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, ..., f_n = 0 is equal to the mixed volume of the Newton polytopes of f₁, ..., f_n, assuming that all non-zero coefficients of f_n 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 each $f_i$ 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 $n! V(\Delta_1, \ldots, \Delta_n)$, 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

1. • 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
2. D. N. Bernstein, "The number of roots of a system of equations", Funct. Anal. Appl. 9 (1975), 183–185
3. A. G. Kouchnirenko, "Polyhedres de Newton et nombres de Milnor", Invent. Math. 32 (1976), 1–31
4. Moscow Mathematical Journal volume in honor of Askold Khovanskii (Mosc. Math. J., 7:2 (2007), 169–171)