Bernstein–Kushnirenko theorem (also known as BKK theorem or Bernstein–Khovanskii–Kushnirenko theorem ), proven by David Bernstein and Anatoli Kushnirenko in 1975, is a theorem in algebra. It claims that the number of non-zero complex solutions of a system of Laurent polynomial equations f1 = 0, ..., fn = 0 is equal to the mixed volume of the Newton polytopes of f1, ..., fn, assuming that all non-zero coefficients of fn are generic. More precise statement is as follows:
Let
A
be a finite subset of
Z
n
. Consider the subspace
L
A
of the Laurent polynomial algebra
C
[
x
1
±
1
,
…
,
x
n
±
1
]
consisting of Laurent polynomials whose exponents are in
A
. That is:
L
A
=
{
f
∣
f
(
x
)
=
∑
α
∈
A
c
α
x
α
}
,
where
c
α
∈
C
and for each
α
=
(
a
1
,
…
,
a
n
)
∈
Z
n
we have used the shorthand notation
x
α
to write the monomial
x
1
a
1
⋯
x
n
a
n
.
Now take
n
finite subsets
A
1
,
…
,
A
n
with the corresponding subspaces of Laurent polynomials
L
A
1
,
…
,
L
A
n
. Consider a generic system of equations from these subspaces, that is:
f
1
(
x
)
=
…
=
f
n
(
x
)
=
0
,
where each
f
i
is a generic element in the (finite dimensional vector space)
L
A
i
.
The Bernstein–Kushnirenko theorem states that the number of solutions
x
∈
(
C
∖
0
)
n
of such a system is equal to
n
!
V
(
Δ
1
,
…
,
Δ
n
)
, where
V
denotes the Minkowski mixed volume and for each
i
,
Δ
i
is the convex hull of the finite set of points
A
i
. Clearly
A
i
is a convex lattice polytope. It can be interpreted as the Newton polytope of a generic element of generic element of the subspace
L
A
i
.
In particular, if all the sets
A
i
are the same
A
=
A
1
=
⋯
=
A
n
, then the number of solutions of a generic system of Laurent polynomials from
L
A
is equal to
n
!
v
o
l
(
Δ
)
where
Δ
is the convex hull of
A
and vol is the usual
n
-dimensional Euclidean volume. Note that even though the volume of a lattice polytope is not necessarily an integer but it is an integer after multiplying by
n
!
.
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.
