The Bernstein–Kushnirenko theorem (or Bernstein–Khovanskii–Kushnirenko (BKK) theorem[1]), proven by David Bernstein[2] and Anatoliy Kushnirenko [ru][3] in 1975, is a theorem in algebra.
It states that the number of non-zero complex solutions of a system of Laurent polynomial equations
is equal to the mixed volume of the Newton polytopes of the polynomials
, assuming that all non-zero coefficients of
A more precise statement is as follows: Let
be a finite subset of
of the Laurent polynomial algebra
consisting of Laurent polynomials whose exponents are in
α = (
we have used the shorthand notation
α
to denote the monomial
finite subsets
, with the corresponding subspaces of Laurent polynomials,
Consider a generic system of equations from these subspaces, that is: where each
is a generic element in the (finite dimensional vector space)
The Bernstein–Kushnirenko theorem states that the number of solutions
of such a system is equal to where
denotes the Minkowski mixed volume and for each
is the convex hull of the finite set of points
is a convex lattice polytope; it can be interpreted as the Newton polytope of a generic element of the subspace
then the number of solutions of a generic system of Laurent polynomials from
is the convex hull of
and vol is the usual
-dimensional Euclidean volume.
Note that even though the volume of a lattice polytope is not necessarily an integer, it becomes an integer after multiplying by
Kushnirenko's name is also spelt Kouchnirenko.
David Bernstein is a brother of Joseph Bernstein.
Askold Khovanskii has found about 15 different proofs of this theorem.