Thom series

Thom polynomials of contact singularities of arbitrary relative dimension

Let Q be a finite-dimensional local algebra and let \mu=\dim(Q)-1. Choose a multiplicative filtration of its maximal ideal, with profile \omega. Repeated filtration steps and zero entries in the profile are suppressed.

The tables below describe the Thom polynomials of the contact singularity \Sigma_Q^l for all relative dimensions l at the same time, in the following two forms.

  1. For the chosen filtration profile \omega, the table gives a rational function F_{Q,\omega} in the variables z_1,\ldots,z_\mu such that

    \text{Tp}(\Sigma_Q^l)=(-1)^\mu \text{Res}_{z_1=\infty}\text{Res}_{z_2=\infty}\ldots\text{Res}_{z_\mu=\infty}\left(F_{Q,\omega}V_\mu P_{\mu,l}\right).

    Here

    • V_\mu=\prod_{j=1}^\mu\prod_{i=1}^{j-1}(z_j-z_i) is the Vandermonde factor;
    • P_{\mu,l}=\prod_{j=1}^\mu\left(\sum_{q=0}^\infty c_q/z_j^q\right)\left(\prod_{j=1}^\mu z_j\right)^l dz_\mu\wedge dz_{\mu-1}\wedge\ldots\wedge dz_1, where c_0=1;
    • every multiplication weight is oriented as W_{ij}^k=z_k-z_i-z_j, that is, output weight minus the two input weights.

    The displayed function F_{Q,\omega} includes the monomial contribution of the chosen filtration. Thus the tables do not list a separate rational factor and a separate filtration monomial. In particular, a factor such as z_1 in F_{Q,\omega} may come from the filtration rather than from the algebra locus.

    Apart from this filtration monomial, the numerator can be written as a polynomial with nonnegative rational coefficients in the multiplication weights W_{ij}^k. When the chosen filtration is not intrinsic, these rational coefficients also include the reciprocal of the generic number of compatible filtrations. If the family of compatible filtrations is positive-dimensional, this scalar normalization does not apply; such entries will be marked explicitly.

    Cancellation between numerator and denominator is allowed, but the tables preferentially display an unreduced expression when this makes the positive weight numerator visible. The rational function representing a given Thom polynomial need not be unique.

    Below every residue formula, a read-only box gives the same rational function as a copyable Maple expression. In these boxes, variables are written as z[1], z[2], and so on, and multiplication signs are explicit.

  2. The Thom series of Q is a formal power series \text{Ts}_Q in variables d_i, i\in Z, characterized by

    \text{Tp}(\Sigma_Q^l)=\text{Ts}_Q|_{d_i=c_{l+1+i}}.

    We use the notation \Delta_{\lambda_1,\ldots,\lambda_r}=\text{det}\left(d_{\lambda_i+j-i}\right)_{i,j=1,\ldots,r}.

All formulas below use these conventions. In particular, signs and orientations may differ from older versions of this page.

\mu=1

Local algebra C[x]/(x^2)
\mu 1
Tp degree l+1
Residue presentations
Profile \omega=(1) F_{Q,(1)}=1

Maple expression:
Explicit Thom series d_0=\Delta_0

\mu=2

Local algebra C[x]/(x^3)
\mu 2
Tp degree 2l+2
Residue presentations
Profile \omega=(1,1) F_{Q,(1,1)}=\frac{1}{z_2-2z_1}

Maple expression:
Explicit Thom series \sum_{i=0}^\infty 2^i\Delta_{i,-i}
Local algebra C[x,y]/(x^2,xy,y^2)
\mu 2
Tp degree 2l+4
Residue presentations
Profile \omega=(2) F_{Q,(2)}=z_1

Maple expression:
Explicit Thom series \Delta_{1,1}

\mu=3

Local algebra C[x]/(x^4)
\mu 3
Tp degree 3l+3
Residue presentations
Profile \omega=(1,1,1) F_{Q,(1,1,1)}=\frac{1}{(z_2-2z_1)(z_3-2z_1)(z_3-z_1-z_2)}

