Bernstein–Kushnirenko theorem

The Bernstein–Kushnirenko theorem (or Bernstein–Khovanskii–Kushnirenko (BKK) theorem ), proven by David Bernstein and 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 $$\Z^n.$$ Consider the subspace $$L_A$$ of the Laurent polynomial algebra $$\Complex \left [ x_1^{\pm 1}, \ldots, x_n^{\pm 1} \right ]$$ consisting of Laurent polynomials whose exponents are in $$A$$. That is:


 * $$L_A = \left \{ f \,\left|\, f(x) = \sum_{\alpha \in A} c_\alpha x^\alpha, c_\alpha \in \Complex \right \}, \right.$$

where for each $$\alpha = (a_1, \ldots, a_n) \in \Z^n $$ 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 $$\Z^n $$, 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 $$x \in (\Complex \setminus 0)^n $$ 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.