Determinantal Singularities and Newton Polyhedra

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c Pleiades Publishing, Ltd., 2007. ISSN 0081-5438, Proceedings of the Steklov Institute of Mathematics, 2007, Vol. 259, pp. 16–34.

Determinantal Singularities and Newton Polyhedra A. Esterov a Received August 2007

Abstract—Topological invariants of determinantal singularities are studied in terms of Newton polyhedra. The approach is based on the notion of a toric resolution of a determinantal singularity. Computations are carried out in the more general setting of “elimination theory in the context of Newton polyhedra.” DOI: 10.1134/S0081543807040037

1. INTRODUCTION Many invariants of isolated singularities of complete intersections have been computed in terms of Newton polyhedra. This paper extends some of these computations (namely, [9, 13, 10, 14]) to isolated singularities of determinantal sets. A determinantal set is an analytic set in Cm deﬁned by the vanishing locus of the maximal minors of a holomorphic n × k matrix, provided that this set has dimension m − |n − k| + 1 (i.e., the minimal one). In this section, we recall the meaning of the phrase “to compute in terms of Newton polyhedra” and present formulas that express some invariants of singularities of determinantal sets in terms of Newton polyhedra. Namely, Theorems 1.9, 1.12, and 1.15 compute the multiplicity of a determinantal singularity, the Milnor number of a function on it, and the Poincar´e–Hopf index of a covector ﬁeld on it, respectively. These formulas are corollaries to Theorems 3.5 and 3.19, which compute the same invariants for a more general class of singularities. We introduce this class in Section 2 (singularities of resultantal sets, see Deﬁnition 2.1). In Section 4, we construct a toric resolution of singularities of a resultantal set, which turns Theorems 3.5 and 3.19 into obvious generalizations of similar facts about complete intersections. One can consider these computations as a further development of the elimination theory in the context of Newton polyhedra (see [18]). Indeed, in the terminology of Deﬁnition 2.1, the paper [18] is concerned with the Newton polyhedron and the leading coeﬃcients of the equation of a resultantal hypersurface. Theorems 2.12, 3.5, 3.16, and 3.19 remain meaningful for resultantal hypersurfaces as well. In particular, Theorems 2.12 and 3.16 describe the dimension of a resultantal set and the dimension of its singular locus, respectively. This was a motivation for proving these theorems in the context of resultantal sets rather than in the context of determinantal sets, which makes them somewhat more complicated. Newton polyhedra. A polyhedron in Rn is the intersection of a ﬁnite number of closed halfspaces. A polyhedron is said to be integer if its vertices are integer lattice points. All polyhedra considered in this paper are assumed to be integer. m Denote the positive orthant in Rm by Rm + . For every a = (a1 , . . . , am ) ∈ Z , denote the a1 a a m m by x . For a polyhedron ∆ ⊂ R+ , denote the space of all complex analytic monomial x1 . . . xm functions of the form a∈∆∩Zm ca xa by C{∆}. For polyhedra ∆1 , . . . , ∆k ⊂ Rm + , let b1 , . . . , bN be all pairs of the form (i, a), where a is an integer lattice point in a bounded face of the polyhedron ∆i . a Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario M5S 2E4, Canada.

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Deﬁnition 1.1. “Almost all collections of germs f1 ∈ C{∆1 }, . . . , fk ∈ C{∆k }” means “all collections of germs (f1 , . . . , fk ), fi = a∈∆i c(i,a) xa , such that P (cb1 , . . . , cbN ) = 0, where P is some nonzero polynomial in N variables.” Example 1.2. If the diﬀerences Rm + \ ∆i are bounded, then almost all collections of germs f1 ∈ C{∆1 }, . . . , fk ∈ C{∆k } deﬁne isolated complete intersection singularities (see [7] for the ﬁrst formulation and [11] for a detailed proof based on the construction of a toric resolution for a complete intersection singularity). Suppose that a map c assigns an element of a set B to every analytic germ f : Cn → C. Then “to compute the characteristic c in terms of Newton polyhedra” means “for each polyhedron ∆, to ﬁnd c∆ ∈ B such that c(f ) = c∆ for almost all f ∈ C{∆}.” Example 1.3. The Milnor number, the ζ-function of monodromy, and many other invariants of singularities of analytic functions are computed in terms of Newton polyhedra (see, e.g., [9, 13]). This terminology comes from the notion of the Newton polyhedron of a germ of an analytic function. Indeed, by deﬁnition, the Newton polyhedron of a germ a∈Zm ca xa : Cm → C near the origin is the convex hull of the union a : ca =0 (a + Rm + ). Thus, ∆ is the Newton polyhedron for almost all functions in C{∆}. Mixed volumes of pairs of polyhedra. The Minkowski sum of sets A and B in Rn is the set A + B = {a + b | a ∈ A, b ∈ B}. Deﬁnition 1.4. Polyhedra ∆1 and ∆2 in Rn are said to be parallel if, for each point a ∈ Rn , a + ∆1 ⊆ ∆1

⇔

a + ∆2 ⊆ ∆2 .

Deﬁnition 1.5. 1. A pair of polyhedra ∆1 , ∆2 in Rn is called bounded if both ∆1 \ ∆2 and ∆2 \ ∆1 are bounded. The set of all bounded pairs of polyhedra parallel to a given convex cone C ⊂ Rn is denoted by BPC . 2. The Minkowski sum (∆1 , ∆2 ) + (Γ1 , Γ2 ) of two pairs (∆1 , ∆2 ) and (Γ1 , Γ2 ) in BPC is the pair (∆1 + Γ1 , ∆2 + Γ2 ) ∈ BPC . 3. The volume Vol(∆1 , ∆2 ) of a bounded pair (∆1 , ∆2 ) is the diﬀerence Vol(∆1 \ ∆2 ) − Vol(∆2 \ ∆1 ). 4. The mixed volume is a symmetric linear (with respect to the Minkowski summation) function MV : BPC × . . . × BPC → R such that MV(A, . . . , A) = Vol(A) for each pair A ∈ BPC . n

The function MV is well-deﬁned and unique. The mixed volume of pairs of n-dimensional integer polyhedra is a rational number with denominator n! (see [15] for the proof and other basic facts about the mixed volume of pairs). One can use Lemma 3.21 to express the mixed volume of pairs in terms of the mixed volumes of polyhedra. Example 1.6. If C consists of one point, then BPC consists of pairs of bounded polyhedra, and MV (∆1 , Γ1 ), . . . , (∆n , Γn ) = MV(∆1 , . . . , ∆n ) − MV(Γ1 , . . . , Γn ), where MV on the right-hand side means the classical mixed volume of bounded polyhedra. If C is not bounded, then both terms on the right-hand side are inﬁnite, but their diﬀerence makes sense. Multiplicities of determinantal singularities in terms of Newton polyhedra. Suppose that A = (Ai,j ) : Cm → Cn×k is a germ of a matrix with holomorphic entries near the origin and n ≤ k. Deﬁnition 1.7. The set [A] = {x | rk A(x) < n} is called the n × k determinantal set deﬁned by A, provided that its dimension equals m − k + n − 1 (i.e., its codimension in Cm equals the codimension of degenerate matrices in the space of all n × k matrices). PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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Example 1.8. A 1 × k determinantal set is the same as a complete intersection of codimension k. m m m For a polyhedron ∆ ⊂ Rm + , denote the pair (R+ , ∆) by ∆. Denote the pair (R+ , R+ \ Sm ) by L, where Sm is the standard m-dimensional simplex. Theorem 1.9. Suppose that ∆i ⊂ Rm + , i = 1, . . . , k, are polyhedra such that the diﬀerences m R+ \ ∆i are bounded. 1. If k − n + 1 < m, then, for almost all collections of germs Ai,j ∈ C{∆j }, i = 1, . . . , n, j = 1, . . . , k, the set [A] is n × k determinantal, and its multiplicity equals

j ,...,∆ j m! MV ∆ , L, . . . , L . 0 k−n 0
m−k+n−1

2. If k − n + 1 = m, then, for almost all collections of germs Ai,j ∈ C{∆j }, i = 1, . . . , n, j = 1, . . . , k, the Buchsbaum–Rim multiplicity dimC OCm ,0 /maximal minors of A is ﬁnite and equals

j ,...,∆ j m! MV ∆ . 0 k−n 0
Proof. Part 1 is a special case of Theorem 3.5, part 1, because the multiplicity of [A] is, by deﬁnition, the intersection number of [A] and a generic plane of the complementary dimension, which is a determinantal set as well. Part 2 is a special case of Theorem 3.5, part 2, because the Buchsbaum–Rim multiplicity equals the intersection number of the germ A(Cm ) and the set of all degenerate matrices in the space of all n × k matrices. The formulas in parts 1 and 2 of Theorem 3.5 need some simpliﬁcation in this case. For example, 1 , . . . , Sn−1 × ∆ k ), where Sn−1 ⊂ Rn−1 is the standard part 2 gives the answer k! MV(Sn−1 × ∆ (n − 1)-dimensional simplex. One can simplify it using the following well-known fact. Lemma 1.10. (p + q)! MV A1 × B1 , . . . , Ap+q × Bp+q

