Bernstein–Kushnirenko theorem

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The Bernstein–Kushnirenko theorem (or Bernstein–Khovanskii–Kushnirenko (BKK) theorem [1]), proven by David Bernstein[2] and Anatoliy 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:


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 of , 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. Vol. 185 (Second ed.). Springer. ISBN 0-387-20706-6. MR 2122859.
  2. ^ Bernstein, David N. (1975), "The number of roots of a system of equations", Funkcional. Anal. i Priložen., 9 (3): 1–4, MR 0435072
  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. ^ Arnold, Vladimir; et al. (2007). "Askold Georgievich Khovanskii". Moscow Mathematical Journal. 7 (2): 169–171. MR 2337876.