数理解析研究所講究録 1233 巻 2001 年 1-17

1
THREE LECTURES ON NEWTON POLYHEDRA

A. KHOVANSKIILECTURE 1. INTRODUCTION TO THE THEORY
1. The Newton polyhedra.
The Newton polyhedron of apolynomial which depends on several variables isthe convex hull of the powers of the monomials appearing in the polynomial with
nonzero coefficients. The Newton polyhedron generalizes the notion of degree andplays an analogous role. It is well known that the number of complex roots of asystem of equations of identical degree $m$ in unknowns is the same for nearly(Bezout’s theorem). Similarly, theall values of the coefficients, and is equal tonumber of complex roots of asystem of equations in unknowns with the same
Newton polyhedron is the same for nearly all values of the coefficients and is equal
to the volume of the Newton polyhedron, multiplied by $n!$ (Kushnirenko’s theorem,see 3.1.1).
The level line of apolynomial in two complex variables is aRiemann surface. For
nearly all polynomials of fixed degree , the topology of this surface (the numberof handles ) is expressed in terms of its degree, and does not depend on the values$n$$\mathrm{n}$$m^{n}$

$n$

$n$

$n$

$g$

of the coefficients of the polynomial: $g= \frac{(n-l)(n-2)}{2}$ . In the more generalcase, where instead of polynomials of fixed degree we consider polynomials with
fixed Newton polyhedra, all the discrete characteristics of the manifold of zeros of
the polynomial (or several polynomials) are expressed in terms of the geometryof the Newton polyhedra. Among these discrete characteristics are the number ofsolutions of asystem of equations in unknowns, the Euler characteristic, thearithmetic and geometric genus of complete inter- sections, and the Hodge numberof amixed Hodge structure on the cohomologies of complete intersections.
The Newton polyhedron is defined not only for polynomials but also for germs ofanalytic functions. For germs of analytic functions in general position, with given
Newton polyhedra, one can calculate the multiplicity of the zero solution of asystem of analytic equations, the Milnor number and 4function of the monodromy
operator, the asymptotics of oscillatory integrals, the Hodge number of the mixed
Hodge structure in vanishing cohomologies, and in the tw0-dimensional and multidimensional quasihomogeneous cases, one can calculate the modality of the germof the function.
In the answers one meets quantities characterizing both the sizes of the polyhedra(volume, number of integer points lying inside the polyhedron) as well as their
$n$

$n$

Typeset by$A\mathcal{M}S-\Pi EX$

2

A. KHOVANSKII

combinatorics (the number of faces of various dimensions, numerical characteristicsof their contacts).
In terms of the Newton polyhedron one can construct explicitly the compactification of complete intersections, and the resolution of singularities by means of asuitable toric manifold.
Thus, the Newton polyhedra connect algebraic geometry and the theory of singularities to the geometry of convex polyhedra. This connection is useful in both
directions. On the one hand, explicit answers are given to problems of algebra andthe theory of singularities in terms of the geometry of polyhedra. We note in thisconnection that even the volume of the convex hull of asystem of points is avery
complicated function of their coordinates. Therefore, the formulation of answers innumerical terms is so opaque that without knowing their geometric interpretationno progress is possible. On the other hand, algebraic theorems of general character(the Hodge theorem on the index of an algebraic surface, the Riemann-Roch the0rem) give significant information about the geometry of polyhedra. In this way one
obtains, for example, asimple proof of the Aleksandrov-Fenchel inequalities in thegeometry of convex bodies.
The Newton polyhedra are also met in the theory of numbers in real analysis, inthe geometry of exponential sums, in the theory of differential equations. In thislecture we present formulations of some theorems about Newton polyhedra.

2. The number of roofs of asystem of equations with agiven Newton
polyhedron.
According to Bezout’s theorem, the number of nonzero roots of the equation$f(z)=0$ is equal to the difference between the highest and the lowest powers ofthe monomials appearing in the polynomial . This difference is the volume of
the Newton polyhedron (in the present case, the length of asegment). In thefollowing two paragraphs we present the generalizations of this theorem to the caseof arbitrary Newton polyhedra.
Let us start with definitions. Amonomial in complex variables is aproductof the coordinates to integer (possibly negative) powers. Each monomial is ass0ciated with its degree, an integer vector, lying in -dimensional real space, whose
components are equal to the powers with which the coordinate functions enter inthe monomial. ALaurent polynomial is alinear combination of monomials. The
support of the Laurent polynomial is the set of powers of monomials entering in theLaurent polynomial with nonzero coefficients. (The Laurent polynomial is an ordinary polynomial if its support lies in the positive octant.) The Newton polyhedronof the Laurent polynomial is the convex hull of its support. It is much more convenient to consider the Laurent polynomials not in
but in the-dimensional
complex space, from which all the coordinate planes have been eliminated. With
each face of the Newton polyhedron of the Laurent polynomial we associateanew Laurent polynomial, which is called the restriction of the polynomial to theface, denoted byand defined as follows: only those monomials appear inthathave powers lying in the face , with the same coefficients that they have in .
Now let us consider asystem of Laurent equations $f_{1}=\cdots=f_{n}=0$ inwith acommon Newton polyhedron. The restricted system,
corresponding to each faceof the polyhedron. The restricted system actuallydepends on asmaller number of variables, and in the case of general position isincompatible in. We say that asystem of equations in unknowns with
$f$

$n$

$n$

$\mathbb{C}^{n}$$(\mathbb{C}\backslash 0)^{n}$$f$$\Gamma$$f^{\Gamma}$$f^{\Gamma}$$f$$\Gamma$$n$$(\mathbb{C}\backslash 0)^{n}$$f_{1}^{\Gamma}=\cdots=f_{n}^{\Gamma}=0$$\Gamma$$(\mathbb{C}\backslash 0)^{n}$

$n$

$n$

3

THREE LECTURES ON NEWTON POLYHEDRA

acommon Newton polyhedron is regular if all the restrictions of this system are. The following theorem of Kushnirenko holds: The number
incompatible in, counted with their multiplicities, of aregular system ofof solutions inequations in unknowns with acommon Newton polyhedron is equal to the volume$(\mathbb{C}\backslash 0)^{n}$$n$$(\mathbb{C}\backslash 0)^{n}$$n$of the Newton polyhedron multiplied by .
Example: The Newton polyhedron of the polynomial of degree $m$ in unknowns, $0\leq x_{n}$ ,is the simplex $0\leq x_{1}$ ,, $\leq m$ (we assume that the polynomial
contains all monomials of degree $\leq m$ ). The volume of such asimplex is $m^{n}/n!$ .
The number of roots of the total system of equations of degree $m$ in unknowns,. This answer agrees withaccording to Kushnirenko’s theorem, is equal topolynomial
does not contain all monomials of degree lessBezout’s theorem. If thethan or equal to $m$ , then the Newton polyhedron can be smaller than the simplex,
so the number of solutions, calculated from Kushnirenko’s theorem, can be smaller
than the number
calculated from Bezout’s theorem. Because of the absenceof the monomials, certain infinitely distant points may be roots of the system of
equations. Bezout’s theorem, which calculates the number of roots of the system
in projective space, takes into account these parasitic roots, while Kushnirenko’s
theorem does not.
$n!$

$n$

$\sum x$$\ldots$$n$$\mathrm{n}$$m^{n}$$m^{n}$

3. How does one find the number of solutions of asystem of equationsin unknowns with different Newton polyhedra.
Here is the answer to this question for asystem in general position with fixed
Newton polyhedra: the number of solutions not lying on the coordinate planes isequal to the mixed volume of the Newton polyhedra, multiplied by $n!$ Below weshall give the definition of mixed volume and describe explicitly the conditions fordegeneracy.
The Minkow ski sum of two subsets of alinear space is the set of sums of allpairs of vectors, in which one vector of the pair lies in one subset and the second
vector in the other. The product of asubset and anumber can be determined in asimilar manner. The Minkowski sum of convex bodies (convex polyhedra, convex
polyhedra with vertices at integer points) is aconvex body (convex polyhedron,convex polyhedron with vertices at integer points). The following theorem holds:$\mathrm{n}$$\mathrm{n}$Minkowski theorem. The volume of a body which is a linear combination withis a homogeneous polynomial
positive coefficients of fixed convex bodies lying inof degree in the coefficients of the linear combination.
$\mathbb{R}^{n}$$n$

Definition. The mixed volume
is the coefficient in the polynomial
by $n!$ (hereis the volume of the body

$V(\Delta_{1}, \ldots, \Delta_{n})$of the convex bodiesof$V(\lambda_{1}\Delta_{1}+\ldots, \lambda_{n}\Delta_{n})$$\mathbb{R}^{n}$$V(\Delta)$$\Delta$$\Delta_{1}$,$\ldots$$\lambda_{1}\cross\cdots\cross\lambda_{n}$,individed$\Delta_{n}$).

