\input amstex \documentstyle{amsppt} \magnification 1200 \NoBlackBoxes \pagewidth{40em} \def \C {\bold{C}} \def \Q {\bold{Q}} \def \Z {\bold{Z}} \def \P {\bold{P}} \def \R {\bold{R}} \def \eps {\varepsilon} \def \|{\,|\,} \def\vol{\operatorname{vol}} %---------- \topmatter \title The volume of the Newton polytope of a discriminant \endtitle %---------- \author S.Yu.~Orevkov \endauthor %---------- \address Steklov Math. Inst. of Russ. Acad. Sci. ul. Gubkina 8, Moscow, Russia \endaddress \endtopmatter %---------- \document \subsubhead 1. Statement of the result \endsubsubhead % Let $D_n=D_n(x_0,...,x_n)$ be the {\it discriminant}, i.e. the polynomial in $x_0,\dots,x_n$, vanishing if and only if the polynomial $\sum_{k=0}^n x_k t^k$ has a multiple root. Example: $D_2(a,b,c)=b^2-4ac$. The {\it Newton polytope} $\Delta(f)$ of a polynomial $f=\sum a_u x_1^{u_1}\dots x_N^{u_N}$, where $u=(u_1,\dots,u_n)$, is the convex hull in $\R^N$ of the set $\{u\in\Z^N\,|\,a_u\ne 0\}$. If $V\in\R^N$ is the affine $k$-plane such that the rank of the lattice $V\cap\Z^N$ equals $k$, then the $k$-dimensional volume $\vol_k$ on the plane $V$ will be normalized so that the volume of the fundamental parallelepiped of the lattice is equal to one. Denote $\Delta(D_n)$ by $Q_n$. Because of the evident homogenity and quasihomogenity of the discriminant, $Q_n$ lies in the $(n-1)$-plane % % $$ % \sum x_k=2(n-1), \qquad \sum kx_k = n(n-1). \eqno(1) % $$ % $$ u_0+\dots+u_n=2(n-1),\qquad u_1+2u_2+\dots+nu_n=n(n-1). \eqno(1) $$ \proclaim{ Theorem 1 } $\vol_{n-1} Q_n = 2^{n-1} n^{n-2}/n!$. \endproclaim \proclaim{ Теорема 2 } $ \vol_{n-2} \Delta(\bar D_n) = (n+6)\,2^{n-3}n^{n-5}/(n-2)!$ for $n\ge 3$, where $\bar D_n(y_0,\dots,y_{n-2})$ is the discriminant of $t^n+y_{n-2} t^{n-2} + \dots + y_0$. \endproclaim Let $A\subset\Z^d$ be an $n$-point set, $P_A$ its convex hull, $\dim P_A=d$. Following [1], denote by $\C^A$ the space of Laurent polynomials of the form $\sum_{a\in A} x_a t^a$, where $t=(t_1,\dots,t_d)$, $a=(a_1,\dots,a_d)$, $t^a=t_1^{a_1}\dots t_d^{a_d}$, and let us define the discriminant $D_A$ as the polynomial in $n$ variables $(x_a)_{a\in A}$, such that the equation $D_A=0$ defines a hyperplane in $\C^A$, which is the closure of the set of all polynomials $f$, for which the hypersurface $\{f=0\}$ has a singularity in the torus $(\C\setminus0)^d$. Respectively, the discriminant $E_A$ defines the closure of the set of polynomials which have a degenerate restriction at least to one face of $P_A$ (see details in [1]). Let $N=n-d-1=\dim\Delta(D_A)=\dim\Delta(E_A)$. \proclaim{ Theorem 3 } $\vol_N \Delta(E_A) > \big(\prod_{k=1}^d (k+1)^{i_k}\big)(N-c)!/N!$, where $c=i_0-d-1$, and $i_k$ is the number of points in $A$, which are the interior points of $k$-planes of $P_A$. \endproclaim \proclaim{ Corollary } For any $d$ there exist $C_0(d),C_1(d)>0$, such that $\log\vol Q_{d,m} \ge C_0(d)+C_1(d) m^d$, where $Q_{d,m}=\Delta(D_A)$ for $A=\{a\in\Z^d\| a_i\ge0,\, \sum a_i \le m\}$. \endproclaim This gives a negative answer to a question of E.I.