THE VOLUME OF THE NEWTON POLYTOPE OF A DISCRIMINANT

S.Yu. Orevkov

1. Statement of the result. Let D_{n} = D_{n}(x_{0}, ..., x_{n}) be the discriminant, i.e. the polynomial in x_{0}, …, x_{n}, vanishing if and only if the polynomial ∑_{k=0}^{n} x_{k}t^{k} has a multiple root. Example: D_{2}(a, b, c) = b^{2} − 4ac.

The Newton polytope Δ(f) of a polynomial f = ∑ a_{u}x_{1}^{u_{1}} … x_{N}^{u_{N}}, where u = (u_{1}, …, u_{n}), is the convex hull in R^{N} of the set {u ∈ Z^{N} | a_{u} ≠ 0}. If V ∈ R^{N} is the affine k-plane such that the rank of the lattice V ∩ 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 Δ(D_{n}) by Q_{n}. Because of the evident homogenity and quasihomogenity of the discriminant, Q_{n} lies in the (n − 1)-plane

u_{0} + ⋯ + u_{n} = 2(n − 1),  u_{1} + 2u_{2} + ⋯ + nu_{n} = n(n − 1).  (1)

Theorem 1. vol_{n−1} Q_{n} = 2^{n−1}n^{n−2}/n!.

2. vol_{n−2} Δ(D̄_{n}) = (n + 6) 2^{n−3}n^{n−5}/(n − 2)! for n ≥ 3, where D̄_{n}(y_{0}, …, y_{n−2}) is the discriminant of t^{n} + y_{n−2}t^{n−2} + ⋯ + y_{0}.

Let A ⊂ 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 ∑_{a∈A} x_{a}t^{a}, where t = (t_{1}, …, t_{d}), a = (a_{1}, …, a_{d}), t^{a} = t_{1}^{a_{1}} … t_{d}^{a_{d}}, and let us define the discriminant D_{A} as the polynomial in n variables (x_{a})_{a∈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 \ 0)^{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 Δ(D_{A}) = dim Δ(E_{A}).

Theorem 3. vol_{N} Δ(E_{A}) > ( ∏_{k=1}^{d}(k + 1)^{i_{k}})(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}.

Corollary. For any d there exist C_{0}(d), C_{1}(d) > 0, such that log vol Q_{d,m} ≥ C_{0}(d) + C_{1}(d)m^{d}, where Q_{d,m} = Δ(D_{A}) for A = {a ∈ Z^{d} | a_{i} ≥ 0, ∑ a_{i} ≤ m}.

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}) ≤ 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 → ∞ (of the same order as the lower bound following from the constructions by Viro’s method).

Typeset by AMS-TEX

2. Notation. For k ∈ Z set k̄ = {1, …, k} (¯0 = ∅). By S_{n} we denote the symmetric group: S_{n} = {σ : n̄ → n̄ | σ(n̄) = n̄}; by π_{n} : R^{n+1} → R^{n−1} we denote the projection (u_{0}, …, u_{n}) ↦ (u_{1}, …, u_{n−1}). For a finite set α, we denote its cardinality by #α, and we set C_{α}^{k} = {β ⊂ α | #β = k} (then #C_{α}^{k} = C_{#α}^{k} is the binomial coeficient). For α ⊂ Z, let us denote by μ_{α} : {1, …, #α} → α, the bijection such that μ_{α}(1) < μ_{α}(2) < . . .. The letter m will always denote n − 1.

3. Q_{n} as the secondary polytope. 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_{α}}_{α⊂m̄}, where the coordinates (q_{0}^{α}, …, q_{n}^{α}) of q_{α} are defined as follows. If α = {k_{1}, …, k_{a}}, 0 = k_{0} < k_{1} < ⋯ < k_{a} < k_{a+1} = n, k_{−1} = 1, k_{a+2} = n − 1, then

q_{k}^{α} = \{ \begin{matrix}k_{i+1} − k_{i−1}, & k = k_{i} ∈ α ∪ \{0, n\} \\ 0, & k ∉ α ∪ \{0, n\} \end{matrix}

4. Triangulation of a skew cube. Let p_{α} = (p_{1}^{α}, …, p_{N}^{α}) be sets in R^{N}, indexed by subsets α ⊂ N̄, such that p_{i}^{α} > 0 for i ∈ α and p_{i}^{α} = 0 for i ∉ α. For a σ ∈ S_{N}, we denote by s_{σ} the simplex spanned on the points p_{σ(k̄)}, k = 0, …, N.

