Thom series

Thom polynomials of contact singularities of arbitrary relative dimension

Let $Q$ be the local algebra of the contact singularity $\Sigma_Q^l \subset \{ f:(C^*,0)\to (C^{*+l},0)$ analytic $\}$ , and let $\mu=dim(Q)-1$ . Here are two ways of presenting the Thom polynomials of $\Sigma_Q^l$ for all relative dimensions $l$ at the same time.

There exists a rational function in the variables such that the Thom polynomial of for relative dimension is
$\text{Tp}(\Sigma_Q^l)=(-1)^\mu \text{Res}_{z_1=\infty} \text{Res}_{z_2=\infty} \ldots \text{Res}_{z_\mu=\infty} \left( F_Q V_\mu K_\mu M_{\mu,l} \right),$
where
- $V_\mu=\prod_{j=1}^\mu \prod_{i=1}^{j-1} (z_j-z_i)$ is the Vandermonde discriminant,
- $K_\mu=\prod_{j=1}^\mu \left( \sum_{i=0}^\infty {c_i}/{z_j^i} \right)$ is the “kernel” function,
- $M_{\mu,l}=(\prod_{j=1}^\mu z_j)^l \cdot dz_\mu \wedge dz_{\mu-1} \wedge \ldots \wedge dz_1$ is a monomial differential form.
The rational function $F_Q$ associated with $Q$ is not unique.
The Thom series of an algebra $Q$ is a formal power series $\text{Ts}_Q$ in $d_i, i\in Z$ (typically given by an explicit formula for its coefficients or at least a recursion), such that
$\text{Tp}(\Sigma_Q^l)=\text{Ts}_Q|_{d_i=c_{l+1+i}}$ .

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

Below we present some known $F_Q$ functions and Thom series.

$\mu=1$

Local algebra	C[x]/(x^2)
$\mu$	1
Tp degree	l+1
$F_Q$	1
Explicit Thom series	$d_0=\Delta_0$

$\mu=2$

Local algebra	C[x]/(x^3)
$\mu$	2
Tp degree	2l+2
$F_Q$	$\frac{1}{z_2- 2 z_1}$
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
$F_Q$	$z_1$
Explicit Thom series	$\Delta_{1,1}$

$\mu=3$

Local algebra	C[x]/(x^4)
$\mu$	3
Tp degree	3l+3
$F_Q$	$\frac{1}{(z_2-2z_1)(z_3-2z_1)(z_3-z_1-z_2) }$
Explicit Thom series	$\sum_{i=0}^\infty 2^i d_{-i}d_0d_i + 1/3 \cdot \sum_{i=1}^\infty \sum_{j=1}^\infty 2^i 3^j d_{-i} d_{-j} d_{i+j} + 1/2 \cdot \sum_{i=0}^\infty \sum_{j=0}^\infty a_{i,j} d_{-i-j}d_id_j$ where $a_{i,j}$ is defined by the recursion $a_{0,0}=0$ , $a_{i,0}=a_{0,i}=3^{i-1}$ 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
$F_Q$	$\frac{z_1}{(z_3-2z_1)(z_3-2z_2)(z_3-z_1-z_2) }$
$F_Q$	$\frac{1}{2(z_3-2z_1)(z_3-z_1-z_2) }$
Explicit Thom series	$\sum_{0\leq j<i} ((i,j)) \Delta_{i,j+1,-i-j}$ where $((i,j))$ is defined by the recursion: $((i,i))=0$ , $((i,0))=2^i-1$ , 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
$F_Q$	$\frac{2z_1}{(z_3 - 2 z_1)(z_3 - 2 z_2)}$
$F_Q$	$\frac{z_2}{(z_3- 2 z_1)(z_2- 2 z_1) }$
$F_Q$	$\frac{1}{z_3- 2 z_1}$
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
$F_Q$	$z_1^2 z_2$
Explicit Thom series	$\Delta_{2,2,2}$

