Next: 2.ii. The Polyhedral Homotopy Algorithm
Up: 2. Polynomial Systems
2.i. Sparse Polynomial Systems
 The BKK Bound
 An Example
Perhaps the most obvious structure of a system of n
polynomials in n variables
f_{1}(x_{1},...,x_{n})
=
f_{2}(x_{1},...,x_{n})
= ... =
f_{n}(x_{1},...,x_{n})
= 0 ,

(2.1) 
is the list of total degrees of the polynomials
f_{1}, f_{2}, ..., f_{n}.
For such a system, we have the degree or Bézout upper bound, which is a
consequence of the refined Bézout Theorem of Fulton and
MacPherson [Fu1, §1.23].
Theorem 2.1 (Bézout Bound)
The system (
2.1), where the polynomial
f_{i} has total degree
d_{i} := deg
f_{i}, has at most
d_{1}d_{2}...
d_{n}
isolated complex solutions.
The Bézout bound on the number of real solutions is sharp.
For example, if
f_{i} = (x_{i}1)
(x_{i}2)...(x_{i}d_{i}) 
(2.2) 
then the system (2.1) has
d_{1}d_{2}...d_{n}
real solutions.
The reader is invited to construct systems with the minimum possible
number (0 or 1) of real solutions.
A system of polynomial equations with only simple solutions, but with fewer
solutions than the Bézout bound is called deficient.
For example, fewer monomials in the polynomials lead to fewer solutions.
We make this idea more precise.
A monomial (or rather its exponent vector) is a point of
N^{n}.
The convex hull of the monomials in f
is its Newton polytope,
New(f), a polytope with integral vertices.
The terms of f are indexed by the lattice points
in its Newton polytope.
Figure 1 displays the monomials (dots)
and Newton polytopes of the polynomials
f  = 
1 + x  y + xyz^{4}

g  = 
1 + x  y + 3z 
z^{3} +
2z^{4}

Figure 1:
Newton Polytopes of f and g.

Subsections
 2.i.a. The BKK Bound
 2.i.b. An Example
Next: 2.ii. The Polyhedral Homotopy Algorithm
Up: 2. Polynomial Systems