Lemma 1. a). {s_{σ}}_{σ∈S_{N}} is a triangulation of some (not necessarily convex) polyhedron P, homeomorphic to a cube (hence, vol P = ∑ vol s_{σ}).

b). If the convex hull P′ of the points p_{α} is combinatorially equivalent to a cube (i.e. for any i, all the points{p_{α}}_{i∈α} lye on the same (N − 1)-face of P′), then P′ = P.

Proof. Projecting from pN̄, let us define P inductively as the union of the cones over the intersections with the coordinate hyperplanes. □

Example: if P is a cube then s_{σ} = {(x_{1}, …, x_{N}) ∈ P | x_{σ(1)} ≥ x_{σ(2)} ≥ ⋯ ≥ x_{σ(N)}}.

5. Recurrent relation. Let {s_{σ}}_{σ∈S_{m}} be a triangulation of Q_{n} from Sect. 4.

Lemma 2. vol π_{n}(s_{σ}) = ( ∏_{k=1}^{n−1} q_{σ(k)}^{σ(k̄)})/(n − 1)!.

Proof. π_{n}(q_{σ(¯0)}) = π_{n}(q_{∅}) = 0. Hence, m! vol π_{n}(s_{σ}) = | det A_{σ}| where A_{σ} is the matrix

composed of the vectors {q_{σ(k̄)}}_{k∈m̄} written as columns. It remains to note that q_{σ(k)}^{σ(k̄)} is the k-th entry in the σ(k)-th row and all the entries to the left of it vanish. □

Lemma 3. v_{n} = n ∑_{k=1}^{n−1} C_{n−2}^{k−1}v_{k}v_{n−k}, where v_{1} = 1, v_{n} = (n − 1)! vol π_{n}(Q_{n}).

Proof. This follows from Lemmas 1 and 2, if we presents ∑_{σ∈S_{m}} as ∑_{k∈m̄} ∑_{σ∈S_{m}^{k}}, where S_{m}^{k} = {σ | σ(1) = k}, and then replace the innermost sum with the triple sum corresponding to the bijection C_{m̄\{1}}^{k−1}×S_{k−1}×S_{m−k} → S_{m}^{k}, (α, σ_{1}, σ_{2}) ↦ σ where σ(1) = k, σ(i) = μ_{α}(σ(i)) for i < k, σ(i) = μ_{m̄\(α∪{k})}(σ_{2}(i − k)) for i > k. □

6. Identity. The Abel binomial identity can be written in the form [2; Sect.1.2.7] αβ ∑_{k=0}^{n} C_{n}^{k}(α+ k)^{k−1}(β + n − k)^{n−k−1} = (α + β)(α + β + n)^{n−1}. Substituting β = −α, dividing by α^{2} and taking the limit as α → 0, we get

∑_{k=1}^{n−1} C_{n}^{k}k^{k−1}(n − k)^{n−k−1} = 2(n − 1)n^{n−2}.  (2)

7. Proof of the theorems. 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} → V ∩ Z^{n+1}, moreover, | det(j_{n}π_{n})| = n. Hence, n vol Q_{n} = vol π_{n}(Q_{n}) = v_{n}/(n − 1)!. Theorem 1 is proved. Theorem 2 is proved similarly: using the recurrent relation v̄_{n} = n ∑_{k=2}^{n−1} C_{n−3}^{k−2}v̄_{k}v_{n−k}, we find v̄_{n} = (n + 6) 2^{n−3} n^{n−4} where v̄_{n}/(n − 2)! is the volume of the projection of Δ(D̄_{n}) onto the plane y_{n} = 0.

Theorem 3 follows from Lemma 1(a). According to [1], Δ(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 α ⊂ A \ V, let us consider any triangulation whose set of vertices is α ∪ V. The corresponding points {q_{α}} ⊂ R^{A} lye on an M-plane (M = N − c = #A \ 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 ≥ ∏(k + 1)^{i_{k}}/M! (this follows from Lemma 2 and the description of Q_{A}, given in [1]). The points {q_{α}} lye on an M-dimensional section of Q_{A} (dim Q_{A} = N), and this gives M!/N!.

References

1. I.M. Gelfand, M.M. Kapranov, A.V. Zelevinskii, Discriminants, resultants and multidimensional determinants, Birkhäuser, Boston, 1994.

2. I.P. Goulden, D.M. Jackson, Combinatirial enumeration, John Wiley and Sons, N.Y., 1983.

Steklov Math. Inst. of Russ. Acad. Sci. ul. Gubkina 8, Moscow, Russia