The mixed volume of identical bodies is equal to the volume of any one ofthem. The mixed volume of bodies is expressed in terms of the usual volumes of
their sums in the same way as the product of numbers is expressed in terms ofthe -th powers of their sums. For example, for $n=2$ ,
$n$

$n$

$n$

$n$

$ab= \frac{1}{2}[(a+b)^{2}-a^{2}-b^{2}]$$V( \Delta_{1}, \Delta_{2})=\frac{1}{2}[V(\Delta_{1}+\Delta_{2})-V(\Delta_{1})-V(\Delta_{2})]$

4

A. KHOVANSKII

Similarly, for $n=3$ ,$V( \Delta_{1}, \Delta_{2}, \Delta_{3})=\frac{1}{3!}[V(\Delta_{1}+\Delta_{2}+\Delta_{3})-\sum_{\dot{|}<j}V(\Delta:+\Delta_{j})+\sum V(\Delta:)]$is theis the rectangle $0\leq x\leq a$ , $0\leq y\leq b$ and
Example: Suppose that$0\leqx\leq
c$rectangle
is
the
,
.
MinkowskisumTherectangle$0\leq x\leq a+c$ , $0\leq y\leq b+d$ . The mixed volume
is equal to $ad+be$ .
$\Delta_{2}$$\Delta_{1}$$0\leq y\leq d\mathrm{O}$$\Delta_{1}+\Delta_{2}$$V(\Delta_{1}, \Delta_{2})$The number $ad+be$ is the permanent of the matrix$\{\begin{array}{ll}a cb d\end{array}\}$(the definition of thepermanent differs from that of the determinant only in that all the terms in thepermanent have aplus sign). In the multidimensional case the mixed volume ofparallelepipeds with sides parallel to the coordinate axes is also equal to thepermanent of the corresponding matrix.
Let us consider asystem of Laurent equations $f_{1}=\cdots=f_{n}=0$ with Newton, An. Below we define the regularity condition for such systems.
,
polyhedra$n$$n$$\Delta_{1}$$\ldots$(which take accountBernshtein’s theorem. The number of solutions inof the multiplicity) of a regular system of equations in unknowns, is equal tothe mixed volume of the Newton polyhedra of the equations of the system, multipliedby .
$(\mathbb{C}\backslash 0)^{n}$

$n$

$n$

$n!$

Example: The number of roots of ageneral system of polynomial equations, inwhich the -th variable enters in the -th equation with apower no higher than, multiplied by $n!$ .
is equal to the permanent of the matrix
Kushnirenko’s theorem coincides with Bernshtein’s theorem for equations withidentical Newton polyhedra. We now proceed to the definition of aregular systemof equations. We first define the truncations of asystem associated with afunction
4. We take an arbitrary linear function 4on the space in which the Newton
the restriction of the Laurent polynomial to thatpolyhedra lie. We denote byface of its Newton polyhedron on which the linear function 4takes its maximum
value. We associate the restricted system $f_{\xi}^{1}=\cdots=f_{n}^{\xi}=0$ with the system of
equations $f_{1}=\cdots=f_{n}=0$ and the linear function (. For anonzero function 4
the restricted system actually depends on asmaller number of variables. Therefore,. Agivenin the case of general position such asystem is inconsistent insystem of equations has only afinite number of restricted systems (if the polyhedraof all the equations coincide, then the truncations correspond to the faces of thecommon polyhedron). Asystem of equations in unknowns is said to be regular. It is just
if its restrictions for all nonzero functions 4are inconsistent insuch asystem to which Bernshtein’s theorem is applicable.
We note that if for each nonzero linear function , the maximum in one of thepolyhedra is attained at avertex then the regularity conditions are automaticallysatisfied. [In this case the truncated system contains an equation which is containedFor
in amonomial set equal to zero; this equation has no solutions in$n=2$
automatically,,aresatisfied
the regularity conditions
example, in the case of
if the two Newton polyhedra on the real plane have no parallel sides.
$i$

$j$

$a_{ij}$$(a_{\dot{|}j})$$\mathbb{R}^{n}$$f$$f^{\xi}$$(\mathbb{C}\backslash 0)^{n}$

$n$

$n$

$(\mathbb{C}\backslash 0)^{n}$$\xi$$(\mathbb{C}\backslash 0)^{n}.]$4. Complete intersections.
asystem of k Laurent equations $f_{1}=\cdots=f_{k}=0$ with
Consider in,
Newton polyhedra$(\mathbb{C}\backslash 0)^{n}$$\Delta_{1}$\ldots ,$\Delta_{k}$5THREE LECTURES ON NEWTON POLYHEDRA

Assume that the system is nondegenerate. It turns out that the nondegeneracy
condition holds for almost all Laurent polynomials with fixed Newton polyhedra.
Many discrete invariants of the set of solutions are identical and are expressed interms of the polyhedra.
Theorem. The Euler characteristic of the nondegenerate complete intersection, $(k\leq n)$ , with the Newton polyhedra,$f1=\cdots=f_{k}=0$ inis equal to, where the sum is taken overall sets $1=i_{1}\leq\cdots\leq in-k\leq k$ .
$(\mathbb{C}\backslash 0)^{n}$$(-1)^{n-k}n!

$\Delta_{1}$$\ldots.\Delta_{k}$\sum V(\Delta_{1}, \ldots, \Delta_{k}, \Delta:_{1}, \ldots\Delta:_{n-k})$

We note one special case of this theorem.
Corollary. The Euler characteristic of a hypersurface indefined by non$N$degenerate equation with fixedewton polyhedron is equal to the volume of the
Newton polyhedron multiplied by $(-1)^{n-1}n!$ .
$(\mathbb{C}\backslash 0)^{n}$Another consequence of this theorem is Bernshtein’s theorem (see par.3.1.2):
for $k=n$ , the nondegenerate complete intersections consist of points, and theEuler characteristic is equal to the number of points (strictly speaking, Bernshtein’s
theorem is slightly stronger than this corollary, since it is applicable to adegenerate
regular system).

5. Genus of complete intersections.
The formulas given below for the genus of complete intersections are generalizations of the following formulas for Abelian and elliptic integrals. Let us consider aRiemann surface $y^{2}=P_{3}(x)$ (the complex phase curve of motion of apoint in afieldwith acubic potential). This Riemann surface, which is diffeomorphic to the torus,
exhibits asingle, everywhere holomorphic, differential form $dx/y$ (the differentialof the time of motion along the phase curve). In the case of apotential of degree, the curve $y^{2}=P_{n}(x)$ is diffeomorphic to the sphere with handles, whereis connected with either by the formula $n=2g+1$ , or the formula $n=2g+2$(depending on the parity of ). The basis of holomorphic forms in this case is given
by forms of the type $x^{m}dx/y$ , $0\leq m<g$ . The number is the genus of the curve.
The Newton polyhedron of the curve $y^{2}==P_{n}(x)$ is atriangle with vertices $(0, 0)$ ,$(0, 2)$ , and $(n, 0)$ . There are exactly
points with integer coordinates strictly inside
this triangle. In terms of these points, one can give abasis for the space of hol0morphic forms: the point $(l, a)$ , which lies inside the triangle, corresponds to theform $x^{a-1}dx/y$ . We give below the generalization of this procedure for constructing
abasis of holomorphic forms for the multi-dimensional case.
The nondegenerate complete intersections $f_{1}=\cdots=f_{k}=0$ , where the , are
Laurent polynomials, are smooth algebraic affine manifolds. In the cohomologies ofsuch manifolds there is an additional structure, namely, the mixed Hodge structure.
The discrete invariants of such astructure are calculated in terms of the Newton
polyhedra. We consider only the calculation of the arithmetic and geometric genus
which are invariants of this kind.
We first recall some definitions and general statements. Suppose thatis anonsingular (possibly noncompact) algebraic manifold. The set of holomorphic
forms on , which extend holomorphically to any nonsingular algebraic compactification, is automatically closed, and it realizes the zero class of homology ofthe manifold only if it is equal to zero. Forms of this kind form asubspace in the-dimensional cohomologies of the manifold Y. We denote the dimension of this
$n$

