# Bernstein–Kushnirenko theorem

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# Bernstein–Kushnirenko theorem

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 inalgebra. It states that the number of non-zero complex solutions of a system ofLaurent polynomialequationsis equal to themixed volumeof theNewton polytopesof the polynomials, assuming that all non-zero coefficients ofare generic. A more precise statement is as follows:Theorem statement

`Letbe a finite subset ofConsider the subspaceof the Laurent polynomial algebraconsisting ofLaurent polynomialswhose exponents are in. That is:`

`where for eachwe have used the shorthand notationto denote the monomial`

`Now takefinite subsetswith the corresponding subspaces of Laurent polynomialsConsider a generic system of equations from these subspaces, that is:`

`where eachis a generic element in the (finite dimensional vector space)`

`The Bernstein–Kushnirenko theorem states that the number of solutionsof such a system is equal to`

`wheredenotes the Minkowskimixed volumeand for eachis theconvex hullof the finite set of pointsClearlyis aconvex lattice polytope. It can be interpreted as theNewton polytopeof a generic element of the subspace`

`In particular, if all the setsare the samethen the number of solutions of a generic system of Laurent polynomials fromis equal to`

`whereis the convex hull ofand 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 Linkopenlibrary.org*Cox, David A.; Little, John; O'Shea, Donal (2005),

*Using algebraic geometry*, Graduate Texts in Mathematics,**185**(Second ed.), Springer, ISBN 0-387-20706-6Sep 24, 2019, 6:05 PM

[2]

Citation Linkopenlibrary.orgBernstein, David N. (1975), "The number of roots of a system of equations",

*Funct. Anal. Appl.*,**9**: 183–185Sep 24, 2019, 6:05 PM

[3]

Citation Link//www.ams.org/mathscinet-getitem?mr=0419433Kouchnirenko, Anatoli G. (1976), "Polyèdres de Newton et nombres de Milnor",

*Inventiones Mathematicae*,**32**(1): 1–31, doi:10.1007/BF01389769, MR 0419433Sep 24, 2019, 6:05 PM

[4]

Citation Linkwww.ams.orgMoscow Mathematical Journal volume in honor of Askold Khovanskii (Mosc. Math. J., 7:2 (2007), 169–171)

Sep 24, 2019, 6:05 PM

[7]

Citation Linkwww.ams.orgMoscow Mathematical Journal volume in honor of Askold Khovanskii (Mosc. Math. J., 7:2 (2007), 169–171)

Sep 24, 2019, 6:05 PM

[8]

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Sep 24, 2019, 6:05 PM