| Russian Mathematical Surveys |
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Brief communications On the adjacency of type V. D. Sedykh National State University of Oil and Gas "Gubkin University"
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     We study the adjacencies of singularities of the front of a proper generic Legendrian map with simple stable singularities. For all necessary definitions and facts from the theory of Legendrian singularities, see [1], [4], and [5]. By Arnold’s theorem on Legendrian singularities, simple stable germs of Legendrian maps are numbered by the symbols $A_{\mu}$ ($\mu=1,2,\dots$), $D_{\mu}^\pm$ ($\mu=4,5,\dots$), $E_6$, $E_7$, and $E_8$ up to Legendrian equivalence. If $\mu$ is odd, then germs of types $D_{\mu}^{+}$ and $D_{\mu}^{-}$ are Legendrian equivalent. Their types are denoted by $D_{\mu}$. Let us consider the unordered set of symbols that are the types of germs of a generic Legendrian map $f\colon L\to V$ at the preimages of a point $y\in V$. The formal commutative product $\mathcal{A}$ of these symbols is called the type of the multisingularity of the map $f$ at the point $y$ (or the type of the monosingularity if $y$ has only one preimage). If $f^{-1}(y)=\varnothing$, then $\mathcal{A}=\mathbf{1}$. The set $\mathcal{A}_f$ of points of $V$ at which $f$ has a multisingularity of type $\mathcal{A}$ is a smooth submanifold in the space $V$. Assume that $f$ has a multisingularity of type $\mathcal{B}$ at a point $y$ lying in the closure of the manifold $\mathcal{A}_f$. From the paper [2] by Looijenga it follows that $y$ has a neighbourhood $O(y)\subset V$ such that its intersection with any connected component of the manifold $\mathcal{A}_f$ whose closure contains $y$ is contractible. The number of the connected components of the intersection $O(y)\cap\mathcal{A}_f$ depends only on $\mathcal{A}$ and $\mathcal{B}$, is denoted by $J_{\mathcal A}(\mathcal{B})$, and is called the adjacency index of a multisingularity of type $\mathcal{B}$ to a multisingularity of type $\mathcal{A}$. The index $J_{\bf 1}(\mathcal{B})$ is equal to the number of the connected components of the complement to the front of the map $f$ in $O(y)$. The adjacency indices of multisingularities are calculated in terms of the adjacency indices of monosingularities (see [3], Proposition 2.5). The adjacency indices of monosingularities of types $A_\mu$ are calculated by the formula in Theorem 2.6 in [3]. The indices of all adjacencies of monosingularities of types $D_\mu^\pm$ and $E_\mu$ for $\mu\leqslant 6$ were listed in [3], Theorems 2.8 and 2.9. Vassiliev [6] calculated the indices $J_{\bf 1}(D_\mu^\pm)$ and $J_{\bf 1}(E_\mu)$ for the other $\mu$. In the present paper we calculate the adjacency indices of monosingularities of types $D_\mu^\pm$ to multisingularities of types $D_\nu^\pm A_{\mu_1}^{k_1}\dots A_{\mu_p}^{k_p}$. Let
$$
\begin{equation*}
f\colon\mathbb{R}^{n-1}\to \mathbb{R}^n,\quad (t,q)\mapsto (x,q,u),\quad x=-\frac{\partial S(t,q)}{\partial t}\,,\quad u=S(t,q)-t\,\frac{\partial S(t,q)}{\partial t}\,,
\end{equation*}
\notag
$$
where $t=(t_1,\dots,t_m)$, $x=(x_1,\dots,x_m)$, $q=(q_{m+1},\dots,q_{n-1})$, and $S=S(t,q)$ is a smooth function of $t$ with parameter $q$. The map $f$ is Legendrian. If $m=1$ and $S$ is given by the formula $S=t_1^{\mu+1}+q_{\mu-1}t_1^{\mu-1}+\cdots+q_2t_1^2$, where $\mu\geqslant 1$, then the germ of $f$ at zero has type $A_\mu$.
Assume that $m=2$, $\delta=\pm1$ and $S=t_1^2t_2+\delta t_2^{\mu-1}+ q_{\mu-1}t_2^{\mu-2}+\cdots+q_3t_2^2$, $\mu\geqslant4$. Then $f$ has a singularity of type $D_\mu^\delta$ at zero ($D_\mu^+$ if $\delta=+1$, and $D_\mu^-$ if $\delta=-1$). Let
$$
\begin{equation*}
H_i=H_i(t,q)=t_2^{i}\, \frac{\partial^{i+1}S}{\partial t_2^{i+1}}-(-1)^{i+1}(i+1)!\,t_1^2,\qquad i=1,\dots,\mu-1.