$g$

$g$

$n$

$n$

$g$

$g$

$g$

$f_{i}$$\mathrm{Y}$$p$$\mathrm{Y}$$\mathrm{Y}$

$p$

6

A. KHOVANSKII

subspace by$h^{p,0}(\mathrm{Y})$. The arithmetic genus of the manifold is the alternating. The geometric genus of the manifoldof the numbers$\mathrm{Y}$sum, where is the complex dimension of the manifold Y.
is the number
Now we turn to the Newton polyhedron. We shall use the characteristic
is adimensionof integer polyhedra. Here is its definition. Suppose thatpolyhedron with vertices at integer points, lying inandis adimension
affine subspace, containing . The number
is defined as the number of integer$\sum(-1)^{p}h^{p,0}(\mathrm{Y})$$h^{p,0}(\mathrm{Y})$$h^{n,0}(\mathrm{Y})$$\mathrm{Y}$$n$$B(\Delta)$$\mathbb{R}^{n}$$\Delta$$\mathrm{a}1$$\mathbb{R}^{q}$$\mathrm{a}1$$B(\Delta)$$\Delta$points lying strictly within he polyhedron A(in the geometry of the subspace.
multiplied by$\mathbb{R}^{q}$),$(-\mathrm{l})^{}$

Theorem. The arithmetic genus of the nondegenerate complete intersection $f_{1}=$$\ldots=f_{k}=0$ in, $(k\leq n)$ , with the Newton polyhedra,,is$(\mathbb{C}\backslash 0)^{n}$$\Delta_{k}$$\Delta_{1}$$\ldots$$1- \sum B(\Delta:)+\sum_{j>\dot{l}}B(\Delta:+\Delta_{j})-\cdots+(-1)^{k}B(\Delta_{1}+\cdots+\Delta_{k})$.

Corollary. The geometr genus of the nondegenerate complete intersection (for$k<n)$ with the polyhedra of full dimensionality is$r\cdot c$$(-1)^{n-k}((- \sum B(\Delta:)+\sum_{j>\dot{1}}B(\Delta:+\Delta_{j})-\cdots+(-1)^{k}B(\Delta_{1}+\cdots+\Delta_{k}))$We now give acomplete description of the holomorphic forms of highest dimension which could be extended holomorphically to the compactification, for the caseof anondegenerate hypersurface $f=0$ inwith the Newton polyhedron
Aof complete dimension. For each integer point, lying strictly within the Newton
polyhedron , we denote by
the -form on the hypersurface $f=0$ , defined by
the formula$(\mathbb{C}\backslash 0)^{n}$$\Delta$$\omega_{a}$$\omega_{a}=z_{1}^{a_{1}}\cross\cdots \mathrm{x}$$n$;$z_{n}^{a_{n}} \frac{dz_{1}}{z_{1}}\wedge\cdots\wedge\frac{dz_{n}}{z_{n}}/df$$a=a_{1}$,$\ldots$,$a_{n}$.

Theorem. The forms
lie in the space offorms extendible holomorphically on thecompactification of the manifold $f=0$ ;they are linearly independent and generatethat space. In particular, the geometric genus of the hypersurface is equal to $|B(\Delta)|$ .
$\omega_{a}$Example: Consider acurve in the plane determined by the equation $y^{2}=P_{n}(x)$ .
The interior monomials for the Newton polyhedron of this curve have the form $yxa$ ,where $1\leq a<n/2$ . The forms
corresponding to the monomials coincide [on the$y^{2}=P_{n}(x)]$curvewith the forms $x^{a-1}dx/2y$ . Thus, for the curve $y^{2}=P_{n}(x)$ we
get the usual description of all Abelian differentials.
$\omega_{a}$6. Affine case.
, but in the
It is customary to consider systems of equations not in. Calculations with Newton polyhedra also are carried out
usual complex spacein this situation. The answers here are more messy. As an example we consider
the calculation of the Euler characteristic of ahypersurface in. Suppose thatis apolynomial in complex variables with anonzero free term and Newton
polyhedron . We introduce the following notation:is the intersection of the, $d(I)$ is the dimension
Newton polyhedron Awith the coordinate planeiof this plane, andis the $d(I)$ -dimensional volume of the polyhedron.
$(\mathbb{C}\backslash 0)^{n}$$\mathbb{C}^{n}$$\mathbb{C}^{n}$

$f$

$n$

$\Delta^{I}$$\Delta$$R^{I}$$\mathbb{R}^{n}$$\mathrm{n}$$V(\Delta^{I})$$\Delta^{I}$

7

THREE LECTURES ON NEWTON POLYHEDRA

Theorem. Let
be a nondegenerate polynomial with the Newton polyhedron,
which has a nonzero free term. Then $f=0$ is a nonsingular hypersurface inintersecting transversally all the coordinate planes in. The Euler characteristic$\sum(-1)^{d(I)-1}d(I)!V(\Delta)$ , where the summation runs
hypersurface
equal
is
to
this
of
over all intersections of the Newton polyhedron
with the (nonzero) coordinate
planes.
$f$$\Delta$$\mathbb{C}^{n}$$\mathbb{C}^{n}$$\Delta^{I}$

This theorem follows from the calculation of the Euler characteristic inand from the additivity of the Euler characteristic. We also note that the formula
given for ahypersurface is analogous to the formula for the Milnor number.
$(\mathbb{C}\backslash 0)^{n}$

LECTURE 2. POLYHEDRA AND INDEX OF APOLYNOMIAL FIELD
In this lecture we will present an estimation for the index of apolynomial vector
field with components of fixed degrees. We also will give examples showing that
this estimation is best possible (see [1], [2]). The proof of the estimates in thenondegenerate case is closely related to the proof in Petrovskii and Oleinik [3],where the Euler characteristics of some algebraic sets are estimated. In addition,
the proof is closely related to the proof in the article by Arnol’d [4] and clarifies
the connection between these two methods of argument. As in [4], the index isassociated with the signature of acertain quadratic form (see also [5, 6]). As in [3],akey factor in the proof is the use of the Euler-Jacobs formula.

1. Notation.
,
Let $V=P_{1}$ ,be avector field inwith polynomial components . We
denote by ind the sum of the indices of all singular points of the field $V$ in.
,, provided
has degree not equal to $m=m_{1}$ ,We will say that the field, , are equal to. We will say
the degrees of all the polynomials , $i=1$ ,that the field
is nondegenerate if the real singular points of the field $V$ havemultiplicity one and “lie in the finite part of the space. (We write the lastcondition out in more detail. Let thebe homogeneous polynomials of degree, , such thatin the variables , ,. Thelast condition means that the system
has only the trivial
solution $x_{0}=x_{1}=\cdots=x_{n}=0.$ )We introduce some notation.
is the parallelepiped indefined by the inequalities $0\leq y_{1}\leq m_{1}$1, . . . ’ $0\leq y_{n}\leq m_{n}-1$
is the number of integer points in the parallelepiped.
$\mu=m_{1}\ldots$$\Pi(m)$ is the number of integer points in the central section $y_{1}+\cdots+y_{n}=$$\ldots$$P_{n}$$\mathbb{R}^{n}$$P_{i}$$\mathbb{R}^{n}$$V$$\ldots$$P_{i}$$\ldots$$m_{n}$$m_{i}$$\mathrm{n}$$V$$\mathbb{R}^{n}"$$P_{i}$$x_{0}$$x_{1}$$\ldots$$m_{i}$$\overline{P}_{i}(1, x_{1}, \ldots, x_{n})\equiv P_{i}(x_{1}, \ldots, x_{n})$$x_{n}$$\overline{P}_{1}=\cdots=\overline{P}_{n}=x_{0}=0$$\Delta(m)$$\mathbb{R}^{n}$-$\Delta(m)$$m_{n}$$\frac{1}{2}(m_{1}+\cdots+m_{n}-n)$of the parallelepiped$\Delta(m)$.