Maple expression:
Explicit Thom series \sum_{i=0}^\infty 2^i d_{-i}d_0d_i+1/3\cdot\sum_{i=1}^\infty\sum_{j=1}^\infty 2^i3^j d_{-i}d_{-j}d_{i+j}+1/2\cdot\sum_{i=0}^\infty\sum_{j=0}^\infty a_{i,j}d_{-i-j}d_id_j where a_{0,0}=0, a_{i,0}=a_{0,i}=3^{i-1} for i>0, and a_{i,j}=a_{i-1,j}+a_{i,j-1} for i,j>0.
Local algebra C[x,y]/(xy,x^2+y^2)
\mu 3
Tp degree 3l+4
Residue presentations
Profile \omega=(2,1) F_{Q,(2,1)}=\frac{z_1}{(z_3-2z_1)(z_3-z_1-z_2)(z_3-2z_2)}

Maple expression:
Profile \omega=(1,1,1) F_{Q,(1,1,1)}=\frac{1}{2(z_3-2z_1)(z_3-z_1-z_2)}

Maple expression:

The factor 1/2 corrects for the two compatible refined filtrations on a generic algebra.
Explicit Thom series \sum_{0\leq j<i}((i,j))\Delta_{i,j+1,-i-j}, where ((i,i))=0, ((i,0))=2^i-1, and ((i,j))=((i-1,j))+((i,j-1)) for 0<j<i.
Local algebra C[x,y]/(x^2,xy,y^3)
\mu 3
Tp degree 3l+5
Residue presentations
Profile \omega=(2,1) F_{Q,(2,1)}=\frac{2z_1}{(z_3-2z_1)(z_3-2z_2)}

Maple expression:
Profile \omega=(1,2) F_{Q,(1,2)}=\frac{z_2}{(z_2-2z_1)(z_3-2z_1)}

Maple expression:
Profile \omega=(1,1,1) F_{Q,(1,1,1)}=\frac{1}{z_3-2z_1}

Maple expression:
Explicit Thom series \sum_{i=0}^\infty 2^{i+1}\Delta_{i+1,1,-i}
Local algebra C[x,y,z]/(x,y,z)^2
\mu 3
Tp degree 3l+9
Residue presentations
Profile \omega=(3) F_{Q,(3)}=z_1^2z_2

Maple expression:
Explicit Thom series \Delta_{2,2,2}

\mu=4 — under construction

Local algebra C[x]/(x^5)
\mu 4
Tp degree 4l+4
Residue presentations
Profile \omega=(1,1,1,1) F_{A_4,(1,1,1,1)}=\frac{z_4-2z_1-z_2}{(z_2-2z_1)(z_3-2z_1)(z_3-z_1-z_2)(z_4-2z_1)(z_4-z_1-z_2)(z_4-z_1-z_3)(z_4-2z_2)}

Maple expression:
Sign check The historical page used the opposite numerator. With the corrected sign, \text{Tp}(A_4^0)=c_1^4+6c_1^2c_2+2c_2^2+9c_1c_3+6c_4.
Explicit Thom series Not presently listed.
Local algebra C[x,y]/(xy,x^2+y^3)
\mu 4
Tp degree 4l+5
Residue presentations
Profile \omega=(2,1,1) F_{I_{23},(2,1,1)}=\frac{z_1((z_4-2z_1-z_2)(z_4-z_1-2z_2)-(z_4-z_2-z_3)(z_4-z_1-z_3))}{(z_3-2z_1)(z_4-2z_1)(z_3-z_1-z_2)(z_4-z_1-z_2)(z_4-z_1-z_3)(z_3-2z_2)(z_4-2z_2)(z_4-z_2-z_3)}

Maple expression:
Profile \omega=(1,1,1,1), graded in degrees (2,3,4,6) F_{I_{23}}=\frac{1}{(z_3-2z_1)(z_4-2z_1)(z_4-z_1-z_2)(z_4-z_1-z_3)(z_4-2z_2)}

