Bernstein–Kushnirenko theorem

Bernstein–Kushnirenko theorem (also known as BKK theorem or Bernstein–Khovanskii–Kushnirenko theorem ), proven by David Bernstein  and Anatoli Kushnirenko  in 1975, is a theorem in algebra. It claims that the number of non-zero complex solutions of a system of Laurent polynomial equations f1 = 0, ..., fn = 0 is equal to the mixed volume of the Newton polytopes of f1, ..., fn, assuming that all non-zero coefficients of fn 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. 

References

• David A. Cox; J. Little; D. O'Shea Using algebraic geometry.
Second edition. Graduate Texts in Mathematics, 185. Springer, 2005. xii+572 pp. ISBN 0-387-20706-6
1. D. N. Bernstein, "The number of roots of a system of equations", Funct. Anal. Appl. 9 (1975), 183–185
2. A. G. Kouchnirenko, "Polyhedres de Newton et nombres de Milnor", Invent. Math. 32 (1976), 1–31
3. Moscow Mathematical Journal volume in honor of Askold Khovanskii (Mosc. Math. J., 7:2 (2007), 169–171)