MV Ai1 , . . . , Aip MV Bj1 , . . . , Bjq , = p! q! {i1 ,...,ip }{j1 ,...,jq }={1,...,p+q}

where A1 , . . . , Ap+q are polyhedra or pairs of polyhedra in Rp and B1 , . . . , Bp+q are polyhedra or pairs of polyhedra in Rq . The description of the nondegeneracy condition from the formulation of Theorem 1.9 is given in Theorem 1.17. More generally, one can compute the intersection number of several determinantal sets in terms of Newton polyhedra, without assuming that the Newton polyhedron of Ai,j does not depend on i (see Theorems 3.5 and 1.19). See also [2] for a generalization of part 2 in another direction (k − n + 1 > m). Topological invariants of functions on isolated determinantal singularities in terms of Newton polyhedra. Suppose that the germ of a complex analytic set M ⊂ Cm is smooth outside the origin. Let f : (Cm , 0) → (C, 0) be a germ of a complex analytic function, such that the restriction f |M has no singular points in a punctured neighborhood of the origin. Deﬁnition 1.11. The Milnor ﬁbration of f |M is the locally trivial ﬁbration f : M ∩ B ∩ f (−1) D∗ , M ∩ ∂B ∩ f (−1) D∗ → D∗ , origin and D∗ ⊂ C is a small punctured ball with where B ⊂ Cm is a small ball with center at the (−1) (δ), M ∩∂B∩f (−1) (δ) of this ﬁbration is called center at the origin. A ﬁber (F, ∂F ) = M ∩B∩f PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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i

a Milnor ﬁber of f |M . The ζ-function of monodromy of f |M is the product i det(t · id − hi )(−1) , where hi : Hi (F, ∂F ; C) → Hi (F, ∂F ; C) is the monodromy of the loop δe2πit , t ∈ [0, 1]. For a set I ⊂ {1, . . . , m}, let RI ⊂ Rm be a coordinate plane given by the equations xi = 0, I m I I i∈ / I. For a polyhedron ∆ ⊂ Rm + , denote the pair (R ∩ R+ , R ∩ ∆) by ∆ . Rm +

Theorem 1.12. Suppose that ∆i ⊂ Rm + , i = 0, . . . , k, are polyhedra such that the diﬀerences \ ∆i are bounded, and m ≤ 2(k − n + 2). Then

(1) for almost all collections of germs Ai,j ∈ C{∆j }, i = 1, . . . , n, j = 1, . . . , k, the determinantal set [A] is smooth outside the origin; (2) for almost all collections of germs f ∈ C{∆0 }, Ai,j ∈ C{∆j }, i = 1, . . . , n, j = 1, . . . , k, the Euler characteristic of a Milnor ﬁber of f |[A] equals

n+q−k−1 (−1)|I|+n+k C|I|+q−a 0 −2

a0 ∈N, I⊂{1,...,m} {j1 ,...,jq }⊂{1,...,k}

×

aj1 ∈N, ..., ajq ∈N aj1 +...+ajq =|I|−a0

I ,...,∆ I, ∆ I ,... ,∆ I ,... ,∆ I ,...,∆ I . |I|! MV ∆ 0 0 j1 j1 jq jq a0

aj1

ajq

/ {0, . . . , n}. Thus, all terms in this sum are In the last formula we assume that Cnk = 0 if k ∈ equal to zero, except the terms with |I| − a0 ≥ q > k − n. Proof. Part (1) is a special case of Theorem 3.16, part 2. Part (2) is a special case of Theorem 3.19, part 2. The formula in part 2 of Theorem 3.19, simpliﬁed by Lemma 1.10, states that the Euler characteristic of a Milnor ﬁber equals the sum of mixed volumes of the form I 0, . . . , ∆ I0 , . . . , ∆ I,... ,∆ I (−1)|I|+n+k |I|! MV ∆ k k a0

with coeﬃcients

ak

(−1)b0 Cnb0 Cba11+a1 . . . Cbakk+ak .

b0 +...+bk =n−1 b0 ≥1−a0 , ..., bk ≥1−ak

One can compute this expression by multiplying the generating functions of the sequences involved. Recall that the generating function of the sequence (−1)b Cnb , b ∈ Z, is (1 − t)n , and the generating a , b ∈ Z, is 1/(1 − t)a+1 . function of the sequence Cb+a The description of the nondegeneracy condition from the formulation of this theorem is given in Theorem 1.17. The Euler characteristic of a Milnor ﬁber equals the degree of the ζ-function of monodromy. More generally, one can compute the Euler characteristic and the ζ-function itself in terms of Newton polyhedra, without assuming that the Newton polyhedron of Ai,j does not depend on i (see Theorems 3.19, 3.20, and 1.19). We also generalize this computation to 1-forms as follows. Radial indices and Newton polyhedra of 1-forms. To generalize Theorem 1.12 to 1-forms, we need corresponding generalizations of the notions of the Newton polyhedron and the Euler characteristic of a Milnor number. m Suppose that ∆ ⊂ Rm + is a polyhedron such that the diﬀerence R+ \∆ is bounded. Let b1 , . . . , bN be all pairs (i, a) such that a = (a1 , . . . , am ) is an integer lattice point in a bounded face of the polyhedron ∆ and ai = 0. Deﬁnition 1.13. Let C∧ {∆} be the space of all germs of complex analytic 1-forms m i=1 fi dxi in Cm such that xi fi ∈ C{∆}. “Almost all 1-forms ω ∈ C∧ {∆}” means “all 1-forms ω = PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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a dxi a∈∆, ai =0 ci,a x xi

i

such that P (cb1 , . . . , cbN ) = 0, where P is some nonzero polynomial in

N variables.” Note that f ∈ C{∆} if and only if df ∈ C∧ {∆}. The polyhedron ∆ is the Newton polyhedron {∆} (the Newton polyhedron of an analytic 1-form fi dxi is the of almost all 1-forms from C ∧ convex hull of the union Ni , where Ni is the Newton polyhedron of xi fi ; see [14]). Suppose that the germ of a real analytic set V ⊂ Rm is smooth outside the origin. Suppose that ω is a germ of a real continuous 1-form in Rm near the origin, and the restriction ω|V has no zeros in some punctured neighborhood U of the origin. Let ω be a 1-form on V \ {0} such that (1) ω has isolated zeros p1 , . . . , pN in U ; (2) ω = ω|V outside U ; and (3) ω = i xi dxi V near the origin.

, where indpj is the Poincar´e– Deﬁnition 1.14 [3]. The radial index of ω|V is 1 + j indpj ω Hopf index at pj . If V is smooth, then the radial index of ω|V equals the Poincar´e–Hopf index ind0 ω|V . If V ⊂ Cm and f : Cm → C are complex analytic, then the radial index of the 1-form d Re f |V equals one minus the Euler characteristic of a Milnor ﬁber of f |V . Theorem 1.15. Suppose that ∆i ⊂ Rm + , i = 0, . . . , k, are polyhedra such that the diﬀerences m R+ \ ∆i are bounded, and m ≤ 2(k − n + 2). Then, for almost all collections of germs ω ∈ C∧ {∆0 }, Ai,j ∈ C{∆j }, i = 1, . . . , n, j = 1, . . . , k, the radial index of Re ω|[A] is well deﬁned and equals 1−

n+q−k−1 (−1)|I|+n+k C|I|+q−a 0 −2

a0 ∈N, I⊂{1,...,m} {j1 ,...,jq }⊂{1,...,k}

×

aj1 ∈N, ..., ajq ∈N aj1 +...+ajq =|I|−a0

I 0, . . . , ∆ I0 , ∆ Ij , . . . , ∆ Ij , . . . , ∆ Ij , . . . , ∆ Ij . |I|! MV ∆ q q 1 1 a0

aj1

ajq

(We write Re ω for a complex 1-form ω on a complex manifold, considering ω as a complex-valued function on the tangent bundle.) This is a special case of Theorem 3.19, part 3. More generally, one can compute the radial index without assuming that the Newton polyhedron of Ai,j does not depend on i (see Theorems 3.5 and 1.19). Nondegeneracy conditions. Keeping track of proofs of Theorems 1.9, 1.12, and 1.15, one can formulate explicitly what “almost all” means in the formulations of these theorems. For a k a the bounded face Γ of a polyhedron ∆ ⊂ R and a function f = a∈∆ ca x ∈ C{∆}, denote dxi a i gi xi ∈ C∧ {∆}, denote the 1-form gi |Γ dx polynomial a∈Γ ca x by f |Γ . For a 1-form ω = xi by ω|Γ . Deﬁnition 1.16. Suppose that A = (Ai,j ) : Cm → Cn×k is a germ of a holomorphic matrix near the origin, n ≤ k, Ai,j ∈ C{∆j }, i = 1, . . . , n, j = 1, . . . , k, where ∆j ⊂ Rm + , j = 1, . . . , k, are m polyhedra such that the diﬀerences R+ \ ∆i are bounded. 1. The for each collection of faces Γi ⊂ ∆i such that matrix A is said to be nondegenerate if, the sum i Γi is a bounded face of the polyhedron i ∆i , the codimension of the set

(x, λ1 : . . . : λn ) ∈ (C \ {0})m × CPn−1 (λ1 , . . . , λn ) · AΓ (x) = 0 in (C \ {0})m × CPn−1 equals k, where AΓ is the n × k matrix with entries Ai,j |Γj , i = 1, . . . , n, j = 1, . . . , k. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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2.The matrix A is said to be strongly nondegenerate if, for each collection of faces Γi ⊂ ∆i such that i Γi is a bounded face of i ∆i , the codimension of the set {x ∈ (C \ {0})m | rk AΓ (x) < n} in (C \ {0})m equals k − n + 1 and the set {x ∈ (C \ {0})m | rk AΓ (x) < n − 1} is empty. m 3. Let ∆0 ⊂ Rm + be a polyhedron such that the diﬀerence R+ \ ∆0 is bounded. A 1-form ω ∈ C∧ {∆0 } is said to be nondegenerate with respect tothe matrix A if A is strongly nondegenerate k k and, for each collection of faces Γi ⊂ ∆i such that i=0 Γi is a bounded face of i=0 ∆i , the m restriction of ω|Γ0 to the set {x ∈ (C \ {0}) | rk AΓ (x) < n} has no zeros.

