Gruber P. Convex and Discrete Geometry

Zeros of Systems of Polynomial Equations,
the Minding–Kouchnirenko–Bernstein Theorem
20 Linear Optimization 335
The problem to determine the number of (positive, negative, real or complex) roots
of a polynomial equation or of a system of such equations has attracted interest
for several centuries. Highlights of the study of this problem are Descartes’s rule
of signs, the fundamental theorem of algebra, and theorems of Sturm, Routh and
Hurwitz. For systems of polynomial equations Bezout’s theorem is as follows: If the
system
(1) p1(z1,...,zd) = 0
p2(z1,...,zd) = 0
.................
pd(z1,...,zd) = 0
of d complex polynomial equations in d complex variables has only finitely many
common complex zeros (z1,...,zd), then the number of these zeros is at most the
product of the degrees of the polynomials p1,...,pd. For an elementary proof,
see the book by Cox, Little and O’Shea [228]. In general this upper estimate is
far too large. In the generic case, the following theorem of Minding–Kouchnirenko–
Bernstein gives the precise answer. We state it without proof and do not explain what
is meant by generic. For detailed information, see the book of Sturmfels [975], or the
original article of Bernstein [100].
Theorem 19.8. For generic systems of polynomial equations over C of the form (1)
the number of common solutions in (C\{0}) d is finite and equals d ! V(N p1 ,...,N pd ).
20 Linear Optimization
A linear optimization (or linear programming) problem entails minimizing or
maximizing a linear form on a convex polytope or polyhedron. A typical form is
the following:
(1) sup � c · x : x ∈ E d , Ax ≤ b � ,
where A is a real m × d matrix, c ∈ E d , b ∈ E m and the inequality is to be
understood componentwise. When writing (1), we mean the problem is to determine
the supremum and, if it is finite, to find a point at which it is attained. It turns out that
all common linear optimization problems are polynomially equivalent to one of the
form (1).
With early contributions dating back to the eighteenth century, a first vague
version of linear programming was given by Fourier [342] at the beginning of the
nineteenth century. Pertinent later results on systems of linear inequalities are due to
Gordan, Farkas, Stiemke, Motzkin and others. Linear optimization, as it is used at
present, started with the work of Kantorovich [565] which won him a Nobel prize in