“Bernstein–Kushnirenko theorem”的意思、由来-开放百科全书

Bernstein–Kushnirenko theorem (also known as BKK theorem or Bernstein–Khovanskii–Kushnirenko theorem ^[1]), proven by David Bernstein^[2] and {{Interlanguage link multi|Anatoli Kushnirenko|ru|3=Кушниренко, Анатолий Георгиевич}}^[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:

Theorem statement

Let

be a finite subset of

Consider the subspace

of the Laurent polynomial algebra

consisting of Laurent polynomials whose exponents are in

. That is:

Now take

finite subsets

with the corresponding subspaces of Laurent polynomials

Consider a generic system of equations from these subspaces, that is:

The Bernstein–Kushnirenko theorem states that the number of solutions

of such a system

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

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. ^*{{citation|first1=David A.|last1=Cox|authorlink1=David A. Cox|first2= John|last2= Little|first3=Donal|last3= O'Shea|authorlink3=Donal O'Shea| title=Using algebraic geometry|edition=Second |series=Graduate Texts in Mathematics|volume= 185|publisher= Springer|year= 2005 |isbn=0-387-20706-6}}
2. ^{{citation|first=David N. |last=Bernstein|title=The number of roots of a system of equations|journal=Funct. Anal. Appl.|volume= 9 |year=1975|pages= 183–185}}
3. ^{{citation|first=Anatoli G. |last =Kouchnirenko|title=Polyèdres de Newton et nombres de Milnor|journal=Inventiones Mathematicae|volume=32 |issue=1| |year= 1976 |pages=1–31|mr=0419433|doi=10.1007/BF01389769}}
4. ^Moscow Mathematical Journal volume in honor of Askold Khovanskii (Mosc. Math. J., 7:2 (2007), 169–171)