Theorem 1.17. 1. Under the assumptions of Deﬁnition 1.16, the matrix A is nondegenerate for almost all collections Ai,j ∈ C{∆j }. The statement of Theorem 1.9 is true for all such matrices A. 2. If m ≤ 2(k − n + 2), then the 1-form df is nondegenerate with respect to A for almost all collections f ∈ C{∆0 }, Ai,j ∈ C{∆j }. The statement of Theorem 1.12 is true for all such matrices A and functions f . 3. If m ≤ 2(k − n + 2), then the 1-form ω is nondegenerate with respect to A for almost all collections ω ∈ C∧ {∆0 }, Ai,j ∈ C{∆j }. The statement of Theorem 1.15 is true for all such matrices A and 1-forms ω. The proof of the ﬁrst part of each of statements 1–3 easily follows from the Bertini–Sard theorem. Mixed volumes of prisms. Suppose that a germ of a matrix A = (Ai,j ) : Cm → Cn×k with holomorphic entries Ai,j ∈ C{∆i,j } near the origin deﬁnes a germ of a determinantal set [A]. Theorems 3.5, 3.19, and 3.20 express topological invariants of singularities of [A] in terms of the Newton polyhedra ∆i,j without the assumption that ∆1,j = . . . = ∆n,j for every j. The answers given by Theorems 3.5, 3.19, and 3.20 involve mixed volumes of prisms over the polyhedra ∆i,j in the following sense. Deﬁnition 1.18. For polyhedra ∆1 , . . . , ∆n ⊂ Rm , the prism ∆1 ∗ . . . ∗ ∆n is deﬁned as the convex hull of the union {bi } × ∆i ⊂ Rn−1 ⊕ Rm , i

where b1 , . . . , bn are the points (1, 0, . . . , 0), (0, 1, . . . , 0), . . . , (0, 0, . . . , 1), and (0, 0, . . . , 0) in Rn−1 . For pairs of polyhedra (Γ1 , ∆1 ), . . . , (Γn , ∆n ) in Rm , the prism (Γ1 , ∆1 ) ∗ . . . ∗ (Γn , ∆n ) is deﬁned as the pair (Γ1 ∗ . . . ∗ Γn , ∆1 ∗ . . . ∗ ∆n ). The following formula simpliﬁes the computation of the mixed volume of integer prisms. For a bounded set ∆ ⊂ Rm , denote the number of integer lattice points in ∆ by I(∆). For a bounded pair I(Γ \ ∆) − I(∆ \ Γ) by I(Γ, ∆). of (closed) integer polyhedra (Γ, ∆) in Rm , denote the diﬀerence m by ⊂ R ∆ Denote the convex hull of the union of polyhedra ∆ i i i . For bounded pairs of polyhedra m (Γi , ∆i ) in R , denote the pair i Γi , i ∆i by i (Γi , ∆i ). Theorem 1.19. If Bi,j , i = 1, . . . , n, j = 1, . . . , k, are bounded integer polyhedra or bounded pairs of integer polyhedra in Rm parallel to each other and m = k − n + 1, then the mixed volume of the prisms B1,j ∗ . . . ∗ Bn,j , j = 1, . . . , k, equals ⎛ 1 k!

⎜ (−1)k−|J| I ⎜ ⎝

J⊂{1,...,k} b1 +...+bn =|J|

⎞

i=1,...,n J1 ...Jn =J |J1 |=b1 , ..., |Jn |=bn j∈Ji

⎟ Bi,j ⎟ ⎠.

The proof is given in [17, Lemmas 13 and 23]. Note that some of Bi,j may be empty. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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2. RESULTANTAL SETS In this section we deﬁne (A1 , . . . , AI )-resultantal sets for arbitrary ﬁnite sets A1 , . . . , AI ⊂ Zn (Deﬁnition 2.1), compute the dimension of an (A1 , . . . , AI )-resultantal set in terms of A1 , . . . , AI (Theorem 2.12), and prove that one can assume without loss of generality that the collection A1 , . . . , AI is nondegenerate in some sense (Theorem 2.15). Deﬁnition of resultantal sets. Let A ⊂ Zn be a ﬁnite set. Denote the set of all Laurent polynomials of the form a∈A ca ta by C[A], where a = (a1 , . . . , an ), t = (t1 , . . . , tn ), and ta = ta11 . . . tann . Let A1 , . . . , AI ⊂ Zn be ﬁnite sets. Let Σ(A1 , . . . , AI ) be the closure of the set of all collections (p1 , . . . , pI ) ∈ C[A1 ] ⊕ . . . ⊕ C[AI ] such that the set {t ∈ (C \ {0})n | p1 (t) = . . . = pI (t) = 0} is not empty. Deﬁnition 2.1. Let A1 , . . . , AI ⊂ Zn be ﬁnite sets. The germ of an analytic set M ⊂ Cn near the origin is called (A1 , . . . , AI )-resultantal if, for some analytic germ f : Cm → C[A1 ] ⊕ . . . ⊕ C[AI ], the set M equals f (−1) (Σ(A1 , . . . , AI )) and its codimension in Cn equals the codimension of Σ(A1 , . . . , AI ) in the space C[A1 ] ⊕ . . . ⊕ C[AI ]. Because of the following lemma, the codimension of the set Σ(A1 , . . . , AI ) is well deﬁned. Lemma 2.2. The set Σ(A1 , . . . , AI ) is irreducible.

Proof. Denote the projections of the space (C \ {0})n × C[A1 ] ⊕ . . . ⊕ C[AI ] to the ﬁrst and the second factors by π1 and π2 , respectively. This space contains the set T of all collections (t, p1 , . . . , pI ) such that p1 (t) = . . . = pI (t) = 0. Since π1 : T → (C \ {0})n is a vector bundle over an irreducible variety, T is irreducible. Since Σ(A1 , . . . , AI ) is the closure of π2 (T ), it is also irreducible. Examples of resultantal sets. Example 2.3. If A1 = . . . = AI = Z0 , then Σ(A1 , . . . , AI ) is the origin of C[A1 ] ⊕ . . . ⊕ C[AI ] = CI , and an (A1 , . . . , AI )-resultantal set is the same as a complete intersection of codimension I. Example 2.4. If I = n + 1 and the convex hulls of sets A1 , . . . , AI ⊂ Zn are n-dimensional, then Σ(A1 , . . . , AI ) is the zero set of an irreducible polynomial, which is usually referred to as the (A1 , . . . , AI )-resultant (see, for example, [12]). In particular, this resultantal set is a hypersurface. Example 2.5. If A1 = . . . = AI is the set of vertices of the standard n-dimensional simplex Rn , n < I, then identify the space of collections of linear functions in one can n a0,1 + i=1 ai,1 ti , . . . , a0,I + ni=1 ai,I ti ∈ C[A1 ]⊕ . . . ⊕ C[AI ] with the space of (n + 1)× I matrices (ai,j ), the set Σ(A1 , . . . , AI ) with the set of all degenerate matrices, and (A1 , . . . , AI )-resultantal sets with (n + 1) × I determinantal sets. Example 2.6. If M ⊂ Cn and N ⊂ Cn are resultantal germs and codim M ∩ N = codim M + codim N , then M ∩ N is also resultantal. Indeed, suppose that M = f (−1) Σ(A1 , . . . , AI ), Ai ⊂ Zm, I , B 1 , . . . , B J ) , 1 , . . . , A and N = g(−1) Σ(B1 , . . . , BJ ), Bj ⊂ Zn . Then M ∩ N = (f, g)(−1) Σ(A m n m n where Ai = Ai × {0} ⊂ Z ⊕ Z and Bj = {0} × Bj ⊂ Z ⊕ Z . Example 2.7. Combining the last two examples, let Ekl ⊂ Zkl be the set of vertices of the standard kl -dimensional simplex, and Il > kl for l = 1, . . . , p. Denote the image of Eki under the k . Then a proper intersection of kl × Il determinantal sets, where l embedding Zki → l Zkl by E i k , . . . , E k , . . . , E k -resultantal set. k , . . . , E runs over 1, . . . , p, is an E p p 1 1 I1

Ip

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Codimension of resultantal sets. The following theorem expresses the codimension of the set Σ(A1 , . . . , AI ) in terms of A1 , . . . , AI . Deﬁnition 2.8. A sublattice L ⊂ Zn is said to be generated by a set A ∈ Zn if it is generated by all vectors of the form a − b, where a ∈ A and b ∈ A. Deﬁnition 2.9. The dimension of a set A ∈ Zn is the dimension of the sublattice generated by A. Deﬁnition 2.10. The sum of sets Ai ∈ Zn is the set of all sums of the form i ai , where ai ∈ Ai . Deﬁnition 2.11. The codimension of a collection of ﬁnite sets A1 , . . . , AI ⊂ Zn is the diﬀerence I − dim i Ai if I = 0. The codimension of the empty collection is 0. The following theorem is a detailed version of Theorem 1.1 from [12] and D. Bernstein’s theorem from [8, § 1.5]. Theorem 2.12. The codimension of an (A1 , . . . , AI )-resultantal set (and, in particular, the codimension of the set Σ(A1 , . . . , AI )) equals the maximum of codimensions of subcollections Ai1 , . . . , AiJ over all subsets {i1 , . . . , iJ } ⊂ {1, . . . , I}. Proof. The codimension of an (A1 , . . . , AI )-resultantal germ equals the codimension of the set Σ(A1 , . . . , AI ) by deﬁnition. The codimension of Σ(A1 , . . . , AI ) is computed at the end of this section. This computation is based on the most important special case of Theorem 2.12: Lemma 2.13. Suppose that the codimensions of all subcollections of a collection of ﬁnite sets A1 , . . . , AI ⊂ Zn are not positive. Then Σ(A1 , . . . , AI ) = C[A1 ] ⊕ . . . ⊕ C[AI ]. Proof. If codim Σ(A1 , . . . , AI ) > 0, then, by D. Bernstein’s formula [1], the mixed volume of the convex hulls ∆1 , . . . , ∆I of the sets A1 , . . . , AI and arbitrary n-dimensional bounded polyhedra ∆I+1 , . . . , ∆n equals zero. Consider points ai ∈ ∆i and bi ∈ ∆i such that the vectors a1 − b1 , . . . , an − bn are in general position in the sense that the dimension of the space generated by ai1 − bi1 , . . . , aiJ − biJ is maximal possible for every subset {i1 , . . . , iJ } ⊂ {1, . . . , n}. By the monotonicity of the mixed volume, the mixed volume of the vectors connecting ai and bi equals zero, which means that the vectors a1 − b1 , . . . , an − bn are linearly dependent. In particular, there exists a minimal subset {i1 , . . . , iJ } ⊂ {1, . . . , n} such that the vectors ai1 − bi1 , . . . , aiJ − biJ are linearly dependent. They generate a proper subspace L ⊂ Rn , and every J − 1 of them form a basis / L, then the vectors ai1 − bi1 , . . . , aiJ − biJ with of L. If there exists b ij ∈ ∆ij such that aij − b ij ∈

