Bernstein–Kushnirenko theorem

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 ${\displaystyle f_{1}=\cdots =f_{n}=0}$ is equal to the mixed volume of the Newton polytopes of the polynomials ${\displaystyle f_{1},\ldots ,f_{n}}$, assuming that all non-zero coefficients of ${\displaystyle f_{n}}$ are generic.

Statement

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

${\displaystyle L_{A}=\left\{f\,\left|\,f(x)=\sum _{\alpha \in A}c_{\alpha }x^{\alpha },c_{\alpha }\in \mathbb {C} \right\},\right.}$

where for each ${\displaystyle \alpha =(a_{1},\ldots ,a_{n})\in \mathbb {Z} ^{n}}$ we have used the shorthand notation ${\displaystyle x^{\alpha }}$ to denote the monomial ${\displaystyle x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}.}$

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

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

where each ${\displaystyle f_{i}}$ is a generic element in the (finite dimensional vector space) ${\displaystyle L_{A_{i}}.}$

The Bernstein–Kushnirenko theorem states that the number of solutions ${\displaystyle x\in (\mathbb {C} \setminus 0)^{n}}$ of such a system is equal to

${\displaystyle n!V(\Delta _{1},\ldots ,\Delta _{n}),}$

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

In particular, if all the sets ${\displaystyle A_{i}}$ are the same, ${\displaystyle A=A_{1}=\cdots =A_{n},}$ then the number of solutions of a generic system of Laurent polynomials from ${\displaystyle L_{A}}$ is equal to

${\displaystyle n!\operatorname {vol} (\Delta ),}$

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

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.^[4]