~Shustin about existence of constants $B_0(d)$, $B_1(d)$, such that $\log( N! \vol Q_{d,m} ) \le B_0(d) + B_1(d) m^d$, where $N = C_{n+d}^d-d-1 = \dim Q_{d,m}$. An affirmative answer would provide an expected asymptotical upper bound for the number of rigid isotopy types of projective real hypersurfaces of degree $m$ as $m\to\infty$ (of the same order as the lower bound following from the constructions by Viro's method). \subsubhead 2. Notation \endsubsubhead % For $k\in\Z$ set $\bar k=\{1,\dots,k\}$ ($\bar 0=\varnothing$). By $S_n$ we denote the symmetric group: $S_n=\{\sigma:\bar n\to\bar n \| \sigma(\bar n)=\bar n\}$; by $\pi_n:\R^{n+1}\to\R^{n-1}$ we denote the projection $(u_0,\dots,u_n)\mapsto(u_1,\dots,u_{n-1})$. For a finite set $\alpha$, we denote its cardinality by $\#\alpha$, and we set $C_\alpha^k=\{\beta\subset\alpha\| \#\beta=k\}$ (then $\#C_\alpha^k=C_{\#\alpha}^k$ is the binomial coeficient). For $\alpha\subset\Z$, let us denote by $\mu_\alpha:\{1,\dots,\#\alpha\}\to\alpha$, the bijection such that $\mu_\alpha(1)<\mu_\alpha(2)<\dots$. The letter $m$ will always denote $n-1$. \subsubhead 3. $Q_n$ as the secondary polytope \endsubsubhead % According to a result due to Gelfand-Kapranov-Zelevinski [1], $Q_n$ is combinatorially equivalent to the $m$-cube (recall that $m=n-1$), and its vertices are the points $\{q_\alpha\}_{\alpha\subset\bar m}$, where the coordinates $(q_0^\alpha,\dots,q_n^\alpha)$ of $q_\alpha$ are defined as follows. % %Рассмотрим разбиение (триангуляцию) отрезка $[0,n]$ на меньшие отрезки %точками множества $\alpha$. Если $00$ for $i\in\alpha$ and $p_i^\alpha=0$ for $i\not\in\alpha$. For a $\sigma\in S_N$, we denote by $s_\sigma$ the simplex spanned on the points $p_{\sigma(\bar k)}$, $k=0,\dots,N$. \proclaim{ Lemma 1 } %%% a). $\{s_\sigma\}_{\sigma\in S_N}$ is a triangulation of some (not necessarily convex) polyhedron $P$, homeomorphic to a cube (hence, $\vol P=\sum\vol s_\sigma$). b). If the convex hull $P'$ of the points $p_\alpha$ is combinatorially equivalent to a cube (i.e. for any $i$, all the points$\{p_\alpha\}_{i\in\alpha}$ lye on the same $(N-1)$-face of $P'$), then $P'=P$. \endproclaim \demo{ Proof } Projecting from $p_{\bar N}$, let us define $P$ inductively as the union of the cones over the intersections with the coordinate hyperplanes. \qed\enddemo Example: if $P$ is a cube then $s_\sigma=\{(x_1,\dots,x_N)\in P \| x_{\sigma(1)} \ge x_{\sigma(2)} \ge\dots\ge x_{\sigma(N)} \}$. \subsubhead 5. Recurrent relation \endsubsubhead % Let $\{s_\sigma\}_{\sigma\in S_m}$ be a triangulation of $Q_n$ from Sect.~4. \proclaim{ Lemma 2 } $\vol \pi_n(s_\sigma) = \big(\prod_{k=1}^{n-1} q_{\sigma(k)}^{\sigma(\bar{k})}\big)/(n-1)!