aij − bij instead of aij − bij generate a subspace L L, which contradicts the condition of general position. Thus, ∆j1 , . . . , ∆jI are contained in a (J − 1)-dimensional subspace L up to a shift. In particular, {j1 , . . . , jI } ⊂ {1, . . . , I}, and the codimension of Ai1 , . . . , AiJ is positive. Essentialization. It turns out that each resultantal set is (A1 , . . . , AI )-resultantal for some “nondegenerate” collection of ﬁnite sets A1 , . . . , AI ⊂ Zn in the following sense (this notion was introduced in [12] for I = n + 1). Deﬁnition 2.14. A collection of ﬁnite sets A1 , . . . , AI ⊂ Zn is said to be essential if its codimension is greater than the codimension of every subcollection Ai1 , . . . , Aik , {i1 , . . . , ik } {1, . . . , I}. . . . , BK )-resultantal for some essential collection Theorem 2.15. Every resultantal set is (B1 , m of ﬁnite sets B1 , . . . , BK ⊂ Z such that the sum i Bi generates Zm . Proof. The proof follows from an explicit construction of an essential collection B1 , . . . , BK (Lemmas 2.16–2.19 below). For a collection of sets A1 , . . . , AI ⊂ Zn and a subset J = {i1 , . . . , ik } ⊂ {1, . . . , I}, denote the subcollection Ai1 , . . . , Aik by AJ . Lemma 2.16. Every collection A1 , . . . , AI ⊂ Zn contains a unique essential subcollection of codimension not less than the codimensions of all other subcollections of A1 , . . . , AI . PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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Proof. Let α be the set of all subsets J ⊂ {1, . . . , I} such that the codimension of the subcollection AJ is not less than the codimensions of all other subcollections of A1 , . . . , AI . The set α is closed under the operations of union and intersection because of the following obvious inequality: codim AJ1 ∩J2 + codim AJ1 ∪J2 ≥ codim AJ1 + codim AJ2 , where J1 and J2 are arbitrary subsets of {1, . . . , I}. The minimal element of α corresponds to an essential subcollection. Lemma 2.17. If the codimension of a subcollection Ai1 , . . . , Aik is not less than the codimensions of all other subcollections of A1 , . . . , AI ⊂ Zn and π is the projection C[A1 ] ⊕ . . . ⊕ C[AI ] → C[Ai1 ] ⊕ . . . ⊕ C[Aik ], then Σ(A1 , . . . , AI ) = π (−1) (Σ(Ai1 , . . . , Aik )). The proof is given at the end of this section. Deﬁnition 2.18. A map ϕ : A → B induces a map ϕ∗ : C[A] → C[B] that assigns the polyno mial a∈A ca sϕ(a) ∈ C[B] to a polynomial a∈A ca ta ∈ C[A]. Lemma 2.19. 1. For each collection of ﬁnite sets A1 , . . . , AI ⊂ Zn , there exist a collection B1 , . . . , BI ⊂ Zm , an injective homomorphism j : Zm → Zn , and points ai ∈ Zn , i = 1, . . . , I, such that the collection B1 , . . . , BI generates Zm and Ai = j(Bi ) + ai . 2. Let p be the isomorphism between C[A1 ] ⊕ . . . ⊕ C[AI ] and C[B1 ] ⊕ . . . ⊕ C[BI ] induced by maps j + ai : Bi → Ai . Then Σ(A1 , . . . , AI ) = p(Σ(B1 , . . . , BI )). The proof is obvious. Fibers of resultantal sets. above.

The rest of this section contains the proof of the facts formulated

1 , . . . , fI ) ∈ Σ(A1 , . . . , AI ), the f -ﬁber is the algebraic set

Deﬁnitionn 2.20. For a point f = (f t ∈ (C \ {0}) | f1 (t) = . . . = fI (t) = 0 ⊂ (C \ {0})n . The dimension of the f -ﬁber for a generic f is given by the following obvious formula. Lemma 2.21. There exists a Zariski open subset S ⊂ Σ(A1 , . . . , AI ) such that the dimension of the f -ﬁber equals n − I + codim Σ(A1 , . . . , AI ) for every f ∈ S. It turns out that the f -ﬁber is a disjoint union of subtori of the complex torus (C \ {0})n for a generic f ∈ Σ(A1 , . . . , AI ), provided that the collection A1 , . . . , AI ⊂ Zn is nondegenerate in the following sense: Deﬁnition 2.22. A collection of ﬁnite sets A1 , . . . , AI ⊂ Zn is said to be weakly essential if its codimension is not less than the codimension of each subcollection Ai1 , . . . , Aik , {i1 , . . . , ik } {1, . . . , I}. Theorem 2.23. Suppose that a collection of ﬁnite sets A1 , . . . , AI ⊂ Zn is weakly essential. Then there exists a Zariski open subset S ⊂ Σ(A1 , . . . , AI ) such that the f -ﬁber is a disjoint union of d(A1 , . . . , AI ) subtori of the complex torus (C \ {0})n for every f ∈ S, where the number d(A1 , . . . , AI ) is deﬁned below (Deﬁnition 2.24). The proof is given at the end of this section. Deﬁnition 2.24. Let A1 , . . . , AI be ﬁnite subsets of Zn . Suppose that the sum Ai1 + . . . + Aik generates a sublattice L ⊂ Zn , and let L ⊂ Zn be the maximal sublattice such that L ⊂ L and dim L = dim L . Denote the projection Zn → Zn /L by π. Then the collection of all sets of the / {i1 , . . . , ik }, is called the quotient collection. form π(Ai ), i ∈ PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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Let Ai1 , . . . , Aik be the essential subcollection of a collection of ﬁnite sets A1 , . . . , AI ⊂ Zn that has the same codimension, and let B1 , . . . , BI−k be the quotient collection. Then the product |L /L|(I − k)! MV(conv B1 , . . . , conv BI−k ) is called the multiplicity d(A1 , . . . , AI ) of the collection A1 , . . . , AI (we denote the convex hull of B by conv B). Proof of Theorems 2.12, 2.23, and Lemma 2.17. Theorem 2.12, Theorem 2.23, and Lemma 2.17 are corollaries to parts 1, 2, and 3 of Theorem 2.26, respectively. Deﬁnition 2.25. For a sublattice L ⊂ Zn , let T (L) be the subtorus of (C \ {0})n generated by all one-parameter subtori of the form (tv1 , . . . , tvn ), where (v1 , . . . , vn ) ∈ (Zn )∗ is a covector orthogonal to L. For every c ∈ (C \ {0})n , the set c · T (L) is called an L-subtorus (here the dot stands for the componentwise multiplication). Theorem 2.26. 1. For every collection of ﬁnite sets A1 , . . . , AI ⊂ Zn , the codimension of the set Σ(A1 , . . . , AI ) equals the maximum of the codimensions of subcollections. 2. Suppose that a collection of ﬁnite sets A1 , . . . , AI ⊂ Zn is weakly essential and generates a sublattice L. Then there exists a Zariski open subset S ⊂ Σ(A1 , . . . , AI ) such that the f -ﬁber is a union of d(A1 , . . . , AI ) distinct L-subtori for every f ∈ S. 3. Let Ai1 , . . . , Aik be a weakly essential subcollection of A1 , . . . , AI ⊂ Zn , and denote the quotient collection by B1 , . . . , BI−k . Then, for each Zariski open subset S ⊂ Σ(B1 , . . . , BI−k ), there exists a Zariski open subset T ⊂ Σ(A1 , . . . , AI ) such that every f -ﬁber is a disjoint union of gi -ﬁbers, where f ∈ T and gi ∈ S for i = 1, . . . , d(Ai1 , . . . , Aik ). We formulate these facts together because it is more convenient to prove them simultaneously by induction on I for each n. Proof. Consider an arbitrary collection of ﬁnite sets A1 , . . . , AI ⊂ Zn . We have the following two options. (A) A1 , . . . , AI ⊂ Zn is essential. Proof of parts 1 and 2. Since the subcollection A1 , . . . , AI−1 is weakly essential, there exists, by induction, a Zariski open subset S ⊂ Σ(A1 , . . . , AI−1 ) such that the g-ﬁber is a union of L-subtori for every g ∈ S. Since A1 + . . . + AI−1 generates a sublattice of L of the same dimension, the restriction of every fI ∈ C[AI ] to an L-subtorus is a constant. Thus, for each g ∈ S, there exists a hypersurface K ⊂ C[Ai ] such that fI ∈ K ⇔ (g, fI ) ∈ Σ(A1 , . . . , AI ) (which proves part 1), and the (g, fI )-ﬁber is a union of L-subtori if fI ∈ K (which proves part 2). Part 3 is not applicable unless I = k. (B) A1 , . . . , AI ⊂ Zn is not essential. Proof of part 3. Since the subcollection Ai1 , . . . , Aik is weakly essential, there exists a Zariski open subset S ⊂ Σ(Ai1 , . . . , Aik ) such that, for each h ∈ S, the h-ﬁber is a union of subtori in / {i1 , . . . , ik }, to these subtori are (C \ {0})n . The restrictions of generic polynomials fi ∈ C[Ai ], i ∈ generic polynomials in C[Bj ], j = 1, . . . , I − k. Proof of parts 1 and 2. By Lemma 2.16, the collection A1 , . . . , AI contains an essential subcollection Ai1 , . . . , Aik of codimension not less than the codimensions of all other subcollections, and k < I. Denote the quotient collection by B1 , . . . , BI−k . Since the quotient subcollection satisﬁes the assumption of Lemma 2.13, Σ(B1 , . . . , BI−k ) = C[B1 ] ⊕ . . . ⊕ C[BI−k ]. By Lemma 2.21, there exists a Zariski open subset S ⊂ Σ(B1 , . . . , BI−k ) such that, for each g ∈ S, the g-ﬁber has dimension n − I + k. Choose this S in statement 3. Then part 1 follows from Lemma 2.21, and part 2 follows from the fact that the collection B1 , . . . , BI−k is weakly essential if the collection A1 , . . . , AI is weakly essential. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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3. RESULTANTAL SINGULARITIES AND NEWTON POLYHEDRA In this section, we express the invariants of singularities of an (A1 , . . . , AI )-resultantal set f (−1) (Σ(A1 , . . . , AI )) in terms of the Newton polyhedra of components of the map f . By Then orem 2.15, we can assume without loss of generality that A1 , . . . , nAI ⊂ Z is an essential collection of ﬁnite sets such that i Ai contains the origin and generates Z . Deﬁnition 3.1. An (m, n)-Newton pile A = (A· , ∆· ) is a collection of ﬁnite sets Ai ⊂ Zn and polyhedra ∆b ⊂ Rm + , where i runs over 1, . . . , I and b runs over all pairs (a, i) such that i = 1, . . . , I and a ∈ Ai . A Newton pile is called essential if the collection {A1 , . . . , AI } is essential and the sum i Ai contains the origin and generates Zn . A Newton pile is called convenient if the diﬀerence Rm + \ ∆(a,i) is bounded for all pairs (a, i). For a point p = (p1 , . . . , pI ) ∈ C[A1 ] ⊕ . . . ⊕ C[AI ], denote the coeﬃcient of the monomial ta in the polynomial pi by c(a,i) (p). The functions c(a,i) form a coordinate system in the space C[A1 ] ⊕ . . . ⊕ C[AI ]. For an (m, n)-Newton pile A = (A· , ∆· ), let C{A} be the set of germs of all analytic maps f : Cm → C[A1 ] ⊕ . . . ⊕ C[AI ] such that the components c(a,i) ◦ f are contained in the spaces C{∆(a,i) }. Deﬁnition 3.2. “Almost all maps in C{A}” means “almost all collections of components c(a,i) ◦ f in i=1,...,I, a∈Ai C{∆(a,i) }” (see Deﬁnition 1.1). Lemma 3.3. If A = (A1 , . . . , AI , ∆· ) is a convenient essential (m, n)-Newton pile, then the set f (−1) Σ(A1 , . . . , AI ) is an (A1 , . . . , AI )-resultantal set of codimension I − n for almost all maps f ∈ C{A}. Proof. It follows from Theorem 2.12 that the codimension of Σ(A1 , . . . , AI ) equals I − n. If n = 0, then the rest is a well-known fact about complete intersections, which was mentioned in Example 1.2 (see [11] for a proof, which is based on the construction of a toric resolution for a complete intersection singularity). Replacing a toric resolution of a complete intersection with a toric resolution of a resultantal set (see Theorem 4.1 for the construction), one can prove this fact for an arbitrary n in the same way as for n = 0. Multiplicity of resultantal sets. For almost all maps f ∈ C{A} and g ∈ C{B}, the intersection of the resultantal sets deﬁned by f and g is also a resultantal set (see Example 2.6). This resultantal set is deﬁned by a map from the space C{D}, where the Newton pile D is the direct sum of the Newton piles A and B in the following sense. Deﬁnition 3.4. The direct sum of an (m, n)-Newton pile A = (A1 , . . . , AI , ∆1· ) and an (m, k)Newton pile B = (B1 , . . . , BJ , ∆2· ) is an (m, n + k)-Newton pile (C1 , . . . , CI+J , ∆· ), where Ci = Ai × {0} ⊂ Zn ⊕ Zk for i = 1, . . . , I, CI+j = {0} × Bj ⊂ Zn ⊕ Zk for j = 1, . . . , J, ∆(a×0,i) = ∆1(a,i) for i = 1, . . . , I, a ∈ Ai , and ∆(0×b,I+j) = ∆2(b,j) for j = 1, . . . , J, b ∈ Bj . For an (m, n)-Newton pile A = (A1 , . . . , AI , ∆· ), denote the convex hull of the union a∈Ai {a}× m ∆(a,i) ⊂ Rn ⊕ Rm by A(i). Denote the (m, n)-Newton pile (A1 , . . . , AI , Rm + , . . . , R+ ) by A0 . Theorem 3.5. 1. Let A = (A1 , . . . , AI , ∆· ) be a convenient essential (m, n)-Newton pile such m that I = m + n. Then, for almost all maps f ∈ C{A}, the intersection number of the germ f (C ) and the set Σ(A1 , . . . , AI ) in the space i C[Ai ] is well deﬁned and equals the mixed volume of pairs I! MV (A0 (1), A(1)), . . . , (A0 (I), A(I)) . 2. Let Al = (Al1 , . . . , AlIl , ∆l· ) be convenient nl )-Newton piles for l = 1, . . . , p, essential (m, nl , then, for almost all collections p > 1, and let A be their direct sum. If Il = m + 1 p of maps (f1 , . . . , fp ) ∈ C{A } × . . . × C{A }, the intersection number of the resultantal germs (−1) Σ(Al1 , . . . , AlIl ) in Cm is well deﬁned and equals I! MV (A0 (1), A(1)), . . . , (A0 (I), A(I)) . fl PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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3. For all maps f ∈ C{A} in the formulation of part 1 and for all collections of maps (f1 , . . . , fp ) ∈ C{A1 } × . . . × C{Ap } in the formulation ofpart 2, the intersection number is greater than or equal to I! MV (A0 (1), A(1)), . . . , (A0 (I), A(I)) , provided that the intersection is zerodimensional. Proof. Parts 1 and 3 were proved in [16, Theorem 5] (see also [15]). Part 2 is a special case of part 1. m m In particular, if p = 2 and A2 = ({0}, . . . , {0}, Rm + \ S, . . . , R+ \ S), where S ⊂ R+ is the (−1) standard m-dimensional simplex, then the intersection number of the sets f1 Σ(A11 , . . . , A1I1 ) and (−1)

