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2.i.a. A Simplex System
Suppose P is an nsimplex 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 R^{n} corresponds
to a multiplicative change of coordinates
(x_{1},x_{2},...,x_{n})
>
(y^{a(1)}, y^{a(2)},
..., y^{a(n)}) .
with y_{1},y_{2},...,y_{n}
nonzero 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
a_{0} +
a_{1}y_{1}^{d1}+
a_{2}y_{2}^{d2}+ ... +
a_{n}y_{n}^{dn} ,

(2.3) 
where
d_{1}d_{2}...d_{n}
and
d_{1}d_{2}...d_{n}
=n!VolP, the normalized volume of P.
(These d_{i} 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
y_{1}^{d1} = b_{1},
y_{2}^{d2} = b_{2},
..., y_{n}^{dn} = b_{n}
,
where b_{1}, b_{2}, ..., b_{n}
are some numbers.
Thus a general system whose Newton polytope is P has
n!VolP simple complex solutions.
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