2. Statement of the result.
Theorem. If
is a nondegenerate field of degree $m$ , the number satisfies the. Conversely, for every
inequality $|a|\leq\Pi(m)$ and the congruencenumber satisfying these conditions, there exists a nondegenerate field of degree$m$.
for which$V$$a\equiv\mu \mathrm{m}\mathrm{o}\mathrm{d} 2$

$V$

$a$

$\mathrm{i}\mathrm{n}\mathrm{d}=a$,Corollary. The index ind of an isolated singular point of the field $V=P_{1}$ ,,with homogeneous components of degree $m=m_{1}$ ,satisfies the inequality. The number ind is not subject
and the congruence
to any other restrictions$\ldots$$\ldots$$|\mathrm{i}\mathrm{n}\mathrm{d}|\leq\square (m)$$\mathrm{i}\mathrm{n}\mathrm{d}\equiv\mu \mathrm{m}\mathrm{o}\mathrm{d} 2$$m_{n}$$P_{n}$

8

A. KHOVANSKII

3. Projective transformations.
defined by alinear inhomogeneous equation $l(x)=$Let be ahyperplane in$l_{1}(x)+l_{0}=0$ . We construct aprojective transformation, taking, where$g(x)=[1/l(x)]A(x)$infinity;
hyperplane
hyperplaneattheinto the$A(x)$ is an affine transformation. Aprojective transformation of afield
with
,
,
,
,
,
,
components
whereof degrees
is afield
for $i=1$ , , . If is asingular point of the field ,$\mathbb{R}^{n}$$\Gamma$$g:\mathbb{R}^{n}arrow \mathbb{R}\mathrm{n}$$\Gamma$$V$$P_{1}$$\ldots$$P_{n}$$m_{1}$$\tilde{P}_{\dot{l}}(x)=l^{m}:(x)P_{\dot{l}}(g(x))$$\ldots$$\ldots$$\tilde{V}=\tilde{P}_{1}$$m_{n}$$n$$P\sim n$’$\ldots$$V$$a$i defined
then the point $\tilde{a}=g^{-1}(a)$ is singular for . The Jacobian
outside the hyperplaneand vanishes nowhere. We say that atransformation$xarrow g(x)=[1/l(x)]A(x)$ is positive if its Jacobian is positive in the region $l(x)>0$ .
is orientable and positive transformations coincide withFor odd , the spaceorientation-preserving transformations. In the general case, positive transformawith positive determitions correspond precisely to linear transformations ofnant.
We will be interested in how the index, of the field $V$ changes under pr0jective transformations. The global characteristic ind is obtained by summing
the corresponding local characteristics over the set $X$ of singular points of ,$\tilde{V}$$\det\frac{\partial g}{\partial x}$$\mathrm{s}$$\Gamma$$\mathbb{R}P^{n}$$n$$\mathbb{R}^{n+1}$$V$.

$\mathrm{i}\mathrm{n}\mathrm{d}=\sum_{a\in X}\mathrm{i}\mathrm{n}\mathrm{d}(\mathrm{V})\mathrm{a}$and aprojective transformation $xarrow g(x)$ , we write $X(g)$ for
Given afield
for which $\tilde{a}=g^{-1}(a)$ is defined. The index of
the set of singular points ofthe field is called projectively invariant if for every positive $g=[1/l(x)]A(x)$ and$a\in X(g)$ the equality
holds. The index is said to be projectively.
antiinvariant if
Asimple verification proves the following assertion.
$V$

$V$

$a$

$V$

$\mathrm{i}\mathrm{n}\mathrm{d}(V)_{a}=\mathrm{i}\mathrm{n}\mathrm{d}(\tilde{V})_{\tilde{a}}$$\mathrm{i}\mathrm{n}\mathrm{d}(V)_{a}=\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}$$l(\tilde{a})\mathrm{i}\mathrm{n}\mathrm{d}(\tilde{V})_{\tilde{a}}$Assertion. 1. The index is projectively invariant if
2. The index is projectively antiinvariant if2.

$m_{1}+\cdots+m_{n}\neq n\mathrm{m}\mathrm{o}\mathrm{d}$$m_{1}+\cdots+m_{n}=n\mathrm{m}\mathrm{o}\mathrm{d}$2.

for the image of the plane at infinity under aprojective transWe write
formation. For $g(x)=[1/l(x)]A(x)$ , where $A(x)$ is alinear transformation and$l(x)=-l_{1}(x)+l_{0}$ the equation of the hyperplane
has the form $l_{1}(A^{-1}(x))=1$ .
We associate with the hyperplane $l_{1}(x)=p$ , where $p>0$ , the transformation$xarrow(px)/(l_{1}(x)+l)$ , for which this plane is. The invariant characteristicsof the singular points of the field are preserved under aprojective transformation.
The antiinvariant characteristics remain unchanged for the singular points lying inand change sign for the singular point lying
one of the halfspaces bounded byin the other half space.
$\Gamma_{\infty}$$\Gamma_{\infty}$$\Gamma_{\infty}$$\Gamma_{\infty}$

4. Examples.
In the construction of our examples, aprincipal role will be played by the simplest,
with components $P_{\dot{l}}= \prod_{0\leq k\leq m.-1}(x:-k)$ ,field $V(m)$ of degree $m=m_{1}$ ,$i=1$ ,
, . We note that all the singular points of $V(m)$ coincide with the integral
points of the polyhedrondefined by the inequalities $0\leq x:\leq m:-1$ , $i=$
, , . The signs of the Jacobian at the singular points alternate in “chessboard
order”. Moreover, at the singular points lying on the single section $\sum x:=k$ , thesign of the Jacobian is constant. Upon passing to the next section $\sum X:=k+1$ ,this sign is replaced by the opposite sign. In this section we often encounter the, which we denote by .
number$\ldots$$\ldots$$m_{n}$$n$$\Delta(m)$$1$$\ldots$$n$$\frac{1}{2}\sum(m_{\dot{l}}-1)$$\rho$

9

THREE LECTURES ON NEWTON POLYHEDRA

Consider the case
. In this case $\Pi(m)=0$ , and by
Theorem 1every nondegenerate field has zero index.
We consider the case
. In this case, the characteristic
ind is antiinvariant. The central section $\sum X:=\rho$ of the rectangular box
contains exact $\Pi(m)$ singular points of the field $V(m)$ . The sections $\sum x:=\rho-k$
and $\sum x:=\rho+k$ contain the same number of singular points with Jacobiansof the same sign. We carry out aprojective transformation $xarrow g(x)$ for which
the plane
has the equation $\sum x_{i}=\rho-1/2$ , e.g., the transformation $xarrow$$( \rho-1/2)x/(\sum x:+1)$ . The sections $\sum x:=\rho-k$ and $\sum x:=\rho-k$ for $k>0$
lie in different half spaces bounded by. The indices of the inverse images ofthe singular points of the field $V(m)$ lying in these sections cancel one another.
Therefore, the absolute value of the index of the field
is equal to the number of$\sum
x_{i}=\rho$$V(m)$
singular points of, i.e., it is equal to
lying on the section. Hereis an explicit formula for the componentof the field : $P_{i}(x)= \prod_{0\leq k\leq m_{i}}[(p-$$m_{0}+\cdots+m_{n}\not\equiv n\mathrm{m}\mathrm{o}\mathrm{d} 2$$V$$m_{1}+\cdots+m_{n}\equiv n\mathrm{m}\mathrm{o}\mathrm{d} 2$$\Delta(m)$$\Gamma_{\infty}$$\Gamma_{\infty}$$\tilde{V}$$\square (m)$$\tilde{V}$$\tilde{P}_{i}$$\frac{1}{2})x_{i}-k(\sum x_{i}+1)]$.

The index of the field
does not change if the position of the planeisperturbed slightly. We defineby the equation $\sum a_{i}x_{i}=t$ , where thearescalars close to unity which are independent over the rationals, and is close to$\rho-1/2$ . We now begin to let
get bigger. This will not change the index of thepasses through asingular point of $V(m)$ from the direction of the
field until
central section. When this occurs, becomes degenerate and its index
changes by one. If the number is increased abit more, the field again becomes
nondegenerate and its index again changes by one. Continuing the motion of the, we obtain examples of nondegenerate fields of degree $m$ with any index
plane,satisfying the conditions. The leading homogeneouscomponents of the fieldform afield with an isolated singular point at zero withthe same index.
$\tilde{V}$$\Gamma_{\infty}$$\Gamma_{\infty}$$a_{i}$$t$$t$$\tilde{V}$$\Gamma_{\infty}$$\tilde{V}$$\sum x_{\dot{\iota}}=\rho$$\tilde{V}$$t$$\tilde{V}$$\Gamma_{\infty}$$|\mathrm{i}\mathrm{n}\mathrm{d}|\leq\Pi(m)$$\mathrm{i}\mathrm{n}\mathrm{d}\equiv\mu \mathrm{m}\mathrm{o}\mathrm{d} 2$$\tilde{V}$

5. Signature and index.

5.1. A Finite Set with Involution. Let be afinite set containing elements,$\tau:Aarrow A$ an involution of, and let $X$ be the set of fixed points of . We consider
the algebra
over the field consisting of all complex value functions on for. Let be afixed function in
which
which is nowhere zero. The numberof points of the set $X$ at which is positive is denoted by, the number of points
at which it is negative, by. We consider the bilinear form
ondefined by$\omega_{\varphi}(f, g)=\sum_{a\in A}\varphi(a)f(a)g(a)$ . The signature of aquadratic form $K$ is denoted by
$A$

$\mu$

$A$

$\tau$$R$$L_{\tau}$$A$$f\tau=\tilde{f}$$L_{\tau}$$\varphi$$\varphi^{+}$$\varphi$$L_{\tau}$$\varphi^{-}$$\sigma K$$\omega_{\varphi}$.