Maple expression:
Filtration note Kazarian lists the dimension vectors (2,1,1) and (0,1,1,1,0,1). In the second presentation, suppressing zero entries gives (1,1,1,1), while the nonzero graded pieces occur in degrees (2,3,4,6).
Explicit Thom series To be added.
Local algebra C[x,y]/(x^2,xy,y^4)
\mu 4
Tp degree 4l+6
Residue presentations
Profile \omega=(1,2,1) F_{III_{24},(1,2,1)}=\frac{z_2}{(z_2-2z_1)(z_3-2z_1)(z_4-2z_1)(z_4-z_1-z_2)(z_4-z_1-z_3)}

Maple expression:
Profile \omega=(1,1,2) F_{III_{24},(1,1,2)}=\frac{z_3}{(z_2-2z_1)(z_3-2z_1)(z_4-2z_1)(z_3-z_1-z_2)(z_4-z_1-z_2)}

Maple expression:
Profile \omega=(1,1,1,1), graded in degrees (2,4,5,6) F_{III_{24}}=\frac{1}{(z_2-2z_1)(z_3-2z_1)(z_4-2z_1)(z_4-z_1-z_2)}

Maple expression:
Profile \omega=(1,1,1,1), graded in degrees (3,5,6,9) F_{III_{24}}=\frac{1}{(z_3-2z_1)(z_4-2z_1)(z_4-z_1-z_2)(z_4-z_1-z_3)}

Maple expression:
Filtration note Kazarian lists the dimension vectors (1,2,1), (1,1,2), (0,1,0,1,1,1), and (0,0,1,0,1,1,0,0,1). The last two both reduce to (1,1,1,1) after zero entries are suppressed, but they retain different quasihomogeneous degree data.
Explicit Thom series To be added.
Local algebra C[x,y]/(x^3,xy,y^3)
\mu 4
Tp degree 4l+6
Residue presentations
Profile \omega=(2,2) F_{III_{33},(2,2)}=\frac{z_1z_3}{(z_3-2z_1)(z_4-2z_1)(z_3-z_1-z_2)(z_4-z_1-z_2)(z_3-2z_2)(z_4-2z_2)}

Maple expression:
Profile \omega=(1,1,1,1), graded in degrees (3,4,6,8) F_{III_{33}}=\frac{1}{2(z_3-2z_1)(z_4-2z_1)(z_4-z_1-z_2)(z_4-2z_2)}

Maple expression:
Filtration note Kazarian lists the dimension vectors (2,2) and (0,0,1,1,0,1,0,1). The second reduces to (1,1,1,1), with nonzero graded pieces in degrees (3,4,6,8). Its factor 1/2 is the normalization appearing in Kazarian’s table.
Explicit Thom series To be added.
Local algebra C[x,y]/(x^2,xy^2,y^3)
\mu 4
Tp degree 4l+7
Residue presentations
Profile \omega=(2,2) F_{\Sigma^{2,1},(2,2)}=\frac{2z_1z_3(z_3+z_4-2z_1-2z_2)}{(z_3-2z_1)(z_4-2z_1)(z_3-z_1-z_2)(z_4-z_1-z_2)(z_3-2z_2)(z_4-2z_2)}

Maple expression:
Profile \omega=(1,1,2), graded in degrees (2,3,5) F_{\Sigma^{2,1}}=\frac{z_3}{(z_3-2z_1)(z_4-2z_1)(z_3-z_1-z_2)(z_4-z_1-z_2)}

Maple expression:
Profile \omega=(1,1,1,1), graded in degrees (2,3,4,5) F_{\Sigma^{2,1}}=\frac{1}{(z_3-2z_1)(z_4-2z_1)(z_4-z_1-z_2)}