$\mu=4$ — under construction

Local algebra	C[x]/(x^5)
$\mu$	4
Tp degree	4l+4
$F_Q$	$\frac{(2z_1+z_2-z_4)}{ ( z_2-2z_1)(z_3-2z_1)(z_3-z_1-z_2)(z_4-2z_1)(z_4-z_1-z_2)(z_4-z_1-z_3)(z_4-2z_2)}$

Local algebra	C[x,y]/(xy,x^2+y^3)
$\mu$	4
Tp degree	4l+5
$F_Q$	…

Local algebra	C[x,y]/(x^2,xy,y^4)
$\mu$	4
Tp degree	4l+6
$F_Q$	…

Local algebra	C[x,y]/(x^3,xy,y^3)
$\mu$	4
Tp degree	4l+6
$F_Q$	…

Local algebra	C[x,y]/(x^2+y^2,x^3,x^2y)
$\mu$	4
Tp degree	4l+6
$F_Q$	…

Local algebra	C[x,y]/(x^2,xy^2,y^3)
$\mu$	4
Tp degree	4l+7
$F_Q$	…

Local algebra	C[x,y,z]/(xy,xz,yz,x^2-y^2,x^2-z^2)
$\mu$	4
Tp degree	4l+7
$F_Q$	…

Local algebra	C[x,y,z]/(x^2,y^2,z^2,xz,yz)
$\mu$	4
Tp degree	4l+8
$F_Q$	…

Local algebra	C[x,y,z]/(x^2,y^2,z^3,xy,yz,xz)
$\mu$	4
Tp degree	4l+10
$F_Q$	…

Local algebra	C[x,y,z,t]/(x,y,z,t)^2
$\mu$	4
Tp degree	4l+16
$F_Q$	$z_1^3z_2^2z_3$
Explicit Thom series	$\Delta_{3,3,3,3}$

$\mu=5$ — under construction

Local algebra	C[x]/(x^6)
$\mu$	5
Tp degree	5l+5
$F_Q$	$=F_1/F_2$ 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 $F_2=\prod_{i+j\leq k \leq 5} (z_k-z_i-z_j)$

Local algebra	C[x,y,z,t,u]/(x,y,z,t,u)^2
$\mu$	5
Tp degree	5l+25
$F_Q$	$z_1^4z_2^3z_3^2z_4$
Explicit Thom series	$\Delta_{4,4,4,4,4}$