Lemma 1. The dimension of the algebra
is equal to . The quadratic formtakes real values and is nondegenerate. The signatureis$\varphi=1$
equal toIn particular,
is equal to the number of fixedforpoints of .
$L_{\tau}$$\mu$$K_{\varphi}(f)=\omega_{\varphi}(f, f)$$\sigma K_{\varphi}$$\varphi^{+}-\varphi^{-}$$\sigma K_{\varphi}$$\tau$

Proof. Under the action of , the set

decomposes into invariant setsconsistingof one or two points. Let
denote the subalgebra ofconsisting of functions,
with support. The subspaces
are orthogonal with respect to$a\in
X$ , then $\dim L_{\tau}^{k}=1$ and the. If
the form
consists of asingle point ,signature of the restriction of the form
tois equal to sign
. For two
point sets $A^{k}=\{a, \tau a\}\dim L_{\tau}^{k}=2$ . In this case, the restriction of the form$\tau$$A$$A^{k}$$L_{\tau}^{k}$$A^{k}$$L_{\tau}$$l_{\tau}= \sum L_{\tau}^{k}$$L_{\tau}^{k}$$A^{k}$$a$$\omega_{\varphi}$$K_{\varphi}$$L_{\tau}^{k}$$\varphi(a)$$K_{\varphi}$

10

A. KHOVANSKII

is equal to $\varphi(a)f^{2}(a)+\varphi(\tau a)f^{2}(\tau a)=2{\rm Re}[\varphi(a)f^{2}(a)]$ . As is easily seen, thetosignature of such aform is zero. The lemma is proved.
$L_{\tau}^{k}$holds. 2. Let
be any linear
Corollary. 1. The congruenceis identically equal to zero. Then
on which the form
subspace of the algebra
the estimate $|\varphi^{+}-\varphi^{-}|\leq\mu-2\dim L_{0}$ is valid. If the null subspace is maximal,
this estimate is best possible.
$\varphi^{+}-\varphi^{-}\equiv\mu \mathrm{m}\mathrm{o}\mathrm{d} 2$$L_{\tau}$$L_{0}$$K_{\varphi}$

Proof. Indeed, the signature of anondegenerate form always has the same parity

as the dimension of the space. In addition, the estimate $|\sigma K|\leq\mu-2\dim L_{0}$ holdswith nullspace . For amaximal subspacefor every nondegenerate form $K$ on, this estimate is an equality.
$L_{0}$$\mathbb{R}^{\mu}$$L_{0}$

We will apply the lemma 1and corollary to the case when is the set of complex
singular points of areal vector field and $\tau:Aarrow A$ is the involution given by complex
conjugation.
$A$with polynomial components V
5.2. We consider areal vector field V in,
. Let Abe the set of complex solutions of the system$\mathbb{R}^{n}$$P_{1}$\ldots ,$P_{n}$$=$$\subset \mathbb{C}^{n}$(1)$P_{1}=\cdots=P_{n}=0$

and let the involution $\tau:Aarrow A$ be complex conjugation. The set $X$ of fixed points
under coincides with the set of real solutions of system (1). We assume that all
the complex solutions $a\in A$ of system (1) have multiplicity one. This means that$\tau$of (1) does not vanish at the points of . Let
the Jacobian
any polynomial with real coefficients which does not vanish at the points of
We obtain the following assertion by applying the preceding lemma.
$A$$j(x)= \det\frac{\partial P}{\partial x}$$P_{0}$

$A$

be

.

Assertion. The signatureof the quadratic form $K_{\varphi}(f)= \sum_{a\in A}\varphi(a)f^{2}(a)$is equal to the number of real singular points of the field
if $\varphi=1$ . If $\varphi=1/j$ ,.
where is the Jacobian of system (1), then$\sigma K_{\varphi}$

$V$

$j$

$\sigma K_{\varphi}=\mathrm{i}\mathrm{n}\mathrm{d}$, it is now necessary to describe the algebra
In order to estimate the number
which is as large as possible for $\varphi=1/j$ .
and exhibit anullspace for the form
The null subspace is obtained with the aid of the Euler-Jacobi formula, which werecall.
, in complex,,
Consider asystem of polynomial equations of degrees
unknowns,$\mathrm{i}\mathrm{n}\mathrm{d}$$L_{\tau}$$K_{\varphi}$$n$$m_{1}$$P_{1}=\cdots=P_{n}=0$$\ldots$$m_{n}$$\mathrm{n}$.

We assume that the set of roots of the system contains exactly $\mu=m_{1}\ldots$
does not vanish on
elements. In this case, the Jacobian of the system
the set . Then for every polynomial of degree less than $\sum m:-n$ , we have the$m_{n}$$j= \det\frac{\partial P}{\partial x}$$A$following Euler-Jacobi formula:$Q$$\sum_{a\in A}\frac{Q(a)}{j(a)}=0$.

Apurely algebraic proof of this formula can be found in [7]. An analytic proof andgeneralization to nondegenerate systems of equations with fixed Newton polyhedra
is given in [8]11THREE LECTURES ON NEWTON POLYHEDRA

6. Convenient systems of equations.
,,will be called
Asystem of equations $P_{1}=\cdots=P_{n}=0$ of degrees
distinct roots.
nondegenerate if it has exactly $\mu=m_{1}\ldots$
by the inequalities $0\leq y_{1}\leq$definedinConsider the parallelepiped$m_{1}-1$ ,, $0\leq y_{n}\leq m_{n}-1$ . Let $M(m)$ be the space of polynomials with Newton. Apolynomial $Q\in M(m)$ , if and only if the degree of withpolyhedron. The dimension of the space $M(m)$ overis less thanrespect to the variables
the field is equal to$m_{n}-n=\mu$ .
Anondegenerate system is said to be convenient if every complex valued function
on the set of roots of the system is the restriction of some polynomial in $M(m)$ .
$m_{1}$$\ldots$$m_{n}$$m_{n}$$\Delta(m)$$\mathbb{R}^{n}$$\ldots$$Q$$\Delta(m)$$m_{i}$$x_{i}$$\mathbb{C}$$f$$m_{1}\ldots$$A$

Lemma 2. The system of equations$\prod_{0\leq k\leq m_{1}}(x_{1}-k)=\cdots=\prod_{0\leq k\leq m_{n}}(x_{n}-k)=0$

is convenient.

elof roots of this system contains precisely $\mu=m_{1}\ldots$
Indeed, the set
ement. In addition, the equations of the system can be rewritten in the form of,,
are polynomials, $x_{n}^{m_{n}}=Q_{n}(x_{n})$ , in whichequalities $x_{1}^{m_{1}}=Q_{1}(x_{1})$ ,, $m_{n}-1$ . Using these equations, it is not hard to show thatof degrees $m_{1}-1$ ,with some polynomial in the space
every polynomial $Q(x)$ coincides on the seton the finite set is the$M(m)$ . This implies Lemma 1, since every function
restriction of some polynomial.
$A$$m_{n}$$Q_{1}$$\ldots$$\ldots$$Q_{n}$$\ldots$

$A$

$A$

$f$

Lemma 3. The inconvenient systems form a hypersurface in the space of all systems of degree $m$ .
Indeed, as is well known, the degenerate systems form ahypersurface in thespace of all systems of degree $m$ . Take any nondegenerate system and enumerate, . Then enumerate in some way the integer points in
its roots arbitrarily ,the Newton polyhedron $\Delta(m)$ . These numerations define bases in the -dimensional
space $C(A)$ of all complex-valued functions on $A$ and in the dimension space$M(m)$ . Let $\det$ denote the determinant of the matrix of the restriction mapping$i:M(m)arrow C(A)$ with respect to these bases. The number det2 does not dependon the choice of enumeration; it depends only on the coefficients of the system, thisdependence being analytic. Anondegenerate system is convenient if and only if thefor it is distinct from zero. The function det2 is not identically equal
numberto zero. Indeed, by Lemma 1, there exist convenient systems of degree $m$ . Lemma
2is proved.
Let $M(m, R)$ denote the space of polynomials with real coefficients with Newton
polyhedron $\Delta(m)$ , $M(m, R)=M(m)\cap R(x)$ .
$a_{1}$$\ldots$$a_{\mu}$

$\mu$

$\mu$

$\det^{2}$

Lemma 4. For a convenient system of equations of degree $m$ with real coefficients,
the restriction of polynomials in the space $M(m, R)$ to the set A defines an isomoronto , where $\tau:Aarrow A$ is the involution
phism of $M(m, R)$ with the algebraof complex conjugation.
$L_{\tau}$

$A$

. In
The restrictions of polynomials in $M(m, R)$ clearly lie in the algebra
addition, for convenient systems anonzero polynomial in $M(m)$ corresponds to anonzero function on . Lemma 4follows from the inclusion $M(m, R)\subset M(m)$ andthe fact that $\dim M(m, R)=\mu$ , and $\dim L_{\tau}=\mu$ .
$L_{\tau}$

