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:
![{\displaystyle \mathbb {C} \left[x_{1}^{\pm 1},\ldots ,x_{n}^{\pm 1}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd32b6be3d5d020b3bd9aba186a37608583c392)
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 0-387-20706-6
Sep 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: 183–185
Sep 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 0419433
Sep 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