Maple expression:
Filtration note Kazarian lists the dimension vectors (2,2), (0,1,1,0,2), and (0,1,1,1,1). After suppressing zero entries, these give the profiles (2,2), (1,1,2), and (1,1,1,1). The first presentation carries the multidegree factor 2(z_3+z_4-2z_1-2z_2).
Explicit Thom series Not presently listed.
Local algebra C[x,y,z]/(xy,xz,yz,x^2-y^2,x^2-z^2)
\mu 4
Tp degree 4l+7
Residue presentations
Profile \omega=(3,1) F_{\Phi_{3,0},(3,1)}=\frac{1}{4(z_4-2z_1)(z_4-z_1-z_2)(z_4-z_2-z_3)}

Maple expression:
Filtration note Section 8.6 uses the refined dimension vector (0,3,0,1). After suppressing zero entries, this is the profile \omega=(3,1). For this refined filtration, a generic algebra is already of type \Phi_{3,0}, so no additional degeneracy-locus numerator is needed.
Explicit Thom series \text{Ts}_{\Phi_{3,0}}=\sum_{i>j>k\geq0}((i,j,k))\Delta_{i,j+1,k+2,-i-j-k}.
Coefficient recursion The coefficients ((i_1,\ldots,i_s)) are zero unless i_1>\cdots>i_s\geq0, and ((0))=1. If i_s>0, then s((i_1,\ldots,i_s))=2\sum_{a=1}^s((i_1,\ldots,i_a-1,\ldots,i_s)). If i_s=0, the left-hand side minus the same sum equals ((i_1,\ldots,i_{s-1})).
Local algebra C[x,y,z]/(x^2,y^2,z^2,xz,yz)
\mu 4
Tp degree 4l+8
Residue presentations
Profile \omega=(2,1,1) F_{\Phi_{3,1},(2,1,1)}=\frac{1}{2(z_4-2z_1)(z_4-z_1-z_2)}

Maple expression:
Filtration note Section 8.6 uses the refined dimension vector (0,2,1,1). After suppressing zero entries, this is the profile \omega=(2,1,1). For this refined filtration, a generic algebra is already of type \Phi_{3,1}, so no additional degeneracy-locus numerator is needed.
Explicit Thom series \text{Ts}_{\Phi_{3,1}}=\sum_{i>j\geq1}((i,j))\Delta_{i,j+1,2,1-i-j}.
Coefficient recursion The coefficients ((i_1,\ldots,i_s)) are zero unless i_1>\cdots>i_s\geq0, and ((0))=1. If i_s>0, then s((i_1,\ldots,i_s))=2\sum_{a=1}^s((i_1,\ldots,i_a-1,\ldots,i_s)). If i_s=0, the left-hand side minus the same sum equals ((i_1,\ldots,i_{s-1})).
Local algebra C[x,y,z]/(x^2,y^2,z^3,xy,yz,xz)
\mu 4
Tp degree 4l+10
Residue presentations
Profile \omega=(1,2,1) F_{\Phi_{3,2},(1,2,1)}=\frac{z_2}{z_4-2z_1}

Maple expression:
Filtration note Section 8.6 uses the refined dimension vector (0,1,2,1). After suppressing zero entries, this is the profile \omega=(1,2,1). For this refined filtration, a generic algebra is already of type \Phi_{3,2}, so no additional degeneracy-locus numerator is needed.
Explicit Thom series \text{Ts}_{\Phi_{3,2}}=\sum_{i=0}^{\infty}2^{i+2}\Delta_{i+2,2,2,-i}.
Local algebra C[x,y,z,t]/(x,y,z,t)^2
\mu 4
Tp degree 4l+16
Residue presentations
Profile \omega=(4) F_{\Sigma^4,(4)}=z_1^3z_2^2z_3

Maple expression:
Explicit Thom series \Delta_{3,3,3,3}

\mu=5 — under construction

Local algebra C[x]/(x^6)
\mu 5
Tp degree 5l+5
Residue presentations
Profile \omega=(1,1,1,1,1) F_{A_5,(1,1,1,1,1)}=F_1/F_2, where F_1=(z_5-2z_1-z_2)(2z_1^2+3z_1z_2-2z_1z_5+2z_2z_3-z_2z_4-z_2z_5-z_3z_4+z_4z_5) and F_2=\prod_{i+j\leq k\leq5}(z_k-z_i-z_j).