$\mu=6$ — under construction

Local algebra	C[x]/(x^7)
$\mu$	6
Tp degree	6l+6
$F_Q$	$=F_1/F_2$ where F_1 = -(8z_1^7+ 44z_1^6z_2+ 84z_1^5z_2^2+ 70z_1^4z_2^3+ 26z_1^3z_2^4+ 4z_1^2z_2^5+ 12z_1^6z_3+ 70z_1^5z_2z_3+ 136z_1^4z_2^2z_3+ 103z_1^3z_2^3z_3+ 29z_1^2z_2^4z_3+ 2z_1z_2^5z_3+ 8z_1^5z_3^2+ 38z_1^4z_2z_3^2+ 71z_1^3z_2^2z_3^2+ 46z_1^2z_2^3z_3^2+ 10z_1z_2^4z_3^2+ 4z_1^4z_3^3+ 16z_1^3z_2z_3^3+ 15z_1^2z_2^2z_3^3+ 4z_1z_2^3z_3^3- 4z_1^6z_4- 18z_1^5z_2z_4- 10z_1^4z_2^2z_4+ 14z_1^3z_2^3z_4+ 14z_1^2z_2^4z_4+ 4z_1z_2^5z_4+ 6z_1^5z_3z_4+ 23z_1^4z_2z_3z_4+ 17z_1^3z_2^2z_3z_4+ 19z_1^2z_2^3z_3z_4+ 11z_1z_2^4z_3z_4+ 2z_2^5z_3z_4- 2z_1^4z_3^2z_4+z_1^3z_2z_3^2z_4+ 14z_1^2z_2^2z_3^2z_4+ 13z_1z_2^3z_3^2z_4+ 4z_2^4z_3^2z_4+ 4z_1z_2^2z_3^3z_4+ 2z_2^3z_3^3z_4- 8z_1^3z_2^2z_4^2- 12z_1^2z_2^3z_4^2- 4z_1z_2^4z_4^2- 16z_1^3z_2z_3z_4^2- 24z_1^2z_2^2z_3z_4^2- 12z_1z_2^3z_3z_4^2- 2z_2^4z_3z_4^2- 8z_1^3z_3^2z_4^2- 12z_1^2z_2z_3^2z_4^2- 12z_1z_2^2z_3^2z_4^2- 4z_2^3z_3^2z_4^2- 4z_1z_2z_3^3z_4^2- 2z_2^2z_3^3z_4^2- 4z_1^4z_4^3- 2z_1^3z_2z_4^3+ 2z_1^2z_2^2z_4^3+ 6z_1^3z_3z_4^3-z_1^2z_2z_3z_4^3-z_1z_2^2z_3z_4^3- 2z_1^2z_3^2z_4^3+z_1z_2z_3^2z_4^3- 8z_1^6z_5- 32z_1^5z_2z_5- 54z_1^4z_2^2z_5- 48z_1^3z_2^3z_5- 22z_1^2z_2^4z_5- 4z_1z_2^5z_5- 20z_1^5z_3z_5- 80z_1^4z_2z_3z_5- 95z_1^3z_2^2z_3z_5- 44z_1^2z_2^3z_3z_5- 11z_1z_2^4z_3z_5- 2z_2^5z_3z_5- 16z_1^4z_3^2z_5- 56z_1^3z_2z_3^2z_5- 48z_1^2z_2^2z_3^2z_5- 8z_1z_2^3z_3^2z_5+ 2z_2^4z_3^2z_5- 4z_1^3z_3^3z_5- 8z_1^2z_2z_3^3z_5- 3z_1z_2^2z_3^3z_5- 12z_1^4z_2z_4z_5- 22z_1^3z_2^2z_4z_5- 14z_1^2z_2^3z_4z_5- 4z_1z_2^4z_4z_5+ 6z_1^3z_2z_3z_4z_5- 13z_1^2z_2^2z_3z_4z_5- 19z_1z_2^3z_3z_4z_5- 6z_2^4z_3z_4z_5- 4z_1^2z_2z_3^2z_4z_5- 10z_1z_2^2z_3^2z_4z_5- 5z_2^3z_3^2z_4z_5- 2z_1z_2z_3^3z_4z_5-z_2^2z_3^3z_4z_5+ 4z_1^4z_4^2z_5+ 14z_1^3z_2z_4^2z_5+ 