| Russian Mathematical Surveys |
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Brief communications Combinatorics of type V. D. Sedykh National State University of Oil and Gas "Gubkin University" We study the adjacencies of singularities of the front of a proper generic Legendrian map with simple stable singularities. This note is a continuation of [5]; the same notation is used. For the necessary information from the theory of Legendrian singularities, see [1] and [4]. A Legendrian monosingularity of type $D_\mu^\delta$, $\delta=\pm1$, can only be adjacent to multisingularities of types $\mathcal{A}=A_{\mu_1}^{k_1}\ldots A_{\mu_p}^{k_p}$ and $D_\nu^\pm\mathcal{A}$. The adjacency indices $J_{D_\nu^\pm\mathcal{A}}(D_\mu^\delta)$ were calculated in the note [5]. The combinatorial formula for the calculation of the indices $J_{\mathcal{A}}(D_\mu^\delta)$ is presented below. Consider the map
$$
\begin{equation*}
f\colon\mathbb{R}^{n-1}\to \mathbb{R}^n,\quad (t,q)\mapsto (x,q,u),\quad x=-\frac{\partial S}{\partial t}\,,\quad u=S-t\,\frac{\partial S}{\partial t}\,,
\end{equation*}
\notag
$$
where $t=(t_1,t_2)$, $x=(x_1,x_2)$, $q=(q_{3},\dots,q_{n-1})$, and
$$
\begin{equation*}
S=S(t,q)=t_1^2t_2+\delta t_2^{\mu-1}+q_{\mu-1}t_2^{\mu-2}+ \dots+q_3t_2^2,\qquad \mu\geqslant4.
\end{equation*}
\notag
$$
It is Legendrian and has a singularity of type $D_\mu^\delta$ at zero. All connected components of the manifold $\mathcal{A}_f\subset\mathbb{R}^n$ of type $\mathcal{A}$ multisingularities of the map $f$ are contractible (see [2]).
Consider the polynomial
$$
\begin{equation}
\delta t_2^{\mu}+q_{\mu-1}t_2^{\mu-1}+\dots+q_3t_2^3+x_2t_2^2-ut_2- \frac{x_1^2}{4}
\end{equation}
\tag{1}
$$
in $t_2$ and the following condition on the roots of a real polynomial of one variable:
(R) the polynomial has a simple real root such that there are no other real roots between it and zero. Theorem 1. Let $\mathcal{A}=A_{\mu_1}^{k_1}\ldots A_{\mu_p}^{k_p}$ and $m=\sum_{i=1}^{p}k_i$. Then the manifold $\mathcal{A}_f$ has the following connected components. (1) The components lying in $\mathbb{R}^n\setminus\{x_1=u=0,\ x_2\leqslant 0\}$; they either lie in the space $x_1\ne0$, or intersect the hyperplane $x_1=0$. The components lying in the space $x_1\ne0$ form pairs of manifolds symmetric with respect to the reflection $x_1\mapsto -x_1$; these pairs are numbered by the relative positions of zero and all real roots of the polynomial (1); in this case the polynomial (1) does not satisfy condition (R) and has $m$ multiple real roots; $k_i$ of them have multiplicity $\mu_i+1$, $i=1,\dots,p$. The components intersecting the hyperplane $x_1=0$ are numbered by the relative positions of all real roots of the polynomial (1); in this case the polynomial (1) satisfies condition (R) and has $m$ multiple non-zero real roots; $k_i$ of them have multiplicity $\mu_i+1$, $i=1,\dots,p$. (2) The components lying in the space $x_1=u=0$, $x_2<0$; they are numbered by the relative positions of all real roots of the polynomial (1); in this case the polynomial (1) has $m-2$ multiple non-zero real roots; $l_i$ of them have multiplicity $\nu_i+1$, $i=1,\dots,r$, where $A_{\nu_1}^{l_1}\ldots A_{\nu_r}^{l_r}A_1^2=\mathcal{A}$. (3) The components lying in the subspace $x_1=u=0$, $x_2=0$; they are numbered by the relative positions of all real roots of the polynomial (1); in this case the polynomial (1) has $m-1$ multiple non-zero real roots; $l_i$ of them have multiplicity $\nu_i+1$, $i=1,\dots,r$, where $A_{\nu_1}^{l_1}\ldots A_{\nu_r}^{l_r}A_3=\mathcal{A}$. Theorem 1 reduces the problem of the calculation of the indices $J_{\mathcal{A}}(D_\mu^\delta)$ to the combinatorics of the possible relative positions of the real roots of the polynomial (1) on the $t_2$-axis, taking into account their multiplicities and the position of zero. Below, for any non-negative integers $n_1,\dots,n_r$ we denote the multinomial coefficient $(n_1+ \dots+n_r)!/(n_1!\cdots n_r!)$ by $\langle n_1,\dots,n_r\rangle$. Theorem 2. Let $\mathcal{A}=A_{\alpha_1}^{i_1}\ldots A_{\alpha_k}^{i_k}A_{\beta_1}^{j_1}\ldots A_{\beta_l}^{j_l}$, where $\alpha_1,\dots,\alpha_k$ are pairwise distinct even numbers and $\beta_1,\dots,\beta_l$ are pairwise distinct odd numbers. Let $a_\mu$ denote the number of factors equal to $A_\mu$ in $\mathcal{A}$. Let $m_1=\sum_{p=1}^{k}i_p$, $m_2=\sum_{r=1}^{l}j_r$, Remark 1. The calculation using formula (2) of the adjacency indices of Legendrian monosingularities of types $D_\mu^\pm$, $\mu\leqslant6$, to multisingularities of types $A_{\mu_1}^{k_1}\ldots A_{\mu_p}^{k_p}$ leads to the same results as Theorem 2.8 in [3]. Remark 2. Using (2), it is easy to find formulae for the number $J_{\bf 1}(D_\mu^\delta)$ of connected components of the complement to the front of the map $f$: The author thanks M. E. Kazarian for helpful discussions. 
Citation in format AMSBIB
\Bibitem{Sed25} |
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