# Bernstein–Kushnirenko theorem

**Bernstein–Kushnirenko theorem** (also known as **BKK theorem** or **Bernstein–Khovanskii–Kushnirenko theorem** ^{[1]}), proven by David Bernstein ^{[2]} and Anatoli Kushnirenko ^{[3]} in 1975, is a theorem in algebra. It claims that the number of non-zero complex solutions of a system of Laurent polynomial equations *f*_{1} = 0, ..., *f*_{n} = 0
is equal to the mixed volume of the Newton polytopes of *f*_{1}, ..., *f*_{n}, assuming that all non-zero coefficients of *f _{n}* are generic.
More precise statement is as follows:

## Theorem statement

Let be a finite subset of . Consider the subspace of the Laurent polynomial algebra consisting of Laurent polynomials whose exponents are in . That is:

where and for each we have used the shorthand notation to write the monomial .

Now take finite subsets with the corresponding subspaces of Laurent polynomials . Consider a generic system of equations from these subspaces, that is:

where each is a generic element in the (finite dimensional vector space) .

The Bernstein–Kushnirenko theorem states that the number of solutions of such a system is equal to , where denotes the Minkowski mixed volume and for each , is the convex hull of the finite set of points . Clearly is a convex lattice polytope. It can be interpreted as the Newton polytope of a generic element of generic element of the subspace .

In particular, if all the sets are the same , then the number of solutions of a generic system of Laurent polynomials from is equal to where is the convex hull of and vol is the usual -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 .

## 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]}

## References

- ↑
- David A. Cox; J. Little; D. O'Shea Using algebraic geometry.

- ↑ D. N. Bernstein, "The number of roots of a system of equations",
*Funct. Anal. Appl.*9 (1975), 183–185 - ↑ A. G. Kouchnirenko, "Polyhedres de Newton et nombres de Milnor",
*Invent. Math.*32 (1976), 1–31 - ↑ Moscow Mathematical Journal volume in honor of Askold Khovanskii (Mosc. Math. J., 7:2 (2007), 169–171)