )
+ g_{4, 2}+ 4g_{5, 1}+ g_{6}
-g_{6, 1}
Remarks This contact singularity is open in its Thom-Boardman class.
Source: R. Rimanyi, A. Szenes: Residuesin Grothendieck polynomials, and K-theroetic Thom polynomials, preprint

codimension 6

Local algebra C[x]/(x^4)
Thom-Boardman class \Sigma^{1,1,1,0}
Codimension 6
K-Thom polynomial in Grothendieck expansion

g_{2,2,2}+ 5g_{3,2,1}+ 19g_{4,2}+ 6g_{4,1,1}+ 30g_{5,1}+ 36g_{6}
– ( 5g_{3,2,2}+ 35g_{4,2,1}+ 91g_{5,2}+ 39g_{5,1,1}+ 141g_{6,1}+ 108g_{7} )
+ 10g_{4,2,2}+ 110g_{5,2,1}+ 195g_{6,2}+ 110g_{6,1,1}+ 283g_{7,1}+ 141g_{8}
– ( 10g_{5,2,2}+ 205g_{6,2,1}+ 249g_{7,2}+ 176g_{7,1,1}+ 316g_{8,1}+ 102g_{9})
+ 5g_{6,2,2}+ 251g_{7,2,1}+ 210g_{8,2}+ 175g_{8,1,1}+ 213g_{9,1}+ 43g_{10}
– (g_{7,2,2}+ 210g_{8,2,1}+ 120g_{9,2}+ 111g_{9,1,1}+ 87g_{10,1}+ 10g_{11} )
+ 120g_{9,2,1}+ 45g_{10,2}+ 44g_{10,1,1}+ 20g_{11,1}+ g_{12}
– (45g_{10,2,1}+ 10g_{11,2}+ 10g_{11,1,1}+ 2g_{12,1} )
+ 10g_{11,2,1}+ g_{12,2}+ g_{12,1,1}
– g_{12,2,1}

 5g_{3,3}
– ( 9g_{3,3,1}+ 33g_{4,3})
+ 4g_{3,3,2}+ 43g_{4,3,1}+ 10g_{4,4}+ 81g_{5,3}
– (10g_{4,3,2}+ 10g_{4,4,1}+ 91g_{5,3,1}+ 20g_{5,4}+ 105g_{6,3})
+ 10g_{5,3,2}+ 20g_{5,4,1}+ 110g_{6,3,1}+ 15g_{6,4}+ 80g_{7,3}
– ( 5g_{6,3,2}+ 15g_{6,4,1}+ 81g_{7,3,1}+ 6g_{7,4}+ 36g_{8,3} )
+ g_{7,3,2}+ 6g_{7,4,1}+ 36g_{8,3,1}+ g_{8,4}+ 9g_{9,3}
– (g_{8,4,1}+ 9g_{9,3,1}+ g_{10,3} )
+ g_{10,3,1}
Remarks This contact singularity is open in its Thom-Boardman class.
Source: unpublished
Local algebra C[x,y]/(x^2,xy,y^2)
Thom-Boardman class \Sigma^{2,0}
Codimension 6
K-Thom polynomial in Grothendieck expansion g_{3,3}
Remarks This contact singularity is open in its Thom-Boardman class.

codimension 7

Local algebra C[x,y]/(x^2,y^2)
Thom-Boardman class \Sigma^{2,0}
Codimension 7
K-Thom polynomial in Grothendieck expansion g_{3, 3, 1}+3g_{4, 3}
– ( 3g_{4, 3, 1}+ 3g_{4, 4} + 3g_{5, 3} )
+ 3g_{4, 4, 1}+ 3g_{5, 3, 1}+ 5g_{5, 4}+ g_{6, 3}
– ( 5g_{5, 4, 1} +g_{6, 3, 1}+ g_{5, 5}+ 2g_{6, 4} )
+ g_{5, 5, 1}+ 2g_{6, 4, 1}+ g_{6, 5}
– g_{6, 5, 1}
Remarks Source: unpublished

codimension 8

Local algebra C[x]/(x^5)
Thom-Boardman class \Sigma^{1,1,1,1,0}
Codimension 8
K-Thom polynomial in Grothendieck expansion ?
Local algebra C[x,y]/(x^2,xy,y^3)
Thom-Boardman class \Sigma^{2,0}
Codimension 8
K-Thom polynomial in Grothendieck expansion 2g_{3, 3, 2}+ 4g_{4, 3, 1}+ 8g_{5, 3}
– ( g_{3, 3, 3}+ 5g_{4, 3, 2}+ 12g_{5, 3, 1}+ 12g_{6, 3} )
+ g_{4, 3, 3}+ 4g_{5, 3, 2}+ 13g_{6, 3, 1}+ 6g_{7, 3}
– ( g_{6, 3, 2}+ 6g_{7, 3, 1}+ g_{8, 3})
+ g_{8, 3, 1}
Remarks Source: unpublished