Maple expression:
Sign check The historical sign is retained. At l=0, \text{Tp}(A_5^0)=c_1^5+10c_1^3c_2+10c_1c_2^2+25c_1^2c_3+12c_2c_3+38c_1c_4+24c_5.
Explicit Thom series Not presently listed.
Local algebra C[x,y]/(xy,x^2+y^4)
\mu 5
Tp degree 5l+6
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y]/(xy,x^3+y^3)
\mu 5
Tp degree 5l+6
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y]/(x^2,xy,y^5)
\mu 5
Tp degree 5l+7
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y]/(x^3,xy,y^4)
\mu 5
Tp degree 5l+7
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y]/(x^2,y^3)
\mu 5
Tp degree 5l+7
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y]/(x^2+y^3,xy^2,y^4)
\mu 5
Tp degree 5l+8
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y]/(x^2,xy^2,y^4)
\mu 5
Tp degree 5l+9
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y]/(x^3,x^2y,xy^2,y^3)
\mu 5
Tp degree 5l+10
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y,z]/(x^2+y^2+z^2,xy,xz,yz)
\mu 5
Tp degree 5l+7
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y,z]/(x^2,y^2,z^2,xy+xz)
\mu 5
Tp degree 5l+8
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y,z]/(x^2,xy+z^3,y^2,xz,yz)
\mu 5
Tp degree 5l+8
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y]/(x^2+yz,xz,y^2,z^2)
\mu 5
Tp degree 5l+9
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y]/(xy,xz,y^2,z^2,x^3)
\mu 5
Tp degree 5l+9
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y]/(y^2+x^3,z^2,xy,xz,yz)
\mu 5
Tp degree 5l+9
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y]/(xy,yz,z^2,y^2-xzx^3)
\mu 5
Tp degree 5l+10
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y]/(x^2,xy,y^2,z^2)
\mu 5
Tp degree 5l+11
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y]/(x^2,xy,y^2,xz,yz,z^4)
\mu 5
Tp degree 5l+11
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y]/(x^2,xy,xz,yz,y^3,z^3)
\mu 5
Tp degree 5l+11
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y]/(x^2,xy,xz,y^2,yz^2,z^3)
\mu 5
Tp degree 5l+12
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y,z,w]/(…)
\mu 5
Tp degree 5l+11
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y,z,w]/(…)
\mu 5
Tp degree 5l+12
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y,z,w]/(…)
\mu 5
Tp degree 5l+14
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y,z,w]/(…)
\mu 5
Tp degree 5l+17
Filtration profile \omega To be added.
F_{Q,\omega} To be checked in the present conventions.
Explicit Thom series To be checked or added.
Local algebra C[x,y,z,t,u]/(x,y,z,t,u)^2
\mu 5
Tp degree 5l+25
Residue presentations
Profile \omega=(5) F_{\Sigma^5,(5)}=z_1^4z_2^3z_3^2z_4

Maple expression:
Explicit Thom series \Delta_{4,4,4,4,4}

\mu=6 — presently only A_6

Local algebra C[x]/(x^7)
\mu 6
Tp degree 6l+6
Residue presentations
Profile \omega=(1,1,1,1,1,1) F_{A_6,(1,1,1,1,1,1)}=F_1/F_2, where F_2=\prod_{i+j\leq k\leq6}(z_k-z_i-z_j). The numerator F_1 is given in the first copy box.

Maple numerator F1:
Maple denominator F2:
Sign check The historical leading minus sign is retained. At l=0, \text{Tp}(A_6^0)=c_1^6+15c_1^4c_2+55c_1^3c_3+30c_1^2c_2^2+141c_1^2c_4+79c_1c_2c_3+202c_1c_5+5c_2^3+55c_2c_4+17c_3^2+120c_6.
Explicit Thom series Not presently listed.

Further entries and general formulas will be added as they are checked in the conventions stated above.