Bernstein–Kushnirenko theorem

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 $f_1=\cdots =f_n=0$ is equal to the mixed volume of the Newton polytopes of such polynomials, assuming that all nonzero coefficients of $f_n$ are generic.

David N. Bernstein, circa 1975

It was proven by David Bernstein and Template:Interlanguage link multi in 1975. Askold Khovanskii has found about 15 different proofs of this theorem.

Statement

Let $A$ be a finite subset of ${\mathbb{Z}}^n.$ Consider the subspace $L_A$ of the Laurent polynomial algebra $ℂ[x_1^{\pm 1},\dots ,x_n^{\pm 1}]$ consisting of Laurent polynomials whose exponents are in $A$. That is:

$L_A=\{f|f(x)=\sum _{\alpha \in A}{c}_{\alpha }{x}^{\alpha },c_{\alpha }\in ℂ\},$

where for each $\alpha =(a_1,\dots ,a_n)\in {\mathbb{Z}}^n$ we have used the shorthand notation $x^{\alpha }$ to denote the monomial $x_1^{a_1}\cdots x_n^{a_n}.$

Now take $n$ finite subsets $A_1,\dots ,A_n$ of ${\mathbb{Z}}^n$, with the corresponding subspaces of Laurent polynomials, $L_{A_1},\dots ,L_{A_n}.$ Consider a generic system of equations from these subspaces, that is:

$f_1(x)=\cdots =f_n(x)=0,$

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 $x\in (ℂ\setminus 0{)}^n$ of such a system is equal to

$n!V({\mathrm{\Delta }}_1,\dots ,{\mathrm{\Delta }}_n),$

where $V$ denotes the Minkowski mixed volume and for each $i,{\mathrm{\Delta }}_i$ is the convex hull of the finite set of points $A_i$. Clearly, ${\mathrm{\Delta }}_i$ is a convex lattice polytope; it can be interpreted as the Newton polytope of a generic element of the subspace $L_{A_i}$.

In particular, if all the sets $A_i$ are the same, $A=A_1=\cdots =A_n,$ then the number of solutions of a generic system of Laurent polynomials from $L_A$ is equal to

$n!\mathrm{vol}\mathrm{\Delta },$

where $\mathrm{\Delta }$ is the convex hull of $A$ and $\mathrm{vol}$ is the usual $n$-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 $n!$.

See also

• Bézout's theorem for another upper bound on the number of common zeros of Template:Mvar polynomials in Template:Mvar indeterminates.