$A$

12

A. KHOVANSKII

7. Inequalities for nondegenerate fields.
We conclude the proof of Theorem (see Sec. 2). Let be anondegenerate fieldof degree $m$ . Under asmall change in the coefficients of the components ,,of the number ind will not change. It can therefore be assumed, without loss ofgenerality, that $P_{1}=\cdots=P_{n}=0$ is aconvenient system of degree $m$ (see Lemma 3)
and that the surface $P_{0}=0$ in
does not intersect the set of roots of the system.
By Lemma 4, every function on in , where the involution $\tau:Aarrow A$ is complex
conjugation, is the restriction of aunique polynomial in $M(m, R)$ . We consider thequadratic form
onwith $\varphi=1/j$ (here). According to theassertion in Sec. 5,. It follows from the Euler-Jacobi formula that for
all polynomials of degree less than $\frac{1}{2}\sum_{:>0}(m:-1)$ , the identity $K_{\varphi}(f)=0$ isvalid. In our case, the inequality
takes the form.
Indeed, is equal to the number of integer points in the polyhedron, and$\dim L_{0}$ is equal to the number of integer points insatisfying the inequality$\sum y:<1/2\sum_{:>0}(m:-1)$ . The inequality
has been proved. The
congruenceis almost obvious. Theorem is proved.
$V$$P_{1}$$\ldots$$P_{n}$$V$$A$$\mathbb{C}^{n}$$A$$K_{\varphi}(f)$$L_{\tau}$$j= \det\frac{\partial P}{\partial x}$$L_{\tau}$$\sigma K_{\varphi}=\mathrm{i}\mathrm{n}\mathrm{d}$$f$$|\sigma K_{\varphi}|\leq\mu-2\dim L_{0}$$|\mathrm{i}\mathrm{n}\mathrm{d}|\leq \mathrm{A}(\mathrm{m})$$\Delta(m)$$\mu$$\Delta(m)$$|\mathrm{i}\mathrm{n}\mathrm{d}|\leq\Pi(m)$$\mathrm{i}\mathrm{n}\mathrm{d}\equiv\mu \mathrm{m}\mathrm{o}\mathrm{d} 2$

REFERENCES
1. Khovanskii A.G., Index of a polynomial vector field, Functional Analysis and its applications,13, (1979), no. 1, 49-58; English transl. in Funct ional Anal. Appl. 13 (1979), no. 1, 38-45.
2. Khovanskii A.G., Boundary indexes of 1-forms with h omogeneous polynomial coefficients,
Algebra and Analysis 10 (1998), no. 3; English transl. in St. Petersburg Math. J. 10 (1999
issue 3pages 553-575).
3. Petrovskii I.G. and Oleinik O.A., On the topology of real algebraic surfaces, Izv. Akad. Nauk
SSSR, Ser. Mat. 13 (1949), 389-402.
4. Arnoi’d V.I., The index of a singular point of a vectorfield, the Petrovskii-Oleinik inequalities,
and mixed Hodge structures, Funkts. Anal. Prilozhen. 12 (1978), no. 1, 1-14.
5. Eisenbud D. and Levine H., The topological degree of a finite-map germ, Ann. Math. 106(1977), no. 1, 19-38.
6. Khimshiashvili N.G., On the local degree of a smooth map, Soob. Akad. Nauk Gruz. SSR 85(1977), no. 2, 309-311.
7. Kronecker L., Uber einige Interpolationsfor mein fur ganze Funktion mehrer Variabein, Werke,1(1895), 133-141.
8. Khovanskii A.G., Newton polyhedra and the Euler-Jacobi fo rmula, Usp. Mat. Nauk 33 (1978),no. 6, 245-246.
$c\infty$

LECTURE 3. SYSTEM OF EQUATIONSWITH GENERIC NEWTON POLYHEDRA
Consider the system of equations $P_{1}=\cdots=P_{n}=0$ in, where ,are Laurent polynomials with the Newton polyhedra ,.
Assume that Newton polyhedra ,are developed (see \S 1), which meansthat they are located sufficiently generally with respect to each other.
The geometrical meaning of being developed is especially clear in the tw0dimensional case: two polygons on aplane are developed if and only if they do
not have parallel sides with identically directed outer normals.
We will present two following results about the system of equations whose Newton polyhedra are developed.
First result: Let Q be aLaurent polynomial. Consider the n-for$(\mathbb{C}\backslash 0)^{n}$$\Delta_{1}$$\Delta_{1}$\ldots ,$\Delta_{n}$$\omega=(Q/P)\frac{dz_{1}}{z_{1}}\wedge\cdots\wedge\frac{dz_{n}}{z_{n}}$,\ldots ,$\Delta_{n}$$P_{1}$\ldots ,$P_{n}$

13

THREE LECTURES ON NEWTON POLYHEDRA. Then the sum of,
are independent variables and $P=P_{1}\cdot\ldots$
where ,
the Grothendieck residues of the form over all roots of the system of equations
can be explicitly evaluated (see [1]).
Second result: Vieta formula for the product of roots of apolynomial can bethegeneralized to the multidimensional case: one can compute in the groupproduct of all roots of asystem of polynomial equations with sufficiently general
Newton polyhedra (see [2]).
$z_{1}$$\cdot P_{n}$$z_{n}$$\ldots$$\omega$$(\mathbb{C}^{*})^{n}$$\mathrm{n}$1. Combinatorial coefficient.
and Abe their Minkowski sum. Each,,be convex polyhedra inLetpolyhedra. Aface will be called
face of the polyhedron is the sum of faces ofwill be called criticallocked if its terms include at least one vertex. Avertex
if all faces adjacent to this vertex are locked.
, such that each its, $F=$
Consider acontinuous map
is nonnegative and vanishes on those and only those faces $1’=$
components.
is apoint-vertex of the polyhedron
whose term$F$polyhedronof theonto the boundaryof the map
The restriction
transfers aneighborhood of acritical vertex into aneighborhood of the zero pointof the positive octant.
on the boundaryis the local degree ofof acritical vertex
The combinatorial coefficientis well-defined and.
coefficientThemap
the germ of the.
depends only on the orientations of the polyhedron and the positive octant,
is called developed if all faces of the sum,
The set of the polyhedra
areof polyhedra in the spacesets
allare locked. Almost
polyhedrondeveloped.
$\mathbb{R}^{n}$$\Delta_{n}$$\Delta_{1}$$\ldots$$\Delta$$\Gamma$$\Delta_{i}$$A\in\Delta$$F:\Deltaarrow \mathbb{R}^{n}$$(f_{1}, \ldots, f_{n})$$f_{\dot{l}}$$\Gamma_{1}+\cdots+\Gamma_{n}$$\Delta_{i}$$\Gamma_{i}$$\tilde{F}$$\Delta$$\partial\Delta$$\partial \mathbb{R}_{+}^{n}$$A\in\Delta$$k_{A}$$k_{A}$$\tilde{F}:(\partial\Delta, A)arrow(\partial \mathbb{R}_{+}^{n}, 0)$$\Delta$$\Delta_{1}$$\ldots$$\mathbb{R}_{+}^{n}$$\Delta_{n}$$\mathbb{R}^{n}$$n$$\Delta$

2. Orientations.
depends on the order of the independent variables
The sign of the formthat
, . This order also determines the orientation of the linear space,$\Delta=\Delta_{+}\cdots+$and the Newton polyhedroncontains me lattice of the monomials.
The order of the equations $P_{1}=\cdots=P_{n}=0$ (or their Newton polyhedra, which is involved in the) determines the orientation of the space,,ofthe equations also deterorderThecoefficient.
definition of the combinatorial
mines the sign of the Grothendieck residue in the roots of the system of equations.
Let us arbitrarily select the orders of the independent variables and the equations.
These determine the signs of the form , the Grothendieck residue, and the signsof the combinatorial coefficients.
$\omega$$\mathbb{R}^{n}$$z_{1}$$\ldots$$z_{n}$$z^{a}$$\Delta_{n}$$\Delta_{1}$$\ldots$$\Delta_{n}$$\mathbb{R}_{+}^{n}$$\omega$

3. Residue of the form in avertex of apolyhedron.
of the Laurent polynomial
For each vertex of the Newton polyhedron$P$ , we construct the Laurent series or the function $Q/P$ , whereis an arbitrary
Laurent polynomial.
isof the polyhedroncorresponding to the vertex
The monomial
$P$
the
of
term
therefore,
free
the
;
coefficientwith some nonzero
included inLaurent polynomial $\tilde{P}=P/(C_{A}z^{a})$ is equal to unity. Let us specify, the Laurent
by the formula $1/\tilde{P}=1+(1-\tilde{P})+(1-\tilde{P})^{2}+\ldots$ . Each monomial
series for.
is included with nonzero coefficients in only afinite number of the terms
Thewell-defined.
isin this series
Therefore, the coefficient of each monomial
polynomial $C_{A}z^{a}Q$ will betheLaurent
andobtainedseriestheproductofformal$A$$\Delta(P)$

