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 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 A be a finite subset of \( \mathbb{Z}^n \). Consider the subspace L_A of the Laurent polynomial algebra \( \mathbb{C}[x_1^{\pm 1}, \ldots, x_n^{\pm 1}] \) consisting of Laurent polynomials whose exponents are in A. That is:

\(L_A = \{ f \mid f(x) = \sum_{\alpha \in A} c_\alpha x^\alpha \} \) , where \(c_\alpha \in \mathbb{C} \) and for each \(\alpha = (a_1, \ldots, a_n) \in \mathbb{Z}^n \) we have used the shorthand notation \(x^\alpha \) to write the monomial \(x_1^{a_1} \cdots x_n^{a_n} . \)

Now take n finite subsets \(A_1, \ldots, A_n \) with the corresponding subspaces of Laurent polynomials \(L_{A_1}, \ldots, L_{A_n} \) . Consider a generic system of equations from these subspaces, that is:

\(f_1(x) = \ldots = f_n(x) = 0, \)

where eachv \( f_i \) is a generic element in the (finite dimensional vector space) \( L_{A_i}. \)

The Bernstein–Kushnirenko theorem states that the number of solutions\( x \in (\mathbb{C} \setminus 0)^n \) of such a system is equal to \( n! V(\Delta_1, \ldots, V_n) \) , where V denotes the Minkowski mixed volume and for each \( i, \Delta_i \) is the convex hull of the finite set of points \( A_i \) . Clearly \( A_i \) is a convex lattice polytope. It can be interpreted as the Newton polytope of a generic element of generic element of the subspace \( L_{A_i}. \)

In particular, if all the sets \( A_i \) are the same \( A = A_1 = \cdots = A_n \) , then the number of solutions of a generic system of Laurent polynomials from L_A is equal to \( n! vol(\Delta) \) where \( \Delta \) is the convex hull of A and vol is the usual n-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 n!.

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. Second edition. Graduate Texts in Mathematics, 185. Springer, 2005. xii+572 pp. ISBN 0-387-20706-6
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)

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