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.

Statement

Let $A$ be a finite subset of $\Z^n.$ Consider the subspace $L_A$ of the Laurent polynomial algebra \Complex \left x_1^, \ldots, x_n^ \right /math> consisting of Laurent polynomials whose exponents are in $A$. That is:

:$L_A = \left \, \right.$

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

Now take $n$ finite subsets $A_1, \ldots, A_n$ of $\Z^n$, with the corresponding subspaces of Laurent polynomials, $L_, \ldots, L_.$ 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) $L_.$

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 $L_$.

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

:$n! \operatorname (\Delta),$

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.

References

See also

* Bézout's theorem for another upper bound on the number of common zeros of ' polynomials in ' indeterminates.
{{DEFAULTSORT:Bernstein-Kushnirenko theorem
Theorems in algebra
Theorems in geometry