$q$

$A$

$z^{a}$$\Delta(P)$$C_{A}$$z^{b}$$1/\tilde{P}$$(1-\tilde{P})^{k}$$z^{b}$

14

A. KHOVANSKII

called the Laurent series of the rational function $Q/P$ at the vertex
polyhedron A(P).
The residue resAW of the rational form$\omega=\frac{Q}{P}\frac{dz_{1}}{z_{1}}\wedge\cdots\wedge\frac{dz_{n}}{z_{n}}$of the Newton$A$at the vertex

$A$

at

the vertex of the Newton polyhedronis the free term of the Laurent series
can explicitly be writtenof the function $Q/P$ at the vertex . The residue
and the coefficients of the Laurent polynomials $P$ and .
as apolynomial of$A$$\Delta(P)$$A$$\mathrm{r}\mathrm{e}\mathrm{s}_{A}\omega$$Q$$C_{A}^{-1}$

4. The first main result.

Main Theorem. If the Newton polyhedra of the equations in the system are developed, then the sum of the Grothendieck residues can be evaluated for a form
, where thewith any Laurent polynomial Q. This sum is equal tosummation is over all critical vertices of the polyhedron .
$\omega$$(-\mathrm{l})^{}$$A$$\sum k_{A}\mathrm{r}\mathrm{e}\mathrm{s}$$A\omega$$\Delta$

Corollary. The sum $\sum R(a)\mu(a)$ of values of an arbitrary Laurent polynomial $R$
over roots of a system of equations with developed Newton polyhedra, where theroots are evaluated while taking into account their multiplicities $\mu(a)$ is
where
$a$

$(-\mathrm{l})^{}$$\sum k_{A}\mathrm{r}\mathrm{e}\mathrm{s}_{A}\omega$$\omega=R\frac{dP_{1}}{P_{1}}\wedge\cdots\wedge\frac{dP_{n}}{P_{n}}=[Rz_{1}\ldots z_{n}\det(\frac{dP}{dz})/(P_{1}\ldots P_{n})]\frac{dz_{1}}{z_{1}}\wedge\cdots\wedge\frac{dz_{n}}{z_{n}}$5. Geometric application.
For each vertex of the polyhedronsuch that $A=A_{1}+\cdots+A_{n}$ is determined. Put $\det$,.
of the matrix formed by the vectors ,$A$, aset of vertices
to be equal to the determinant$\Delta=\Delta_{1}+\cdots+\Delta_{n}$$A_{1}$

Theorem. For the mixed volume$nal$ vertices,$V$$\ldots$$A$.

$A_{:}\in\Delta_{i}$$A_{n}$of developed polyhedra$n!V=(-1)^{n} \sum k_{A}\det A$$\Delta_{1}$,$\ldots$,$\Delta_{n}$with valid.

For polyhedra with integer vertices, this theorem is proved by comparing theBernstein formula for the number of the roots of the system of equations with
Corollary for $R=1$ . One can prove the theorem geometrically and eliminate thecondition that the vertices are rational [2].
6. Algebraic application.
Corollary makes it possible to construct an explicit theory of elimination for asystem of equations inwith developed Newton polyhedra. Let us explain,for example, how an equation for the first coordinate of the roots of the system isobtained. For this purpose it is sufficient to evaluate the sums $\sum R(a)\mu(a)$ , where$R$ are polynomials equal to 1,, , , with $N=n!V-1$ , and use the Newton
formulas expressing the coefficients in an equation in terms of the sums of powersof its roots.
$(\mathbb{C}\backslash 0)^{n}$$z_{1}$$z_{1}$$z_{1}^{N}$$\ldots$

7. Local version.
Consider asystem of analytic equations $p_{1}=\cdots=p_{n}=0$ in aneighborhoodof the point. Let the Newton diagrams of , , of these equations beconvenient. The definitions of developed Newton diagrams and the combinatorial
coefficients of the vertices of the diagram of the sum
and theresidues
almost literally repeat the definitions given above. The following
theorem is valid.
$\mathrm{O}\in \mathbb{C}\mathrm{n}$$\Gamma_{1}$$\ldots$$\Gamma_{n}$$\Gamma=\Gamma_{1}+\cdots+\Gamma_{n}$$\mathrm{r}\mathrm{e}\mathrm{s}A\omega$

15

THREE LECTURES ON NEWTON POLYHEDRA

Theorem. For a system of analytic equations $p_{1}=\cdots=p_{n}=0$ with developed
Newton diagrams, the point 0is an isolated solution. The Grothendieck residue atzero of the form$\omega=\frac{q}{p_{1}\ldots p_{n}}\frac{dz_{1}}{z_{1}}\wedge\cdots\wedge\frac{dz_{n}}{z_{n}}$tion divisible byis over all vertices
the positive octant.

, is equal toof the Newton diagram$z_{l}\cdot\ldots\cdot$$A$, where$(-\mathrm{l})^{}$$z_{n}$

is an arbitrary analytic func-$q$$\sum k_{A}\mathrm{r}\mathrm{e}\mathrm{s}_{A}\omega)$, where the summationthat lie strictly inside$\Gamma=\Gamma_{1}+\cdots+\Gamma_{n}$8. Vieta formula.
According to the classical Vieta formula, the product of the nonzero roots ofan equation $a_{n}x^{n}+\cdots+a_{k}x^{k}=0$ with $a_{n}\neq 0$ , $a_{k}\neq 0$ is equal to the number$(-1)^{n-k}a_{k}a_{n}^{-1}$ .
We generalize the Vieta formula to the multidimensional case.
More precisely, we compute in the groupthe product of all the roots of thesystem of equations$(\mathbb{C}^{*})^{n}$$P_{1}(x)=\cdots=P_{n}(x)=0$ ,$x\in(\mathbb{C}^{*})^{n}$,,,whose Newton polyhedra
are developed.
We will present two different formulas for this product. In the first formula
we use the so called Parshin symbols, in the second formula we use derivatives ofthe mixed volume with respect to vertices of all polyhedra. Let us start with the
one-dimensional case.
The Vieta formula has an interpretation connected with Weil’s theorem. First
let us recall Weil’s theorem (see, for example, [3]). Let $X$ be acomplex algebraic
curve, and and be two nonzero meromorphic functions on $X$ . With each point$a\in X$ is associated the s0-called the Weil symbol $[f, g]_{a}$ . Here is its definition. Let
be alocal parameter about the point , $u(a)=0$ , and let $f=c_{1}u^{k_{1}}+\ldots$ and$g=c_{2}u^{k_{2}}+\ldots$ be the leading terms of the expansion of the functions
and at$[f,g]_{a}$point.
the
The Weil symbol
is the number
For all the
points on the curve $X$ , except for afinite number of them, the Weil symbol is equal
to one. The following holds:$\Delta_{1}$$f$$\ldots$$\Delta_{n}$

$g$

$u$

$a$

$f$

$g$

$(-1)^{k_{1}k_{2}}c_{1}^{k_{2}}c_{2}^{-k_{1}}$$a$Weil’s theorem.
$\prod_{a\in X}[f,g]_{a}=1$.

