Bernstein–Kushnirenko theorem

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, ..., f_n = 0 is equal to the mixed volume of the Newton polytopes of f₁, ..., f_n, assuming that all non-zero coefficients of f_n are generic. More precise statement is as follows:

Theorem statement

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

$L_{A}=\{f\mid f(x)=\sum _{\alpha \in A}c_{\alpha }x^{\alpha }\},$ where $c_{\alpha }\in \mathbb {C}$ and for each $\alpha =(a_{1},\ldots ,a_{n})\in \mathbb {Z} ^{n}$ we have used the shorthand notation $x^{\alpha }$ to write the monomial $x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}$.

Now take $n$ finite subsets $A_{1},\ldots ,A_{n}$ with the corresponding subspaces of Laurent polynomials $L_{A_{1}},\ldots ,L_{A_{n}}$. Consider a generic system of equations from these subspaces, that is:

$f_{1}(x)=\ldots =f_{n}(x)=0,$

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 $x\in (\mathbb {C} \setminus 0)^{n}$ of such a system is equal to $n!V(\Delta _{1},\ldots ,\Delta _{n})$, where $V$ denotes the Minkowski mixed volume and for each $i$, $\Delta _{i}$ 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 $A=A_{1}=\cdots =A_{n}$, then the number of solutions of a generic system of Laurent polynomials from $L_{A}$ is equal to $n!vol(\Delta )$ where $\Delta$ 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 $n!$.

Trivia

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.