\end{equation*}
\notag
$$
Lemma ([3], Lemma 5.1). The germ of the map $f$ at a point $(t,q)\ne 0$ has the following type: (1) $A_\nu$, $1\leqslant \nu\leqslant \mu-1$, if $H_1=\dots=H_{\nu-1}=0$, $H_\nu\ne 0$; (2) $A_3$ if $t_1=t_2=0$, $q_3\ne 0$; (3) $D_{\nu}^\pm$, $4\leqslant \nu\leqslant \mu-1$, if $t_1=t_2=q_3=\dots=q_{\nu-1}=0$ and $\pm q_\nu>0$. It follows from the lemma that Legendrian monosingularities of types $D_\mu^\pm$ can only be adjacent to multisingularities of types $D_\nu^\pm A_{\mu_1}^{k_1}\dots A_{\mu_p}^{k_p}$ and $A_{\mu_1}^{k_1}\dots A_{\mu_p}^{k_p}$. Theorem. Let $\mu>\nu\geqslant4$ be integers, and let $\mathcal{A}=A_{\alpha_1}^{i_1}\dots A_{\alpha_k}^{i_k}A_{\beta_1}^{j_1}\dots A_{\beta_l}^{j_l}$, where $\alpha_1,\dots,\alpha_k$ are pairwise distinct even numbers and $\beta_1,\dots,\beta_l$ are pairwise distinct odd numbers. Then the number $J_{D_\nu^{\pm\delta}\mathcal{A}}(D_\mu^\delta)$ is equal to Proof. By the lemma the map $f$ has a multisingularity of type $D_\nu^{\pm\delta}\mathcal{A}$ at a point $y=(x_1,x_2,q_3,\dots,q_{n-1},u)$ if and only if the following three conditions hold:
(1) $x_1=x_2=q_3=\cdots=q_{\nu-1}=u=0$, $\pm q_\nu/\delta>0$; (2) if $(t_1,t_2,q_3,\dots,q_{n-1})$ is a preimage of the point $y$ under the map $f$, where $t_1^2+t_2^2\ne0$, then $t_1=0$ and $t_2$ is a multiple real root of the polynomial
$$
\begin{equation}
t_2^{\mu-\nu}+\frac{q_{\mu-1}}{\delta}\,t_2^{\mu-\nu-1}+ \cdots+\frac{q_\nu}{\delta}\,;
\end{equation}
\tag{1}
$$
(3) the polynomial (1) has $m_1=i_1+\cdots+i_k$ geometrically distinct real roots with odd multiplicity greater than $1$ and $m_2=j_1+\cdots+j_l$ geometrically distinct real roots with even multiplicity; namely it has $i_p$ real roots with odd multiplicity $\alpha_p+1$ for each $p=1,\dots,k$ and $j_r$ real roots with even multiplicity $\beta_r+1$ for each $r=1,\dots,l$. Let $i_0$ be the number of simple real roots of the polynomial (1). Then $0\leqslant i_0\leqslant N$ and $i_0\equiv N$ ($\text{mod } 2)$. The number of positions of all real roots with odd multiplicity on the $t_2$-axis is equal to $C_1=(i_0+m_1)!/(i_0!\,i_1!\cdots i_k!)$. The position of zero on this axis can be chosen in $C_2=[(\varepsilon+i_0+m_1)/2]$ ways, where $\varepsilon=1$ if $q_\nu/\delta<0$ and $\varepsilon=2$ if $q_\nu/\delta>0$. Now, the number of possible locations of real roots with even multiplicity on the $t_2$-axis is equal to $C_3=(1+i_0+m_1+m_2)!/((1+i_0+m_1)!\,j_1!\dotsb j_l!)$. The product of the numbers $C_1$, $C_2$ and $C_3$ is equal to the number of locations of all real roots of the polynomial (1). The number $J_{D_\nu^{\pm\delta}\mathcal{A}}(D_\mu^\delta)$ is equal to the sum of these products over all possible $i_0$. If $\nu$ is odd, then $J_{D_\nu\mathcal{A}}(D_\mu^\delta)= J_{D_\nu^{\delta}\mathcal{A}}(D_\mu^\delta)+ J_{D_\nu^{-\delta}\mathcal{A}}(D_\mu^\delta)$. $\Box$ Example. From the theorem we easily obtain the following formulae presented in [3], Theorem 2.8: $J_{D_4^\pm}(D_5)=1$, $J_{D_4^\delta}(D_6^\delta)=3$, $J_{D_4^{-\delta}}(D_6^\delta)=1$, $J_{D_4^\delta A_1}(D_6^\delta)=2$, and $J_{D_5}(D_6^\pm)=2$.
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\Bibitem{Sed24} |
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