Bernstein–Kushnirenko theorem

The Bernstein–Kushnirenko theorem (or Bernstein–Khovanskii–Kushnirenko (BKK) theorem), proven by David Bernstein and Anatoliy Kushnirenko in 1975, is a theorem in algebra. It states that the number of non-zero 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 the polynomials $f_{1},\ldots ,f_{n}$, assuming that all non-zero coefficients of $f_{n}$ are generic. A more precise statement is as follows:

Statement

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

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

where for each $\alpha =(a_{1},\ldots ,a_{n})\in \mathbb {Z} ^{n}$ 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 $A_{1},\ldots ,A_{n}$ of $\mathbb {Z} ^{n}$, 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:

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

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 $x\in (\mathbb {C} \setminus 0)^{n}$ 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, $\Delta _{i}$ 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

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

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, it becomes 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.

See also

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