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2.i.a. A Simplex System
Suppose P is an n-simplex which meets
the integral lattice only at its vertices.
Translating one vertex to the origin, the others are linearly independent.
(Translating corresponds to division by a monomial.)
Let M be the n by n integer matrix whose columns are
these vertices.
Taking those columns to be a basis for Rn corresponds
to a multiplicative change of coordinates
(x1,x2,...,xn)
|---->
(ya(1), ya(2),
..., ya(n)) .
with y1,y2,...,yn
non-zero complex numbers and
a(1),a(2),...,a(n)
linearly independent integer vectors.
This transforms a polynomial f with Newton polytope P
into a polynomial of the form
|
a0 +
a1y1d1+
a2y2d2+ ... +
anyndn ,
|
(2.3) |
where
d1|d2|...|dn
and
d1d2...dn
=n!VolP, the normalized volume of P.
(These di are the invariant factors of the integral matrix
M.)
A system consisting of n general polynomials of the
form (2.3) is equivalent to the system of
binomials
y1d1 = b1,
y2d2 = b2,
..., yndn = bn
,
where b1, b2, ..., bn
are some numbers.
Thus a general system whose Newton polytope is P has
n!VolP simple complex solutions.
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