The Bernstein–Kushnirenko theorem (or Bernstein–Khovanskii–Kushnirenko (BKK) theorem ), proven by David Bernstein and Anatoliy Kushnirenko 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 .
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