1} + \cdots + c_0$, where the $c_j$'s are polynomials in the coefficients of $f$ and $g$. This gives us maps

$$\mu_{n,r}: \C^n \times \C^r \rightarrow \C^{n+r}$$

\noindent defined by 

$$(a_0, \cdots, a_{n-1}, b_0, \cdots, b_{r-1}) \rightarrow (c_{n+r-1}, \cdots, c_0).$$


\noindent Assume $n \ge r$ and consider the Euclidean algorithm applied to $n$ and $r$:


$$\begin{array}{lr} n=q_1 r+r_1,  \hskip.44 cm 0 < r_1 < r\\
r=q_2 r_1+r_2,  \hskip.35 cm 0 < r_2 < r_1 \\
\vdots\\
r_{k-1}=q_{k+1} r_k, \hskip.3 cm  0 < r_k <r_{k-1}.\end{array}$$


\noindent Let $I(n,r)$ be the tuple given by the Euclidean algorithm on $n$ and $r$:

$$I(n,r)=(r, \cdots, r, r_1, \cdots, r_1, \cdots, r_k, \cdots, r_k, 0, \cdots)$$


\noindent where $r$ is repeated $q_1$ times, and $r_i$ is repeated $q_{i+1}$ times.

\medskip

Let $I(\mu_{n,r})$ be the ideal in the algebra $\CA$ of germs at origin generated by $c_{j}$'s in the map $\mu_{n,r}: \C^n \times \C^r \rightarrow \C^{n+r}$. Denote $TB(I(\mu_{n,r}))$ the Thom-Boardman symbol of this ideal, Robert Varley conjectured that $TB(I(\mu_{n,r}))=I(n,r)$ for any $n \ge r$. In \cite{LinWeth}, J. Lin and J. Wethington confirmed Varley's Conjecture. For the reader's convenience, let us state their result here.

\begin{thm}
$TB(I(\mu_{n,r}))=I(n,r)$ for any $n \ge r$.
\end{thm}


Professor J. M. Boardman provides us the following example which has constant Thom-Boardman symbol.

\begin{ex} The zero map-germ at origin $F: \C^a \rightarrow \C^b$ has Thom-Boardman symbol $(a, a, \cdots, a, \cdots )$. 
\end{ex}


Denote $\mu_{\infty,a}$ the map-germ in Example $1.2$. We have the following theorem.

\begin{thm} Let $(i_0, \cdots, i_0, i_1, \cdots, i_1, \cdots, i_k, \cdots, i_k, \cdots)$ be a non-increasing sequence of nonnegative integers, where $i_0 >i_1 > \cdots > i_k \ge 0$ for some $k$, $i_j$ repeats $l_j$ times for each $j <k$ and $i_k$ repeats infinitely many times. Denote $\mu$ as the Cartesian product of the map-germs of 

$$\mu_{(i_0-i_1) l_0,i_0-i_1} \times \mu_{(i_1-i_2) (l_0+l_1),i_1-i_2} \times \cdots \times \mu_{(i_{k-1}-i_k) (l_0+l_1+\cdots+l_{k-1}),i_{k-1}-i_k} \times \mu_{\infty,i_k}$$

\noindent at origin. Then $(i_0, \cdots, i_0, i_1, \cdots, i_1, \cdots, i_k, \cdots, i_k, \cdots)$ is the Thom-Boardman symbol of the map-germ $\mu$.


\end{thm}

\begin{rem}
Historically there were only sporadic known Thom-Boardman symbols for some map-germs. J. Lin and J. Wethington \cite{LinWeth} gave infinitely many examples of map-germs with distinct Thom-Boardman symbols.  However, Theorem $1.3$ provides us a complete list of representatives of map-germs classified by their Thom-Boardman symbols. We should warn the reader that the classification of map-germs by their Thom-Boardman symbols is not complete. Two non- left-right equivalent map-germs may have the same Thom-Boardman symbols. Under such a circumstance, other invariants are needed (e.g. see page $67$ in \cite{Arnold} and \cite{Adams}). 
\end{rem}

The proof of Theorem $1.3$ is very simple. We first prove that Thom-Boardman symbols have additive property under Cartesian product. Combining this property with Theorem $1.1$, we immediately obtain Theorem $1.3$. 

{\bf Acknowledgments} The authors thank Professor J. M. Boardman for providing us Example $1.2$, which is even not well-known among experts. We are grateful for his generosity to allow us to use his example in Theorem $1.3$. Without it, Theorem $1.3$ would be incomplete.

\section{Proof of Theorem $1.3$}


We first prove that Thom-Boardman symbols have additive property under Cartesian product.


\begin{prop}
Let $F_t: M_t \rightarrow N_t, t=1,2$ be two differentiable maps. Then the Thom-Boardman symbol $TB(F)$ of $F=(F_1,F_2): M_1 \times M_2 \rightarrow N_1 \times N_2$ is equal to the sum of $TB(F_1)$ and $TB(F_2)$, where $F=(F_1,F_2): M_1 \times M_2 \rightarrow N_1 \times N_2$  is the Cartesian product of $F_t: M_t \rightarrow N_t, t=1,2$.
\end{prop}

\begin{proof}
Denote $TB(F)=(s_1, s_2, \cdots, s_p, \cdots)$ and $TB(F_t)=(s_1^{(t)}, s_2^{(t)}, \cdots, s_p^{(t)}, \cdots), t=1,2$. We will show that $s_p=s_p^{(1)}+s_p^{(2)}$ for all integer $p \ge 1$.


Let $x_1^{(t)}, \cdots, x_{m_t}^{(t)}$ be local coordinates of origin on the differential manifolds $M_t, t=1,2$ and $y_1^{(t)}, \cdots, y_{n_t}^{(t)}$ be local coordinates on the differential manifolds $N_t, t=1,2$. Denote $F_t: M_t \rightarrow N_t, F_t=(f_1^{(t)}, \cdots, f_{n_t}^{(t)})$ and $J_{F_t}$ the ideal generated by $f_1^{(t)}, \cdots, f_{n_t}^{(t)}$ in $\CA^{(t)}, t=1,2$, the local ring of germs of differentiable functions at origin in $M_t$. Denote $J_F$ the ideal generated by $f_1^{(1)}, \cdots, f_{n_1}^{(1)}; f_1^{(2)}, \cdots, f_{n_2}^{(2)}$ in $\CA$, the local ring of germs of differentiable functions at origin in $M_1 \times M_2$. Note that in general $\CA^{(1)} \times \CA^{(2)} \varsubsetneq \CA $ and $J_{F_1} \times J_{F_2} \varsubsetneq J_F $. However, it does not affect our conclusion on the additive property of Thom-Boardman symbols. Using induction, we will prove that $\Delta^{s_p} \Delta^{s_{p-1}} \cdots \Delta^{s_1} J_F=(\Delta^{s_p^{(1)}} \Delta^{s_{p-1}^{(1)}} \cdots \Delta^{s_1^{(1)}} J_{F_1}; \Delta^{s_p^{(2)}} \Delta^{s_{p-1}^{(2)}} \cdots \Delta^{s_1^{(2)}} J_{F_2})$ in $\CA $, which immediately implies the conclusion in Proposition $2.1$. 


When $p=1$, it is easy to see that the