$. \endproclaim \demo{ Proof } $\pi_n(q_{\sigma(\bar0)})=\pi_n(q_\varnothing)=0$. Hence, $m! \vol \pi_n(s_\sigma)=|\det A_\sigma|$ where $A_\sigma$ is the matrix composed of the vectors $\{q_{\sigma(\bar k)}\}_{k\in\bar m}$ written as columns. It remains to note that $q_{\sigma(k)}^{\sigma(\bar{k})}$ is the $k$-th entry in the $\sigma(k)$-th row and all the entries to the left of it vanish. \qed\enddemo \proclaim{ Lemma 3 } $v_n = n \sum_{k=1}^{n-1} C_{n-2}^{k-1} v_k v_{n-k}$, where $v_1=1$, $v_n = (n-1)! \vol \pi_n(Q_n)$. \endproclaim \demo{ Proof } This follows from Lemmas 1 and 2, if we presents $\sum_{\sigma\in S_m}$ as $\sum_{k\in\bar m}\sum_{\sigma\in S_m^k}$, where $S_m^k = \{\sigma\|\sigma(1)=k\}$, and then replace the innermost sum with the triple sum corresponding to the bijection $ C_{\bar m\setminus\{1\}}^{k-1} \times S_{k-1} \times S_{m-k} \to S_m^k $, $(\alpha,\sigma_1,\sigma_2)\mapsto\sigma$ where $\sigma(1)=k$, $\sigma(i)=\mu_\alpha(\sigma(i))$ for $ik$. \qed\enddemo \subsubhead 6. Identity \endsubsubhead % The Abel binomial identity can be written in the form [2; Sect.1.2.7] $ \alpha\beta\sum_{k=0}^n C_n^k (\alpha+k)^{k-1}(\beta+n-k)^{n-k-1} =(\alpha+\beta)(\alpha+\beta+n)^{n-1}. $ Substituting $\beta=-\alpha$, dividing by $\alpha^2$ and taking the limit as $\alpha\to 0$, we get $$ \sum_{k=1}^{n-1} C_n^k k^{k-1} (n-k)^{n-k-1}=2(n-1)n^{n-2}. \eqno(2) $$ \subsubhead 7. Proof of the theorems \endsubsubhead % From (2) and Lemma 3, we get by induction $v_n=2^{n-1}n^{n-2}$. Let $V$ be the plane defined by (1). Solving (1) with respect to $u_0$, $u_1$, we get a bijection $j_n:\Z^{n-1}\to V\cap\Z^{n+1}$, moreover, $|\det(j_n\pi_n)|=n$. Hence, $n\vol Q_n=\vol \pi_n(Q_n) = v_n/(n-1)!$. Theorem 1 is proved. % Theorem 2 is proved similarly: using the recurrent relation $\bar v_n = n\sum_{k=2}^{n-1} C_{n-3}^{k-2}\bar v_k v_{n-k}$, we find $\bar v_n = (n+6)\,2^{n-3}\,n^{n-4}$ where $\bar v_n/(n-2)!$ is the volume of the projection of $\Delta(\bar D_n)$ onto the plane $y_n=0$. Theorem 3 follows from Lemma 1(a). According to [1], $\Delta(E_A)$ is the convex hull of the points in $\R^A$, corresponding to all triangulations of $P_A$. Let $V$ be the set of the vertices of $P_A$ ($i_0=\#V$). For each $\alpha\subset A\setminus V$, let us consider any triangulation whose set of vertices is $\alpha\cup V$. The corresponding points $\{q_\alpha\}\subset\R^A$ lye on an $M$-plane ($M=N-c=\#A\setminus V$) and satisfy the hypothesis of Lemma 1(a). Hence, one can span $M!$ simplices on them so that the volume of each one is $\ge\prod(k+1)^{i_k}/M!$ (this follows from Lemma 2 and the description of $Q_A$, given in [1]). The points $\{q_\alpha\}$ lye on an $M$-dimensional section of $Q_A$ ($\dim Q_A=N$), and this gives $M!/N!$. \Refs \ref\no1 \by I.M.~Gelfand, M.M.~Kapranov, A.V.~Zelevinskii \book Discriminants, resultants and multidimensional determinants \publ Birkh\"auser \publaddr Boston \yr 1994 \endref \ref\no2\by I.P.~Goulden, D.M.~Jackson \book Combinatirial enumeration \publ John Wiley and Sons \publaddr N.Y. \yr 1983 \endref \endRefs \enddocument