(−1)

(−1)

f2 Σ(A21 , . . . , A2I2 ) is the multiplicity of f1 Σ(A11 , . . . , A1I1 ) at the origin, as f2 is a generic plane of complementary dimension.

Σ(A21 , . . . , A2I2 )

Cones and fans. To study the singularities of resultantal sets, we need the following combinatorial notions and notation. Deﬁnition 3.6. The (rational ) cone in Rn generated by (rational) vectors v1 , . . . , vm ∈ Qn is the set of all linear combinations of v1 , . . . , vm with positive coeﬃcients. A cone in Rn is said to be simple if it is generated by a part of a basis of Zn . A cone is said to be pointed if it does not contain a line. A cone is not a closed set unless it is a subspace of Rn . A cone is not an open set unless it is n-dimensional. All cones considered in this paper are assumed to be rational. Deﬁnition 3.7. There is a unique minimal decomposition of the closure of a cone C into cones. The elements of this decomposition are called faces of C. Deﬁnition 3.8. A fan Φ in Rn is a nonempty ﬁnite set of nonoverlapping cones in Rn such that (1) each face of each cone in Φ is in Φ; (2) each cone in Φ is a face of an n-dimensional cone in Φ. A fan is said to be simple if it consists of simple cones. A fan is said to be pointed if it consists of pointed cones. A fan Φ is called a subdivision of a fan Γ if each cone in Γ is a union of cones from Φ. Deﬁnition 3.9. Let A ⊂ Rn be a polyhedron or a ﬁnite set, and let B be a face or a subset of A, respectively. The dual cone Γ of the set B is the set of all covectors γ ∈ (Rn )∗ such that {a ∈ A | γ(a) = min γ(A)} = B. If a cone Γ is contained in Γ, then B is called the support set of Γ and is denoted by AΓ . The dual fan of a polyhedron is the set of dual cones of all its faces. A fan is said to be compatible with A if it is a subdivision of the dual fan of A. Codimension of singularities of resultantal sets. The codimension of singularities of almost all (A1 , . . . , AI )-resultantal sets is greater than or equal to the number codim sing(A1 , . . . , AI ) deﬁned below. Since the deﬁnition of codim sing(A1 , . . . , AI ) is somewhat lengthy, we ﬁrst give its values for some important collections of sets A1 , . . . , AI as an example. See Examples 2.4, 2.5, and 2.7 for the meaning of these collections of sets A1 , . . . , AI . Example 3.10. If I = n + 1 and the convex hulls of sets A1 , . . . , AI ∈ Zn are n-dimensional, then codim sing(A1 , . . . , AI ) = 2. Example 3.11. If A1 = . . . = AI is the set of vertices of the standard n-dimensional simplex, then codim sing(A1 , . . . , AI ) = 2I − 2n + 2. Example 3.12. Let Ekl ⊂ Zkl be the set of vertices of the standard kl -dimensional simplex k . and Il > kl for l = 1, . . . , p. Denote the image of Eki under the embedding Zki → l Zkl by E i PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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Then

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k , . . . , E k , . . . , E k = k , . . . , E (Ii − ki ) + min(Ii − ki ) + 2. codim sing E p p 1 1 i I1

