Next: 2.ii.b. Regular Triangulations

## 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.

Next: 2.ii.b. Regular Triangulations