16z_1^2z_2^2z_4^2z_5+ 6z_1z_2^3z_4^2z_5+ 6z_1^3z_3z_4^2z_5+ 17z_1^2z_2z_3z_4^2z_5+ 16z_1z_2^2z_3z_4^2z_5+ 5z_2^3z_3z_4^2z_5+ 10z_1z_2z_3^2z_4^2z_5+ 6z_2^2z_3^2z_4^2z_5+ 2z_1z_3^3z_4^2z_5+z_2z_3^3z_4^2z_5- 4z_1^3z_4^3z_5- 2z_1^2z_2z_4^3z_5+ 2z_1z_2^2z_4^3z_5+ 6z_1^2z_3z_4^3z_5-z_1z_2z_3z_4^3z_5-z_2^2z_3z_4^3z_5- 2z_1z_3^2z_4^3z_5+z_2z_3^2z_4^3z_5+ 4z_1^4z_2z_5^2+ 10z_1^3z_2^2z_5^2+ 10z_1^2z_2^3z_5^2+ 4z_1z_2^4z_5^2+ 8z_1^4z_3z_5^2+ 18z_1^3z_2z_3z_5^2+ 11z_1^2z_2^2z_3z_5^2+ 5z_1z_2^3z_3z_5^2+ 2z_2^4z_3z_5^2+ 8z_1^3z_3^2z_5^2+ 18z_1^2z_2z_3^2z_5^2+z_1z_2^2z_3^2z_5^2- 2z_2^3z_3^2z_5^2+ 4z_1^4z_4z_5^2+ 10z_1^3z_2z_4z_5^2+ 6z_1^2z_2^2z_4z_5^2- 6z_1^3z_3z_4z_5^2- 7z_1^2z_2z_3z_4z_5^2+ 7z_1z_2^2z_3z_4z_5^2+ 4z_2^3z_3z_4z_5^2+ 2z_1^2z_3^2z_4z_5^2+ 3z_1z_2z_3^2z_4z_5^2- 4z_1^2z_2z_4^2z_5^2- 4z_1z_2^2z_4^2z_5^2- 4z_1^2z_3z_4^2z_5^2- 4z_1z_2z_3z_4^2z_5^2- 2z_2^2z_3z_4^2z_5^2- 2z_2z_3^2z_4^2z_5^2- 24z_1^6z_6- 112z_1^5z_2z_6- 180z_1^4z_2^2z_6- 117z_1^3z_2^3z_6- 29z_1^2z_2^4z_6- 2z_1z_2^5z_6- 36z_1^5z_3z_6- 160z_1^4z_2z_3z_6- 233z_1^3z_2^2z_3z_6- 129z_1^2z_2^3z_3z_6- 23z_1z_2^4z_3z_6- 14z_1^4z_3^2z_6- 63z_1^3z_2z_3^2z_6- 89z_1^2z_2^2z_3^2z_6- 42z_1z_2^3z_3^2z_6- 6z_2^4z_3^2z_6- 6z_1^3z_3^3z_6- 15z_1^2z_2z_3^3z_6- 11z_1z_2^2z_3^3z_6- 2z_2^3z_3^3z_6+ 8z_1^5z_4z_6+ 24z_1^4z_2z_4z_6+ 14z_1^3z_2^2z_4z_6- 12z_1^2z_2^3z_4z_6- 11z_1z_2^4z_4z_6- 2z_2^5z_4z_6- 12z_1^4z_3z_4z_6- 16z_1^3z_2z_3z_4z_6- 3z_1^2z_2^2z_3z_4z_6- 7z_1z_2^3z_3z_4z_6- 3z_2^4z_3z_4z_6+ 12z_1^3z_3^2z_4z_6+ 8z_1^2z_2z_3^2z_4z_6- 3z_1z_2^2z_3^2z_4z_6- 3z_2^3z_3^2z_4z_6+ 4z_1z_2z_3^3z_4z_6+ 4z_1^4z_4^2z_6+ 6z_1^3z_2z_4^2z_6+ 14z_1^2z_2^2z_4^2z_6+ 13z_1z_2^3z_4^2z_6+ 2z_2^4z_4^2z_6- 2z_1^3z_3z_4^2z_6+ 29z_1^2z_2z_3z_4^2z_6+ 26z_1z_2^2z_3z_4^2z_6+ 6z_2^3z_3z_4^2z_6+ 14z_1^2z_3^2z_4^2z_6+ 