Bernstein–Kushnirenko theorem

Short description: On the number of common zeros of Laurent polynomials

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 [math]\displaystyle{ f_1= \cdots = f_n=0 }[/math] is equal to the mixed volume of the Newton polytopes of the polynomials [math]\displaystyle{ f_1, \ldots, f_n }[/math], assuming that all non-zero coefficients of [math]\displaystyle{ f_n }[/math] are generic. A more precise statement is as follows:

Statement

Let [math]\displaystyle{ A }[/math] be a finite subset of [math]\displaystyle{ \Z^n. }[/math] Consider the subspace [math]\displaystyle{ L_A }[/math] of the Laurent polynomial algebra [math]\displaystyle{ \Complex \left [ x_1^{\pm 1}, \ldots, x_n^{\pm 1} \right ] }[/math] consisting of Laurent polynomials whose exponents are in [math]\displaystyle{ A }[/math]. That is:

[math]\displaystyle{ L_A = \left \{ f \,\left|\, f(x) = \sum_{\alpha \in A} c_\alpha x^\alpha, c_\alpha \in \Complex \right \}, \right. }[/math]

where for each [math]\displaystyle{ \alpha = (a_1, \ldots, a_n) \in \Z^n }[/math] we have used the shorthand notation [math]\displaystyle{ x^\alpha }[/math] to denote the monomial [math]\displaystyle{ x_1^{a_1} \cdots x_n^{a_n}. }[/math]

Now take [math]\displaystyle{ n }[/math] finite subsets [math]\displaystyle{ A_1, \ldots, A_n }[/math] of [math]\displaystyle{ \Z^n }[/math], with the corresponding subspaces of Laurent polynomials, [math]\displaystyle{ L_{A_1}, \ldots, L_{A_n}. }[/math] Consider a generic system of equations from these subspaces, that is:

[math]\displaystyle{ f_1(x) = \cdots = f_n(x) = 0, }[/math]

where each [math]\displaystyle{ f_i }[/math] is a generic element in the (finite dimensional vector space) [math]\displaystyle{ L_{A_i}. }[/math]

The Bernstein–Kushnirenko theorem states that the number of solutions [math]\displaystyle{ x \in (\Complex \setminus 0)^n }[/math] of such a system is equal to

[math]\displaystyle{ n!V(\Delta_1, \ldots, \Delta_n), }[/math]

where [math]\displaystyle{ V }[/math] denotes the Minkowski mixed volume and for each [math]\displaystyle{ i, \Delta_i }[/math] is the convex hull of the finite set of points [math]\displaystyle{ A_i }[/math]. Clearly, [math]\displaystyle{ \Delta_i }[/math] is a convex lattice polytope; it can be interpreted as the Newton polytope of a generic element of the subspace [math]\displaystyle{ L_{A_i} }[/math].

In particular, if all the sets [math]\displaystyle{ A_i }[/math] are the same, [math]\displaystyle{ A = A_1 = \cdots = A_n, }[/math] then the number of solutions of a generic system of Laurent polynomials from [math]\displaystyle{ L_A }[/math] is equal to

[math]\displaystyle{ n! \operatorname{vol} (\Delta), }[/math]

where [math]\displaystyle{ \Delta }[/math] is the convex hull of [math]\displaystyle{ A }[/math] and vol is the usual [math]\displaystyle{ n }[/math]-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 [math]\displaystyle{ n! }[/math].

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.

See also

• Bézout's theorem for another upper bound on the number of common zeros of n polynomials in n indeterminates.