Ip

i

Example 3.13. If n-dimensional integer polyhedra B1 , . . . , BI ⊂ Rn have the same dual fan and this fan is simple, then codim sing(B1 ∩ Zn , . . . , BI ∩ Zn ) = 2I − 2n unless, up to a shift, the 1 . polyhedra B1 , . . . , BI are equal to the same integer simplex of volume n! For sets B1 , . . . , Bp ⊂ Rn , the number C(B1 , . . . , Bp ) is deﬁned to be 1 if at least one of these sets contains more than one point and, up to a shift, all these sets are contained in the same integer 1 . Otherwise, C(B1 , . . . , Bp ) is deﬁned to be 0. Denote the sublattice of Zn simplex of volume n! orthogonal to a cone Γ ⊂ (Zn )∗ by Γ⊥ . Recall that, for a set A ⊂ Rn and a cone Γ ∈ (Rn )∗ , the support set of Γ is denoted by AΓ (see Deﬁnition 3.9). of ﬁnite sets such that i Ai Deﬁnition 3.14. Let A1 , . . . , AI ⊂ Zn be an essential collection generates Zn , and let Γ be a cone from the dual fan of the sum i Ai (see Deﬁnition 3.9). sing(A1 , . . . , AI , Γ) If Γ is simple and the collection AΓ1 , . . . , AΓI is weakly essential, then codim is deﬁned as the minimum of 2I − 2 dim(Γ⊥ ) − codim AΓi1 , . . . , AΓiJ + 2C AΓj1 , . . . , AΓjI−J over all partitions {1, . . . , I} = {i1 , . . . , iJ }{j1 , . . . , jI−J } such that the sum AΓi1 +. . .+AΓiJ +a is contained in a proper sublattice of Γ⊥ for some a ∈ Zn . then the number codim sing(A1, . . . , AI , Γ) If Γ is not simple or AΓ1, . . . , AΓI is not weakly essential, Γ Γ is deﬁned as the maximum of codim Ai1 , . . . , AiJ over all subsets {i1 , . . . , iJ } ⊂ {1, . . . , I}. The number codim sing(A1 , . . . , AI ) is deﬁned as the minimum ofcodim sing(A1 , . . . , AI , Γ) over all cones Γ (including the zero-dimensional one) in the dual fan of i Ai . Deﬁnition 3.15. A point of an m-dimensional analytic set M ⊂ Cn is said to be weakly regular if its neighborhood is homeomorphic to the 2m-dimensional disk. The weakly singular locus wsing M of M is the set of all points that are not weakly regular. We are interested in the codimension of the weakly singular locus of a resultantal set rather than of the classical singular locus, because topological invariants of isolated singular points make sense for weakly isolated singularities as well and an estimate for the codimension of the weakly singular locus is simpler. This estimate is formulated in part 1 of the following theorem, and part 2 extends this estimate to the classical singular locus in some important special cases. Theorem 3.16. 1. If A = (A1 , . . . , AI , ∆· ) is a convenient essential (m, n)-Newton pile, then the codimension of the weakly singular locus of the resultantal set f (−1) Σ(A1 , . . . , AI ) in Cm is greater than or equal to codim wsing(A1 , . . . , AI ) for almost all maps f ∈ C{A}. 2. Suppose that either integer polyhedra B1 , . . . , BI ⊂ Rn are n-dimensional, have the same dual fan, and this fan is simple, or C(B1 , . . . , Bp ) = 1. Denote Bi ∩ Zn by Ai . If a Newton pile A = (A1 , . . . , AI , ∆· ) is essential and convenient, then codim sing f (−1) Σ(A1 , . . . , AI ) = codim wsing f (−1) Σ(A1 , . . . , AI ) = codim sing(A1 , . . . , AI ) for almost all germs f ∈ C{A}. The proof is given at the end of Section 4. The estimate in part 1 is not always sharp, but it is sharp in many important cases (see the examples above). It is clear from the proof how to compute the codimension of the weakly singular locus precisely and how to compute the codimension of the singular locus in the general case, but the answer is too complicated to be formulated explicitly in general. In Section 1, we apply part 2 of this theorem to determinantal sets as follows. m Example 3.17. Let ∆i,j,l ⊂ Rm + be polyhedra such that R+ \ ∆i,j,l is bounded for all triples (i, j, l), i = 0, . . . , kl , j = 1, . . . , Il , l = 1, . . . , p, and Il > kl for l = 1, . . . , p. Then the germ

sing x ∈ Cm rk(f·,·,l ) < kl , l = 1, . . . , p

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has codimension i (Ii − ki ) + mini (Ii − ki ) + 2 for almost all collections of germs fi,j,l ∈ C{∆i,j,l }, where (f·,·,l ) is a kl × Il matrix with entries fi,j,l , i = 0, . . . , kl , j = 1, . . . , Il . Isolated singularities of resultantal sets. Deﬁnition 3.18. The integer volume form in a rational subspace L ⊂ Rk is a volume form such that the volume of the torus L/(L ∩ Zk ) equals 1. If l polyhedra or pairs of polyhedra are contained in an l-dimensional rational subspace L ⊂ Rk up to a shift, then their mixed volume is understood in the sense of the integer volume form in L. k Suppose that integer polyhedra ∆0 , . . . , ∆I ⊂ R are parallel to each other, the dimension of ∆i equals p + 1, and ∆0 is contained in a pointed cone C such that C \ ∆0 is bounded. Then denote the sum of the mixed volumes of pairs

(−1)I−p−1

(p + 1)!

i0 +...+iI =p+1 i0 ,...,iI ∈N

× MV (C, ∆0 ), . . . , (C, ∆0 ), (∆1 , ∆1 ), . . . , (∆1 , ∆1 ), . . . , (∆I , ∆I ), . . . , (∆I , ∆I ) i0

i1

iI

by χ(∆0 , . . . , ∆I ). Note that the (p + 1)-dimensional mixed volume in this expression makes sense since, up to a shift, all pairs of polyhedra are contained in the same (p + 1)-dimensional rational subspace of Rk . In particular, χ(∆0 , . . . , ∆I ) = 0 if p < I. For a set Q ⊂ {1, . . . , m}, let RQ ⊂ Rm be a coordinate plane deﬁned by the equations xi = 0, Q Q i∈ / Q. For a polyhedron ∆ ⊂ Rm + , denote the polyhedron R ∩ ∆ by ∆ . For an (m, n)-Newton n ∗ the (|Q|, n)-Newton pile pile A = (A1 , . . . , AI , ∆· ) (see Deﬁnition 3.1) and a cone Γ ⊂ (Z ) , denote Q Γ Γ Γ,Q (A1 , . . . , AI , ∆· ) by A , and denote the convex hull of the union a∈Ai {a} × ∆(a,i) ⊂ Rn ⊕ Rm by A(i). Recall that, for a set A ⊂ Rn and a cone Γ ∈ (Rn )∗ , the support set of Γ is denoted by AΓ (see Deﬁnition 3.9). Theorem 3.19. 1. Let A = (A1 , . . . , AI , ∆· ) be a convenient essential (m, n)-Newton pile and m ≤ codim sing(A1 , . . . , AI ). Then the germ of the resultantal set f (−1) Σ(A1 , . . . , AI ) is homeomorphic to a cone over a smooth manifold for almost all germs f ∈ C{A}. m 2. Under the assumption of part 1, let ∆ ⊂ Rm + be a polyhedron such that R+ \ ∆ is bounded. n m Denote the polyhedron {0} × ∆ ⊂ R ⊕ R by ∆. Let Φ be the dual fan of the convex hull conv i Ai . Then the Euler characteristic of a Milnor ﬁber of g|f (−1) Σ(A1 ,...,AI ) equals

Q Γ,Q , A (1), . . . , AΓ,Q (I) χ ∆

Γ∈Φ Q⊂{1,...,m}

for almost all pairs (g, f ) ∈ C{∆} × C{A}. 3. Under the assumption of part 2, the radial index of a 1-form Re ω|f (−1) Σ(A1 ,...,AI ) equals 1−

Q Γ,Q , A (1), . . . , AΓ,Q (I) χ ∆

Γ∈Φ Q⊂{1,...,m}

for almost all pairs (ω, f ) ∈ C∧ {∆} × C{A}. Proof. Part 1 is a corollary of Theorem 3.16. Since the Euler characteristic of a Milnor ﬁber equals the degree of the ζ-function of monodromy, part 2 is a corollary of Theorem 3.20 (see below). To prove that the expression in part 2 equals the degree of the expression in Theorem 3.20, PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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apply Lemma 3.21 to each term of the expression in part 2. If n = 0, then the resultantal set f (−1) Σ(A1 , . . . , AI ) is a complete intersection and part 3 is a corollary of part 2 (see [14] for a proof, which is based on the construction of a toric resolution for a complete intersection singularity). Replacing a toric resolution of a complete intersection with a toric resolution of a resultantal set (see Theorem 4.1 below), one can deduce part 3 from part 2 for an arbitrary n in the same way as for n = 0. Monodromy of functions on isolated singularities of resultantal sets. Consider integer polyhedra ∆0 , . . . , ∆I ⊂ Rk . For every (k −1)-dimensional bounded face B of the sum ∆0 +. . .+∆I , let ∆0,B , . . . , ∆I,B be the faces of ∆0 , . . . , ∆I , respectively, such that ∆0,B + . . . + ∆I,B = B. Let ∆i (B) be the product γ · a, where a ∈ ∆i,B and γ ∈ (Zk )∗ is the primitive covector that attains its minimum on the face B as a linear function on the polyhedron ∆0 + . . . + ∆I . The rational function ζk (∆0 , . . . , ∆I )(t) is deﬁned as the product B (1 − t∆0 (B) )d(B) , where

d(B) = (−1)I+k−1

i0 +...+iI =k−1 i0 ≥0, i1 ≥1, ..., iI ≥1

(k − 1)! MV ∆0,B , . . . , ∆0,B , . . . , ∆I,B , . . . , ∆I,B i0

iI

and B runs over all (k − 1)-dimensional bounded faces of the sum ∆0 + . . . + ∆I . Note that the (k − 1)-dimensional mixed volume on the right-hand side makes sense since, up to a shift, all polyhedra are contained in some rational hyperplane. In particular, ζ(∆0 , . . . , ∆I )(t) = 1 if k ≤ I. Theorem 3.20. Under the assumptions of part 2 of Theorem 3.19, for almost all pairs (g, f ) ∈ C{∆} × C{A}, the ζ-function of monodromy of the restriction g|f (−1) Σ(A1 ,...,AI ) equals

