Bernstein–Kushnirenko theorem

The Bernstein–Kushnirenko theorem (or Bernstein–Khovanskii–Kushnirenko (BKK) theorem[1]), proven by David Bernstein[2] and Anatoliy Kushnirenko [ru][3] 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:

Statement

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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  .

Trivia

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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

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  1. ^ Cox, David A.; Little, John; O'Shea, Donal (2005). Using algebraic geometry. Graduate Texts in Mathematics. Vol. 185 (Second ed.). Springer. ISBN 0-387-20706-6. MR 2122859.
  2. ^ Bernstein, David N. (1975), "The number of roots of a system of equations", Funkcional. Anal. i Priložen., 9 (3): 1–4, MR 0435072
  3. ^ Kouchnirenko, Anatoli G. (1976), "Polyèdres de Newton et nombres de Milnor", Inventiones Mathematicae, 32 (1): 1–31, doi:10.1007/BF01389769, MR 0419433
  4. ^ Arnold, Vladimir; et al. (2007). "Askold Georgievich Khovanskii". Moscow Mathematical Journal. 7 (2): 169–171. MR 2337876.

See also

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