11z_1z_2z_3^2z_4^2z_6+ 6z_2^2z_3^2z_4^2z_6+ 2z_2z_3^3z_4^2z_6+ 4z_1^3z_4^3z_6+ 2z_1^2z_2z_4^3z_6- 2z_1z_2^2z_4^3z_6- 6z_1^2z_3z_4^3z_6+z_1z_2z_3z_4^3z_6+z_2^2z_3z_4^3z_6+ 2z_1z_3^2z_4^3z_6-z_2z_3^2z_4^3z_6+ 32z_1^5z_5z_6+ 114z_1^4z_2z_5z_6+ 143z_1^3z_2^2z_5z_6+ 78z_1^2z_2^3z_5z_6+ 19z_1z_2^4z_5z_6+ 2z_2^5z_5z_6+ 54z_1^4z_3z_5z_6+ 161z_1^3z_2z_3z_5z_6+ 146z_1^2z_2^2z_3z_5z_6+ 46z_1z_2^3z_3z_5z_6+ 5z_2^4z_3z_5z_6+ 28z_1^3z_3^2z_5z_6+ 58z_1^2z_2z_3^2z_5z_6+ 38z_1z_2^2z_3^2z_5z_6+ 4z_2^3z_3^2z_5z_6+ 6z_1^2z_3^3z_5z_6+ 7z_1z_2z_3^3z_5z_6+z_2^2z_3^3z_5z_6- 8z_1^4z_4z_5z_6- 12z_1^3z_2z_4z_5z_6+ 6z_1^2z_2^2z_4z_5z_6+ 14z_1z_2^3z_4z_5z_6+ 4z_2^4z_4z_5z_6- 2z_1^2z_2z_3z_4z_5z_6+ 5z_1z_2^2z_3z_4z_5z_6+ 6z_2^3z_3z_4z_5z_6- 2z_1^2z_3^2z_4z_5z_6- 6z_1z_2z_3^2z_4z_5z_6+ 2z_2^2z_3^2z_4z_5z_6- 2z_1z_3^3z_4z_5z_6- 4z_1^3z_4^2z_5z_6- 20z_1^2z_2z_4^2z_5z_6- 20z_1z_2^2z_4^2z_5z_6- 4z_2^3z_4^2z_5z_6- 16z_1^2z_3z_4^2z_5z_6- 22z_1z_2z_3z_4^2z_5z_6- 11z_2^2z_3z_4^2z_5z_6- 3z_1z_3^2z_4^2z_5z_6- 8z_2z_3^2z_4^2z_5z_6-z_3^3z_4^2z_5z_6- 8z_1^4z_5^2z_6- 26z_1^3z_2z_5^2z_6- 27z_1^2z_2^2z_5^2z_6- 13z_1z_2^3z_5^2z_6- 2z_2^4z_5^2z_6- 18z_1^3z_3z_5^2z_6- 33z_1^2z_2z_3z_5^2z_6- 15z_1z_2^2z_3z_5^2z_6- 5z_2^3z_3z_5^2z_6- 14z_1^2z_3^2z_5^2z_6- 11z_1z_2z_3^2z_5^2z_6+ 3z_2^2z_3^2z_5^2z_6- 4z_1^3z_4z_5^2z_6- 2z_1^2z_2z_4z_5^2z_6-z_1z_2^2z_4z_5^2z_6- 2z_2^3z_4z_5^2z_6+ 10z_1^2z_3z_4z_5^2z_6- 4z_2^2z_3z_4z_5^2z_6- 2z_1z_3^2z_4z_5^2z_6+ 2z_2z_3^2z_4z_5^2z_6+ 4z_1^2z_4^2z_5^2z_6+ 7z_1z_2z_4^2z_5^2z_6+ 2z_2^2z_4^2z_5^2z_6+ 3z_1z_3z_4^2z_5^2z_6+ 5z_2z_3z_4^2z_5^2z_6+z_3^2z_4^2z_5^2z_6+ 28z_1^5z_6^2+ 108z_1^4z_2z_6^2+ 133z_1^3z_2^2z_6^2+ 61z_1^2z_2^3z_6^2+ 9z_1z_2^4z_6^2+ 36z_1^4z_3z_6^2+ 122z_1^3z_2z_3z_6^2+ 130z_1^2z_2^2z_3z_6^2+ 48z_1z_2^3z_3z_6^2+ 4z_2^4z_3z_6^2+ 6z_1^3z_3^2z_6^2+ 