Q Γ,Q , A (1), . . . , AΓ,Q (I) (t). ζ|Q|+n−dim Γ ∆

Γ∈Φ Q⊂{1,...,m}

Q , AΓ,Q (1), . . . , AΓ,Q (I) ⊂ Rn ⊕ RQ This expression makes sense since, up to a shift, the polyhedra ∆ ⊥ Q are contained in the rational subspace Γ ⊕ R of dimension |Q| + n − dim Γ. Proof. If n = 0, then this formula is a well-known fact (see [13, 10, 11] for a proof, which is based on the construction of a toric resolution for a complete intersection singularity). Replacing a toric resolution of a complete intersection with a toric resolution of a resultantal set (see Theorem 4.1 below for the construction), one can prove it for an arbitrary n in the same way as for n = 0. One can use the following formula for the mixed volumes of pairs of polyhedra to express the degree of the ζ-function of monodromy in terms of the mixed volumes of pairs. Lemma 3.21 [16]. Let (C1 , D1 ), . . . , (Ck , Dk ) be parallel pairs of integer polyhedra in Rk . Then the mixed volume of these pairs equals k 1 Di (B) − Ci (B) MV C1,B , . . . , Ci−1,B , Di+1,B , . . . , Dk,B , k B i=1

where B runs over all bounded (k − 1)-dimensional faces of the sum C1 + D1 + . . . + Ck + Dk . The quantities Di (·) and MV(·, . . . , ·) are introduced above in terms of primitive covectors and integer volume forms. Replacing primitive covectors and integer volume forms with unit covectors and volume forms in the sense of some metric in Rn , one can drop the integrality assumption for polyhedra in this formula. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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4. TORIC RESOLUTIONS OF RESULTANTAL SINGULARITIES Toric varieties. See Deﬁnitions 3.6–3.9 for the notions and notation related to cones and fans. For a pointed fan Φ in (Rn )∗ , denote the corresponding complex toric variety by TΦ (see, for example, [4]). For a cone Γ ∈ Φ, denote the corresponding orbit of TΦ by TΓ . Denote the maximal Γ compact union of orbits of TΦ by TΦ c (it consists of the orbits T such that the cone Γ is not in the boundary of Φ). Denote the set of primitive generators of all one-dimensional cones in the fan Φ by Φ1 ⊂ (Rn )∗ , and denote the cone generated by a covector γ by γ . If an integer polyhedron ∆ ⊂ Rn is compatible I∆ and its (meromorphic) section s∆ such with Φ, then the variety TΦ carries a unique line bundle

γ that the divisor of s∆ equals − γ∈Φ1 min(γ(∆)) · T . For every polynomial f ∈ C[∆] on (C \ {0})n , the product f · s∆ is a section of I∆ (we identifya n Φ the torus (C \ {0}) with the open orbit of T ). Let C{∆} be the set of formal series a∈∆∩Zn ca t such that ca (ta ·s∆ ) converges absolutely in a neighborhood of TΦ c (this deﬁnition does not depend on the choice of a fan Φ compatible with ∆). Toric resolutions of resultantal sets. Let A = (A1 , . . . , AI , ∆· ) be a convenient essential m (m, n)-Newton pile. Denote the Newton pile (A1 , . . . , AI , Rm + , . . . , R+ ) by A0 , and denote the n m convex hull of the union a∈Ai {a} × ∆(a,i) ⊂ R ⊕ R by A(i). Denote the minimal fan compatible with polyhedra A0 (1), . . . , A0 (I) ⊂ Rn ⊕ Rm by Φ0 , and denote its simple subdivision compatible with A(1), . . . , A(I) ⊂ Rn ⊕ Rm by Φ. The toric variety TΦ0 carries the line bundles IA0 (i) and their sections sA0 (i) for i = 1, . . . , I. The smooth toric variety TΦ carries the line bundles IA(i) and their sections sA(i) for i = 1, . . . , I. Let r be the natural projection of toric varieties: (C \ {0})n+m ⊂ TΦ id

(C \

{0})n+m

r

⊂ T Φ0

Denote the projection TΦ0 = TΨ × Cm → Cm by p, where Ψ is the minimal fan compatible with the convex hull of A1 + . . . + AI . Denote the standard coordinate systems on (C \ {0})n and Cm by t = (t1 , . . . , tn ) and x = (x1 , . . . , xm ), respectively. For a point p = (p1 , . . . , pI ) ∈ C[A1 ] ⊕ . . . ⊕ C[AI ], denote the coeﬃcient of the monomial ta in the polynomial pi by c(a,i) (p). The functions c(a,i) , where a runs over Ai and i runs over {1, . . . , I}, form a coordinate system on the space C[A1 ] ⊕ . . . ⊕ C[AI ]. For a germ of an analytic map f ∈ C{A}, denote the sum a∈Ai (c(a,i) ◦ f (x))ta by f (i) ∈ C{A(i)} ⊂ C{A0 (i)} for each i = 1, . . . , I. Theorem 4.1. Let A = (A1 , . . . , AI , ∆· ) be a convenient essential (m, n)-Newton pile. Then (1) for every germ of an analytic map f ∈ C{A},

f (−1) Σ(A1 , . . . , AI ) = p ◦ r f (1) · sA(1) = 0 ∩ . . . ∩ f (I) · sA(I) = 0 ; (2) for almost all germs f ∈ C{A}, the set

f (1) · sA(1) = 0 ∩ . . . ∩ f (I) · sA(I) = 0 ⊂ TΦ is smooth and transversal to the orbits of TΦ . Thus, the projection

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is a resolution of singularities for almost all (A1 , . . . , AI )-resultantal sets. In particular, if A1 = . . . = AI = {0} ⊂ R0 , then this construction is the classical toric resolution of a complete intersection f(0,1) = . . . = f(0,I) = 0, where f(0,i) ∈ C{∆(0,i) } for i = 1, . . . , I. The proof of part (2) is the same as for complete intersections (see [7] for the ﬁrst formulation and [11] for a detailed proof). Part (1) is obvious from deﬁnitions. Proof of Theorem 3.16, part 1. Theorem 3.16 is a corollary of the following three lemmas. For a ﬁnite set A ⊂ Zn , a cone Γ ⊂ (Rn )∗ , and a polynomial p(t) = a∈A ca ta ∈ C[A], denote the polynomial a∈AΓ ca ta by pΓ . Let Ψ be the dual fan of the sum A1 + . . . + AI , and let B(A1 , . . . , AI ) be the set of all collections (p1 , . . . , pI ) ∈ C[A1 ] ⊕ . . . ⊕ C[AI ] such that (1) there exists a simple cone Γ ∈ Ψ such that the set {t | pΓ1 (t) = . . . = pΓI (t) = 0} ⊂ (C \ {0})n is a subtorus of dimension dim Γ, and

(2) the set {t | pΓ1 (t) = . . . = pΓI (t) = 0} is empty for all other cones Γ ∈ Ψ. Denote the diﬀerence Σ(A1 , . . . , AI ) \ B(A1 , . . . , AI ) by S(A1 , . . . , AI ). Lemma 4.2. Let A = (A1 , . . . , AI , ∆· ) be an essential convenient Newton pile. Then, for almost all germs f ∈ C{A}, wsing f (−1) Σ(A1 , . . . , AI ) ⊂ f (−1) (S(A1 , . . . , AI )). Proof. We use the notation from Theorem 4.1. For almost all germs f ∈ C{A}, the set

f (1) · sA0 (1) = 0 ∩ . . . ∩ f (I) · sA0 (I) = 0 (−1) Φ0 (−1) Σ(A , . . . , A ) = is smooth outside the 1 I set p (0) ∪ sing T . Thus, if a point x ∈ f p f (1) · sA0 (1) = 0 ∩ . . . ∩ f (I) · sA0 (I) = 0 outside the origin is not weakly regular, then at least one of the following conditions is satisﬁed:

(1) p(−1) (x) ∩ f (1) · sA0 (1) = 0 ∩ . . . ∩ f (I) · sA0 (I) = 0 contains more than one point, or

(2) p(−1) (x) ∩ f (1) · sA0 (1) = 0 ∩ . . . ∩ f (I) · sA0 (I) = 0 intersects sing TΦ0 .

Each of these conditions implies that the point f (x) = (f (1)(x), . . . , f (I)(x)) ∈ C[A1 ] ⊕ . . . ⊕ C[AI ] is contained in S(A1 , . . . , AI ). Lemma 4.3. The codimension of S(A1 , . . . , AI ) in i C[Ai ] is greater than or equal to codim sing(A1 , . . . , AI ). The proof is given below. m Lemma 4.4. Let ∆i ⊂ Rm + be polyhedra such that R+ \ ∆i is bounded for each i = 1, . . . , k, k and let M ⊂ C be an analytic set of codimension p. Then, for almost all collections fi ∈ C{∆i }, the codimension of f (−1) (M ) in Cm equals p, where f = (f1 , . . . , fk ) : Cm → Ck .

Proof. Apply Lemma 4.5 to the graph of f = (f1 , . . . , fk ) : Cm → Ck . Namely, let Cl be ⊕ Ck , let N ⊂ Cm ⊕ Ck be Cm × M , and let gi be yi − fi (x), where y1 , . . . , yk are the standard coordinates on Ck . Recall that, for a set Q ⊂ {1, . . . , l} and a polyhedron ∆ ⊂ Rl+ , the subspace RQ ⊂ Rl is deﬁned / Q, and ∆Q = RQ ∩ ∆. by the equations xi = 0, i ∈

