Bernstein–Kushnirenko theorem explained

The Bernstein–Kushnirenko theorem (or Bernstein–Khovanskii–Kushnirenko (BKK) theorem^[1]), proven by David Bernstein and in 1975, is a theorem in algebra. It states that the number of non-zero complex solutions of a system of Laurent polynomial equations

f _{1=} … = f _{n=0} is equal to the mixed volume of the Newton polytopes of the polynomials f _{1,} \ldots, f _{n} , assuming that all non-zero coefficients of f _{n} are generic. A more precise statement is as follows:

Statement

Let

A be a finite subset of \Z ^{n.} Consider the subspace L _{A} of the Laurent polynomial algebra \Complex \left [ x ^{\pm 1} _{1} , \ldots, x ^{\pm 1} _{n} \right ] consisting of Laurent polynomials whose exponents are in A . That is:L _{A} = \left \{ f \left| f(x) = \sum _{\alpha} c _{\alpha} x ^{\alpha,} c _{\alpha} \in \Complex \right \}, \right.

where for each

\alpha = (a _{1,} \ldots, a _{n)} \in \Z ^{n} we have used the shorthand notation x ^{\alpha} to denote the monomial x ^{a _{1}} _{1} … x ^{a _{n}} _{n} .

Now take

n finite subsets A _{1,} \ldots, A _{n} of \Z ^{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)} = … = 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 (\Complex \setminus 0 ) ^{n} of such a system is equal ton!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, \Delta _{i} is a convex lattice polytope; it can be interpreted as the Newton polytope of a generic element of the subspace L _{A _{i}} .

In particular, if all the sets

A _{i} are the same, A = A _{1} = … = A _{n,} then the number of solutions of a generic system of Laurent polynomials from L _{A} is equal ton! \operatorname{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, it becomes 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.^[2]

See also

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

Notes and References

1. Book: David A.. Cox. David A. Cox. John. Little. Donal. O'Shea. Donal O'Shea. Using algebraic geometry. Second . Graduate Texts in Mathematics. 185. Springer. 2005 . 0-387-20706-6. 2122859.
2. Askold Georgievich Khovanskii. Vladimir. Arnold. Vladimir Arnold. F.. Borodich. 1. Moscow Mathematical Journal. 7. 2 . 2007. 169–171. 2337876.