37z_1^2z_2z_3^2z_6^2+ 36z_1z_2^2z_3^2z_6^2+ 10z_2^3z_3^2z_6^2+ 2z_1^2z_3^3z_6^2+ 3z_1z_2z_3^3z_6^2+ 2z_2^2z_3^3z_6^2- 4z_1^4z_4z_6^2- 10z_1^3z_2z_4z_6^2- 10z_1^2z_2^2z_4z_6^2- 2z_1z_2^3z_4z_6^2+z_2^4z_4z_6^2+ 2z_1^3z_3z_4z_6^2- 17z_1^2z_2z_3z_4z_6^2- 14z_1z_2^2z_3z_4z_6^2- 2z_2^3z_3z_4z_6^2- 14z_1^2z_3^2z_4z_6^2- 9z_1z_2z_3^2z_4z_6^2- 3z_2^2z_3^2z_4z_6^2- 2z_2z_3^3z_4z_6^2- 4z_1^3z_4^2z_6^2- 8z_1^2z_2z_4^2z_6^2- 7z_1z_2^2z_4^2z_6^2- 3z_2^3z_4^2z_6^2- 14z_1z_2z_3z_4^2z_6^2- 6z_2^2z_3z_4^2z_6^2- 6z_1z_3^2z_4^2z_6^2-z_2z_3^2z_4^2z_6^2- 42z_1^4z_5z_6^2- 115z_1^3z_2z_5z_6^2- 104z_1^2z_2^2z_5z_6^2- 36z_1z_2^3z_5z_6^2- 5z_2^4z_5z_6^2- 52z_1^3z_3z_5z_6^2- 106z_1^2z_2z_3z_5z_6^2- 64z_1z_2^2z_3z_5z_6^2- 8z_2^3z_3z_5z_6^2- 12z_1^2z_3^2z_5z_6^2- 20z_1z_2z_3^2z_5z_6^2- 10z_2^2z_3^2z_5z_6^2- 2z_1z_3^3z_5z_6^2-z_2z_3^3z_5z_6^2+ 12z_1^3z_4z_5z_6^2+ 18z_1^2z_2z_4z_5z_6^2+ 4z_1z_2^2z_4z_5z_6^2- 2z_2^3z_4z_5z_6^2+ 4z_1^2z_3z_4z_5z_6^2+ 17z_1z_2z_3z_4z_5z_6^2+ 6z_2^2z_3z_4z_5z_6^2+ 7z_1z_3^2z_4z_5z_6^2+ 4z_2z_3^2z_4z_5z_6^2+z_3^3z_4z_5z_6^2+ 6z_1^2z_4^2z_5z_6^2+ 12z_1z_2z_4^2z_5z_6^2+ 6z_2^2z_4^2z_5z_6^2+ 7z_1z_3z_4^2z_5z_6^2+ 7z_2z_3z_4^2z_5z_6^2+ 2z_3^2z_4^2z_5z_6^2+ 14z_1^3z_5^2z_6^2+ 27z_1^2z_2z_5^2z_6^2+ 17z_1z_2^2z_5^2z_6^2+ 5z_2^3z_5^2z_6^2+ 16z_1^2z_3z_5^2z_6^2+ 18z_1z_2z_3z_5^2z_6^2+ 4z_2^2z_3z_5^2z_6^2+ 6z_1z_3^2z_5^2z_6^2-z_2z_3^2z_5^2z_6^2- 4z_1^2z_4z_5^2z_6^2- 4z_1z_2z_4z_5^2z_6^2+z_2^2z_4z_5^2z_6^2- 3z_1z_3z_4z_5^2z_6^2- 3z_2z_3z_4z_5^2z_6^2-z_3^2z_4z_5^2z_6^2- 3z_1z_4^2z_5^2z_6^2- 3z_2z_4^2z_5^2z_6^2- 2z_3z_4^2z_5^2z_6^2- 16z_1^4z_6^3- 48z_1^3z_2z_6^3- 40z_1^2z_2^2z_6^3- 10z_1z_2^3z_6^3- 12z_1^3z_3z_6^3- 36z_1^2z_2z_3z_6^3- 27z_1z_2^2z_3z_6^3- 6z_2^3z_3z_6^3- 8z_1z_2z_3^2z_6^3- 4z_2^2z_3^2z_6^3+ 6z_1^2z_2z_4z_6^3+ 7z_1z_2^2z_4z_6^3+ 