Cm

Lemma 4.5. Let ∆i ⊂ Rl+ be polyhedra, i = 1, . . . , k, and let N ⊂ Cl be an analytic set of Q codimension p. Suppose that, for every subset Q ⊂ {1, . . . , l}, either all polyhedra ∆Q 1 , . . . , ∆k are Q nonempty or at least one of them equals RQ + + a for some a ∈ R+ . Then, for almost all collections gi ∈ C{∆i }, the codimension of N ∩ {g1 = . . . = gk = 0} equals k + p. PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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Proof. Consider a c-fan Φ of the set N and use the fact that, for almost all gi ∈ C{∆i }, the c-fan of the intersection N ∩ {g1 = . . . = gk = 0} equals the stable intersection of fans Φ, Φ1 , . . . , Φk , where Φi is the c-fan of the polyhedron ∆i (see [5] and [6] for the theory of c-fans). This stable intersection has codimension k + p. Proof of Lemma We use the notation from Deﬁnition 3.14. Let Ψ be the dual fan of the 4.3. Ψ Ψ Ψ convex hull conv i Ai , let T be the corresponding toric variety, and let S ⊂ T be its singular Ψ locus. Denote the section sconv(Ai ) of the line bundle Iconv(Ai ) on T by si . Recall that, for every Laurent polynomial p ∈ C[Ai ], the product psi deﬁnes a section of Iconv(Ai ) as well. The set S(A1 , . . . , AI ) consists of the following two pieces S1 (A1 , . . . , AI ) and S2 (A1 , . . . , AI ): S1 (A1 , . . . , AI ) is the set of all collections (p1 , . . . , pI ) ∈ i C[Ai ] such that p1 s1 (x) = 0, . . . , Ψ \ S Ψ ; S (A , . . . , A ) is the set of all collections more than one point x ∈ T pI sI (x) = 0 for 2 1 I (p1 , . . . , pI ) ∈ i C[Ai ] such that p1 s1 (x) = 0, . . . , pI sI (x) = 0 for some x ∈ S Ψ . Ψ Ψ Ψ Ψ Denote the set (T \ S ) × (T \ S ) \ (the diagonal) by P . Denote the projections of the product P × i C[Ai ] to the ﬁrst and the second factors by π1 and π2 , respectively. The set S1 (A1 , . . . , AI ) is the π2 -projection of the following set T1 ⊂ P × i C[Ai ]:

T1 = (t, t , g1 , . . . , gI ) gi si (t) = gi si (t ) = 0 for i = 1, . . . , I . The set {g1 s1 = . . . = gI sI = 0} contains a curve if it contains more than one point and C(A1 , . . . , AI ) = 1. Thus, codim S1 (A1 , . . . , AI ) ≥ codim T1 − 2n + 2C(A1 , . . . , AI ), and it is enough to estimate codim T1 . To estimate it, we subdivide T1 into pieces π1−1(P (i1 , . . . , iJ , Γ)) ∩ P over all subsets {i1 , . . . , iJ } ⊂ {1, . . . , I} and all simple cones Γ ∈ Ψ, where the set P (i1 , . . . , iJ , Γ) is deﬁned as follows. Let pi1 ,...,iJ ,Γ : (C \ {0})n → (C \ {0})q be an epimorphism of complex tori such that the image of the induced inclusion of their character lattices Zq → Zn is the lattice generated by AΓi1 + . . . + AΓiJ (see Deﬁnition 2.8). Then P (i1 , . . . , iJ , Γ) is deﬁned as the set of all pairs (t, t ) ∈ P such that (1) one of t and t is in TΓ and the other is in the closure of TΓ ; and (2) both t and t are in the closure of the same ﬁber of the projection pi1 ,...,iJ ,Γ : (C \ {0})n → / {i1 , . . . , iJ } (recall (C \ {0})q but not in the closure of the same ﬁber of pi1 ,...,iJ ,i,Γ for i ∈ n Ψ that we identify (C \ {0}) with the open orbit of T ). The codimension of P (i1 , . . . , iJ , Γ) in P equals 2 dim Γ + q. It is easy to see that the projection of the set

(t, t , g) (t, t ) ∈ P (i1 , . . . , iJ , Γ), g ∈ C[AΓi ], gsi (t) = gsi (t ) = 0 to the set P (i1 , . . . , iJ , Γ) is a vector bundle, and the codimension of a ﬁber of this bundle in C[Ai ] equals 1 or 2, depending on whether i ∈ {i1 , . . . , iJ } or not. Thus, the codimension of the set π1−1 (P (i1 , . . . , iJ , Γ)) ∩ P in π1−1 (P (i1 , . . . , iJ , Γ)) equals 2I − J. The set S2 (A1 , . . . , AI ) is the union of the pieces S2 (A1 , . . . , AI ) ∩ TΓ over all nonsimple cones (−1) Γ Γ Γ Γ ∈ Ψ. of S2Γ(A1 , . . . , AI ) ∩ T is pΓ (Σ(A1 , . .Γ. , AI )),Γwhere pΓ is the natural projec The closure tion i C[Ai ] → i C[Ai ]. The codimension of the set Σ(A1 , . . . , AI ) is computed in Theorem 2.12. Proof of Theorem 3.16, part 2. Case 1: C(A1 , . . . , AI ) = 1. Let S3 (A1 , . . . , AI ) be the set of all points (p1 , . . . , pI ) ∈ i C[Ai ] such that p1 s1 (t) = . . . = pI sI (t) = 0 for some t ∈ TΨ \ S Ψ and d(p1 s1 )(v) = . . . = d(pI sI )(v) = 0 for some tangent vector v ∈ Tt TΨ \ {0}. Recall that if s is a section of a line bundle I on a manifold M , then ds is a section of Hom(T M, I). PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS

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The diﬀerence sing f (−1) Σ(A1 , . . . , AI ) \ wsing f (−1) Σ(A1 , . . . , AI ) is contained in the set f (−1) S3 (A1 , . . . , AI ). Since, by Lemma 4.4, codim f (−1) S3 (A1 , . . . , AI ) = codim S3 (A1 , . . . , AI ) for almost all f ∈ C{A}, it is enough to estimate codim S3 (A1 , . . . , AI ). simplex, then both S3 (A1 , . . . , AI ) and S1 (A1 , . . . , AI ) consist of If conv i Ai is the standard all points (p1 , . . . , pI ) ∈ i C[Ai ] such that the set {t | p1 s1 (t) = . . . = pI sI (t) = 0} contains a line. Thus, S3 (A1 , . . . , AI ) = S1 (A1 , . . . , AI ), and Lemma 4.3 estimates codim S3 (A1 , . . . , AI ). Case 2: A1 , . . . , AI have the same dual fan, and this fan is simple. In this case, the set S3 (A1 , . . . , AI ) can be estimated in the same way as the set S1 (A1 , . . . , AI ) was estimated in the proof of Lemma 4.3, but the answer is more complicated. To apply the reasoning from the proof of Lemma 4.3, one can study the subset

T3 = (t, v, f1 , . . . , fI ) v ∈ Tt TΦ , fi (t) = 0, dfi (v) = 0 for i = 1, . . . , I of the variety T (TΨ \ S Ψ ) × i C[Ai ] in the same way as the set T1 ⊂ P × i C[Ai ] was studied. Then S3 (A1 , . . . , AI ) is the projection of T3 . ACKNOWLEDGMENTS The work was partially supported by the Russian Foundation for Basic Research (project no. 0701-00593), RFBR–JSPS (project no. 06-01-91063), and INTAS (project no. 05-7805). REFERENCES 1. D. N. Bernshtein, “The Number of Roots of a System of Equations,” Funkts. Anal. Prilozh. 9 (3), 1–4 (1975) [Funct. Anal. Appl. 9, 183–185 (1975)]. 2. C. Bivi` a-Ausina, “The Integral Closure of Modules, Buchsbaum–Rim Multiplicities and Newton Polyhedra,” J. London Math. Soc., Ser. 2, 69, 407–427 (2004). 3. W. Ebeling, S. M. Gusein-Zade, and J. Seade, “Homological Index for 1-Forms and a Milnor Number for Isolated Singularities,” Int. J. Math. 15 (9), 895–905 (2004). 4. W. Fulton, Introduction to Toric Varieties (Princeton Univ. Press, Princeton, NJ, 1993), Ann. Math. Stud. 131. 5. B. Ya. Kazarnovskii, “Truncation of Systems of Polynomial Equations, Ideals and Varieties,” Izv. Ross. Akad. Nauk, Ser. Mat. 63 (3), 119–132 (1999) [Izv. Math. 63, 535–547 (1999)]. 6. B. Ya. Kazarnovskii, “c-Fans and Newton Polyhedra of Algebraic Varieties,” Izv. Ross. Akad. Nauk, Ser. Mat. 67 (3), 23–44 (2003) [Izv. Math. 67, 439–460 (2003)]. 7. A. G. Khovanskii, “Newton Polyhedra and Toroidal Varieties,” Funkts. Anal. Prilozh. 11 (4), 56–64 (1977) [Funct. Anal. Appl. 11, 289–296 (1977)]. 8. A. G. Khovanskii, “Newton Polyhedra and the Genus of Complete Intersections,” Funkts. Anal. Prilozh. 12 (1), 51–61 (1978) [Funct. Anal. Appl. 12, 38–46 (1978)]. 9. A. G. Kouchnirenko, “Poly`edres de Newton et nombres de Milnor,” Invent. Math. 32, 1–31 (1976). 10. M. Oka, “Principal Zeta-Function of Non-degenerate Complete Intersection Singularity,” J. Fac. Sci. Univ. Tokyo, Sect. IA 37, 11–32 (1990). 11. M. Oka, Non-degenerate Complete Intersection Singularity (Hermann, Paris, 1997), Actualit´es Math´ematiques. 12. B. Sturmfels, “On the Newton Polytope of the Resultant,” J. Algebr. Combin. 3, 207–236 (1994). 13. A. N. Varchenko, “Zeta-Function of Monodromy and Newton’s Diagram,” Invent. Math. 37, 253–262 (1976). 14. A. Esterov, “Indices of 1-Forms and Newton Polyhedra,” Rev. Mat. Complut. 18 (1), 233–242 (2005). 15. A. I. Esterov, “Indices of 1-Forms, Resultants and Newton Polyhedra,” Usp. Mat. Nauk 60 (2), 181–182 (2005) [Russ. Math. Surv. 60, 352–353 (2005)]. 16. A. I. Esterov, “Indices of 1-Forms, Intersection Indices, and Newton Polyhedra,” Mat. Sb. 197 (7), 137–160 (2006) [Sb. Math. 197, 1085–1108 (2006)]. 17. A. Esterov, “Multiplicities of Degenerations of Matrices and Newton Polyhedra,” RIMS Kˆ okyˆ uroku (in press). 18. A. Esterov and A. G. Khovanskii, “Elimination Theory and Newton Polytopes,” Funct. Anal. Other Math. (in press); math.AG/0611107.

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Indices of 1-forms, resultants and Newton polyhedra