The Bernstein–Kushnirenko theorem, also called Bernstein–Khovanskii–Kushnirenko theorem (BKK theorem), states that the number of nonzero complex solutions of a system of Laurent polynomial equations f 1 = ⋯ = f n = 0 {\displaystyle f_{1}=\cdots =f_{n}=0} is equal to the mixed volume of the Newton polytopes of such polynomials, assuming that all nonzero coefficients of f n {\displaystyle f_{n}} are generic.
It was proven by David Bernstein and Anatoliy Kushnirenko in 1975. Askold Khovanskii has found about 15 different proofs of this theorem.
Statement Let A {\displaystyle A} be a finite subset of Z n . {\displaystyle \mathbb {Z} ^{n}.} Consider the subspace L A {\displaystyle L_{A}} of the Laurent polynomial algebra C [ x 1 ± 1 , … , x n ± 1 ] {\displaystyle \mathbb {C} \!\left[x_{1}^{\pm 1},\ldots ,x_{n}^{\pm 1}\right]} consisting of Laurent polynomials whose exponents are in A {\displaystyle A} . That is:
L A = { f | f ( x ) = ∑ α ∈ A c α x α , c α ∈ C } , {\displaystyle L_{A}={\biggl \{}f\mathrel {\bigg |} f(x)=\sum _{\alpha \in A}c_{\alpha }x^{\alpha },c_{\alpha }\in \mathbb {C} {\biggr \}},}
where for each α = ( a 1 , … , a n ) ∈ Z n {\displaystyle \alpha =(a_{1},\ldots ,a_{n})\in \mathbb {Z} ^{n}} we have used the shorthand notation x α {\displaystyle x^{\alpha }} to denote the monomial x 1 a 1 ⋯ x n a n . {\displaystyle x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}.}
Now take n {\displaystyle n} finite subsets A 1 , … , A n {\displaystyle A_{1},\ldots ,A_{n}} of Z n {\displaystyle \mathbb {Z} ^{n}} , with the corresponding subspaces of Laurent polynomials, L A 1 , … , L A n . {\displaystyle L_{A_{1}},\ldots ,L_{A_{n}}.} Consider a generic system of equations from these subspaces, that is:
f 1 ( x ) = ⋯ = f n ( x ) = 0 , {\displaystyle f_{1}(x)=\cdots =f_{n}(x)=0,}
where each f i {\displaystyle f_{i}} is a generic element in the (finite-dimensional vector space) L A i . {\displaystyle L_{A_{i}}.}
The Bernstein–Kushnirenko theorem states that the number of solutions x ∈ ( C ∖ 0 ) n {\displaystyle x\in (\mathbb {C} \setminus 0)^{n}} of such a system is equal to
n ! V ( Δ 1 , … , Δ n ) , {\displaystyle n!V(\Delta _{1},\ldots ,\Delta _{n}),}
where V {\displaystyle V} denotes the Minkowski mixed volume and for each i , Δ i {\displaystyle i,\Delta _{i}} is the convex hull of the finite set of points A i {\displaystyle A_{i}} . Clearly, Δ i {\displaystyle \Delta _{i}} is a convex lattice polytope; it can be interpreted as the Newton polytope of a generic element of the subspace L A i {\displaystyle L_{A_{i}}} . In particular, if all the sets A i {\displaystyle A_{i}} are the same, A = A 1 = ⋯ = A n , {\displaystyle A=A_{1}=\cdots =A_{n},} then the number of solutions of a generic system of Laurent polynomials from L A {\displaystyle L_{A}} is equal to
n ! vol Δ , {\displaystyle n!\operatorname {vol} \Delta ,}
where Δ {\displaystyle \Delta } is the convex hull of A {\displaystyle A} and vol {\displaystyle \operatorname {vol} } is the usual n {\displaystyle n} -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 n ! {\displaystyle n!} .
References
See also Bézout's theorem for another upper bound on the number of common zeros of n polynomials in n indeterminates.