2z_2^3z_4z_6^3+ 6z_1^2z_3z_4z_6^3+ 11z_1z_2z_3z_4z_6^3+ 4z_2^2z_3z_4z_6^3+ 4z_1z_3^2z_4z_6^3+ 2z_2z_3^2z_4z_6^3+ 2z_1z_2z_4^2z_6^3+z_2^2z_4^2z_6^3+ 2z_1z_3z_4^2z_6^3+z_2z_3z_4^2z_6^3+ 24z_1^3z_5z_6^3+ 48z_1^2z_2z_5z_6^3+ 28z_1z_2^2z_5z_6^3+ 4z_2^3z_5z_6^3+ 18z_1^2z_3z_5z_6^3+ 23z_1z_2z_3z_5z_6^3+ 7z_2^2z_3z_5z_6^3+ 4z_2z_3^2z_5z_6^3- 6z_1^2z_4z_5z_6^3- 10z_1z_2z_4z_5z_6^3- 4z_2^2z_4z_5z_6^3- 7z_1z_3z_4z_5z_6^3- 6z_2z_3z_4z_5z_6^3- 2z_3^2z_4z_5z_6^3- 2z_1z_4^2z_5z_6^3- 2z_2z_4^2z_5z_6^3-z_3z_4^2z_5z_6^3- 8z_1^2z_5^2z_6^3- 10z_1z_2z_5^2z_6^3- 4z_2^2z_5^2z_6^3- 6z_1z_3z_5^2z_6^3-z_2z_3z_5^2z_6^3+ 3z_1z_4z_5^2z_6^3+ 2z_2z_4z_5^2z_6^3+ 2z_3z_4z_5^2z_6^3+z_4^2z_5^2z_6^3+ 4z_1^3z_6^4+ 8z_1^2z_2z_6^4+ 3z_1z_2^2z_6^4+ 4z_1z_2z_3z_6^4+ 2z_2^2z_3z_6^4- 2z_1z_2z_4z_6^4-z_2^2z_4z_6^4- 2z_1z_3z_4z_6^4-z_2z_3z_4z_6^4- 6z_1^2z_5z_6^4- 7z_1z_2z_5z_6^4-z_2^2z_5z_6^4- 2z_2z_3z_5z_6^4+ 2z_1z_4z_5z_6^4+ 2z_2z_4z_5z_6^4+z_3z_4z_5z_6^4+ 2z_1z_5^2z_6^4+z_2z_5^2z_6^4-z_4z_5^2z_6^4) and $F_2=\prod_{i+j\leq k \leq 6} (z_k-z_i-z_j)$

Local algebra	C[x,y,z,t,u,v]/(x,y,z,t,u,v)^2
$\mu$	6
Tp degree	6l+36
$F_Q$	$z_1^5 z_2^4 z_3^3 z_4^2 z_5$
Explicit Thom series	$\Delta_{5,5,5,5,5,5}$

General $\Sigma^{i,j}$ and $\Phi_{m,r}$ to be added.

Thom polynomials of contact singularities of arbitrary relative dimension

A_1 aka Sigma^1

A_2 aka Phi_{1,0}

III_{2,2} aka Sigma^2

A_3

I_{2,2} aka Phi_{2,0}

III_{2,3} aka Phi_{2,1}

Sigma^3

— under construction

A_4

I_{2,3}

III_{2,4}

III_{3,3}

V_3

Sigma^{2,1}

Phi_{3,0}

Phi_{3,1}

Phi_{3,2}

Sigma^4

— under construction

A_5

--- classification is unknown

Sigma^5

— under construction

A_6

--- classification is unknown

Sigma^6

$\mu=4$ — under construction

$\mu=5$ — under construction

$\mu=6$ — under construction