Bernstein–Kushnirenko theorem

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Bernstein–Kushnirenko theorem (also known as BKK theorem or Bernstein–Khovanskii–Kushnirenko theorem [1]), proven by David Bernstein[2] and Anatoli Kushnirenko [ru][3] in 1975, is a theorem in algebra. It states that the number of non-zero complex solutions of a system of Laurent polynomial equations is equal to the mixed volume of the Newton polytopes of the polynomials , assuming that all non-zero coefficients of are generic. A more precise statement is as follows:

Theorem statement[edit]

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 for each we have used the shorthand notation to denote 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 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, it becomes an integer after multiplying by .


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]


  1. ^ *Cox, David A.; Little, John; O'Shea, Donal (2005), Using algebraic geometry, Graduate Texts in Mathematics, 185 (Second ed.), Springer, ISBN 0-387-20706-6
  2. ^ Bernstein, David N. (1975), "The number of roots of a system of equations", Funct. Anal. Appl., 9: 183–185
  3. ^ Kouchnirenko, Anatoli G. (1976), "Polyèdres de Newton et nombres de Milnor", Inventiones Mathematicae, 32 (1): 1–31, doi:10.1007/BF01389769, MR 0419433
  4. ^ Moscow Mathematical Journal volume in honor of Askold Khovanskii (Mosc. Math. J., 7:2 (2007), 169–171)