Let us apply Weil’s theorem in the case that the curve $X$ coincides with the
Riemann sphere, the function
is equal to the coordinate function , and thefunction is equal to apolynomial $P$ . We will get
$f$

$x$

$g$

$\prod x(a)=[x, P]_{0}^{-1}[x, P]_{\infty}^{-1}$,where the product is taken over all nonzero roots of the polynomial $P$ . This
formula coincides with the Vieta formula.
The Vieta formula also has acompletely different interpretation.
The Newton polygon of the polynomial $P(x)=a_{n}x^{n}+\cdots+akx^{k}$ is asegment$I(n, k)$ on the real line with vertices
and , where $n>k\geq 0$ . The product of thenonzero roots of the polynomial $P$ is equal, up to asign, to the monomial
inthe coefficients an,at the vertices and of the Newton polyhedron $I(n, k)$ of
the polynomial $P$ . The coefficiententers in this monomial to the power given by
$a$

$n$

$k$

$a_{k}a_{n}^{-1}$$n$$a_{k}$$a_{n}$$k$

16

A. KHOVANSKII

the value, taken with opposite sign, of the derivative of the length of the segment$I(n, k)$ by the vertex , i.e. to the power minus one. The coefficiententers inthis monomial to the power also given by the value, taken with opposite sign, ofthe derivative of the length of the segment $I(n, k)$ by the vertex , i.e. to the power$n$$a_{k}$$k$one.
9. Parshin symbols and the product of roots.
Parshin-Kato theory gives afar-reaching generalization of Weil’s theorem. In
this theory to $(n+1)$ meromorphic functions on an -dimensional algebraic manifold$X$ and aflag of submanifolds in the manifold $X$ is associated the s0-called Parshin
symbol which generalizes the Weil symbol. According to Parshin-Kato theory, theproduct of Parshin symbols over certain flags of submanifolds also turns out to byequal to one (see [4], [5], [6]).
Let $M(P_{1}, \ldots, P_{n})$ be the product in the groupof the roots of the systemof equations (1). The computation of the point $M(P_{1}, \ldots, P_{n})$ in the groupis equivalent to the computation of the value $\chi(M(P_{1}, \ldots, P_{n}))$ of each character’at this point.
, A $=\Delta_{1}+\cdots+\Delta_{n}$ . With
Let Abe the Minkowski sum of the polyhedra
each vertexof the polyhedron
is associated an integer –the combinatorial
coefficientof the vertex .
of the polyhedron
it is possible to associate anumber
With each vertex, which we will call the Parshin symbol of the functions ,, ,at the vertex of the polyhedron A(see [2]).
$n$$(\mathbb{C}^{*})^{n}$$(\mathbb{C}^{*})^{n}$$\chi:(\mathbb{C}^{*})^{n}arrow \mathbb{C}$$\Delta_{:}$$A$$\Delta$$A$$C_{A}$$A$$\Delta$$[P_{1}, \ldots, P_{n}, \chi]$$P_{1}$$\ldots$$P_{n}$$\chi$$A$

Theorem. The following equality holds:$\chi(M(P_{1},$ \ldots ,$P_{n}))= \prod_{A\in\Delta}[P_{1},$\ldots ,$P_{n}, \chi]_{A}^{(-1)^{n}C_{A}}$.

Here the product is taken over all the vertices A of the polyhedron A.
The equality from the theorem is analogous to the interpretation of Vieta formulawith the help of Weil’s theorem. It could be explained in the framework of ParshinKato theory. Our proof, however, is elementary and does not require this theory(see [2]).

10. Volume and the product of roots.
, let
Let Abe apolyhedron in
be one of its vertices, let be the set ofthe other vertices of this polyhedron A. For every vector we can consider the
point $(A+h)$ and define the polyhedron $\Delta(A+h)$ as the convex hull of the set$L\cup(A+h)$ . For example, by definition, the polyhedron $\Delta(A+0)$ coincides withthe polyhedron .
, ,, and let
Now letbe acollection of polyhedra in
be one of. Let us consider the mixed volume $V(h)$ of thethe vertices of the polyhedron$\Delta:(A+h)$ ,,,,,polyhedra
as afunction of the vector .
Suppose that the function $V(h)$ is differentiate by at the point 0, and let $dV$
be its differential. We will call this differential the derivative of the mixed volume
by the vertex of the polyhedron
and we will denote this differential by the.
symbolOne can prove that the mixed volume of adeveloped collection of the polyhedra,,
is differentiate with respect to every vertex of every polyhedron$\mathbb{R}^{n}$

$A$

$L$

$h$

$\Delta$$\Delta_{1}$$\ldots$$\mathbb{R}^{n}$$\Delta_{n}$$A_{\dot{l}}$$\Delta_{:}$$\Delta_{1}$$\ldots$$\Delta:-1,$$\Delta_{:+1}$$\ldots$$h$$\Delta_{n}$

$h$

$A$

$d_{A:}\mathrm{V}\mathrm{o}\mathrm{l}$$\Delta_{1}$$\ldots$$\Delta_{n}$$(\Delta_{1}, \ldots, \Delta_{n})$$\Delta_{:}$

17

THREE LECTURES ON NEWTON POLYHEDRAThe Newton polyhedra are located in the spaceof characters of the group’; to every integral point
corresponds the character:
which maps the point to the number. The dual space
to the spaceis the space of one-parameter subgroups in the group; to every integral
point $on=m_{1}$ ,,corresponds the
subgroup: C’
which assigns to every nonzero number the point $x=x_{1}$ ,, , where $x_{1}=$,, $x_{n}=t^{m_{n}}$ .
For adeveloped collection of integral polyhedra,,
every derivative
is an integral covector on the space. Therefore to thederivative
corresponds aone-parameter group in the space:. For every nonzero complex number $t\neq 0$ the elementinthe groupis defined. In the new version of the formula for the product ofroots $M(P_{1}, \ldots, P_{n})$ we will use these notations.
$\mathbb{R}^{n}$$k\in \mathbb{R}^{n}$$(\mathbb{C}^{*})^{n}$$(\mathbb{C}^{*})^{n}arrow \mathbb{C}$$\chi_{k}$$x^{k}$$x$$(\mathbb{R}^{n})^{*}$$\mathbb{R}^{n}$$(\mathbb{C}^{*})^{n}$$\ldots$$m_{n}$$t^{m}$$\mathrm{o}\mathrm{n}\mathrm{e}\sim \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}$$t$$\ldots$$arrow(\mathbb{C}^{*})^{n}$$x_{n}$$t^{m_{1}}$$\ldots$$\Delta_{1}$$\ldots$$n!d_{A_{i}}\mathrm{V}\mathrm{o}\mathrm{l}$$(\Delta_{1}, \ldots, \Delta_{n})$$\Delta_{n}$$\mathbb{R}^{n}$$n!d_{A}\mathrm{V}o1$ $(\Delta_{1}, \ldots, \Delta_{n})$$t^{n!d_{A_{i}}\mathrm{V}\mathrm{o}\mathrm{l}(\Delta_{1},\ldots,\Delta_{n})}$$(\mathbb{C}^{*})^{n}$$(\mathbb{C}^{*})^{n}$

Theorem. For every system of equations $P_{1}=\cdots=P_{n}=0$ with the developed
collection of Newton polyhedra,,, up to signs the element $M(P_{1}, \ldots, P_{n})$of the groupis defined by the formula$\Delta_{1}$$\ldots$$\Delta_{n}$$(\mathbb{C}^{*})^{n}$$M(P_{1}, \ldots, P_{n})=q\prod_{1\leq i\leq n}\prod_{A_{i}\in\Delta_{\mathrm{i}}}P_{\dot{l}}(A_{i})^{-n!d_{A_{i}}\mathrm{V}\mathrm{o}\mathrm{l}(\Delta_{1},\ldots,\Delta_{n})}$,where the inner product is conducted over all the verticesof the polyhedra;
is the number equal to the coefficient of Laurent polynomialat themonomial corresponding to the vertex ;is aone-charam eter
subgroup in the groupwhich corresponds to the derivative of the mixed vol$ume$ VO1
by the vertex Ai, is an element inwhich coordinates
are equal $to\pm 1$ .
$A_{i}$$P_{i}(A_{i})$$\Delta_{i}$$P_{i}$$A_{i}$$n!d_{A_{i}}\mathrm{V}\mathrm{o}\mathrm{l}$$(\Delta_{1}, \ldots, \Delta_{n})$$(\mathbb{C}^{*})^{n}$$(\Delta_{1}, \ldots, \Delta_{n})$$q$$(\mathbb{C}^{*})^{n}$

REFERENCES[1] Gelfond O.A. and Khovanskii A.G., Newton polytopes and Grothendieck residues, Dokl. Russian Akad. Nauk 3(1996), no. 350, 298-300; English transl. in Russian Acad. Sci. Dokl.
Math.
[2] Khovanskii A.G., Newton polyhedrons, a new fomula for mixed volume, product of roots ofsystem of equations, The Arnoldfest, Proceedings of aConference in Honour of V.I. Arnold for
his Sixtieth Birthday, vol. 24, Amer. Math. Soc, Fields Institute Communicatins, Providence,
RI, 1999, PP.325-364.
[3] Khovanskii A.G., New ton polygons, curves on toric surfaces and Weil theorem, “Uspekhi
matematicheskikh nauk” 52 (1997), no. 6, 113-142.
[4] Parshin A.N., Local class field theory, Proceedings of the Steklov Institute of Mathematics(1985), no. 3.
[5] Parshin A.N., Galois cohomology and the Brauer group of local fields, Proceedings of the
Steklov Institute of Mathematics (1991), no. 4.
[6] Brylinski J.-L. and McLaughlin D. A., Characreristic classes and multidimensional reciprocity
laws, Mathematical Research Letters 3(1996), 19-30.

$a$

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF TORONTO, TORONTO, CANADA M5S3G3-rnail address: askold@math.toronto.edu$E$