Английская Википедия:Bernstein–Kushnirenko theorem
Шаблон:Short description The Bernstein–Kushnirenko theorem (or Bernstein–Khovanskii–Kushnirenko (BKK) theorem[1]), proven by David Bernstein[2] and Шаблон:Interlanguage link multi[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 <math>f_1= \cdots = f_n=0</math> is equal to the mixed volume of the Newton polytopes of the polynomials <math>f_1, \ldots, f_n</math>, assuming that all non-zero coefficients of <math>f_n</math> are generic. A more precise statement is as follows:
Statement
Let <math>A</math> be a finite subset of <math>\Z^n.</math> Consider the subspace <math>L_A</math> of the Laurent polynomial algebra <math>\Complex \left [ x_1^{\pm 1}, \ldots, x_n^{\pm 1} \right ]</math> consisting of Laurent polynomials whose exponents are in <math>A</math>. That is:
- <math>L_A = \left \{ f \,\left|\, f(x) = \sum_{\alpha \in A} c_\alpha x^\alpha, c_\alpha \in \Complex \right \}, \right.</math>
where for each <math>\alpha = (a_1, \ldots, a_n) \in \Z^n </math> we have used the shorthand notation <math>x^\alpha</math> to denote the monomial <math> x_1^{a_1} \cdots x_n^{a_n}.</math>
Now take <math>n</math> finite subsets <math> A_1, \ldots, A_n</math> of <math>\Z^n </math>, with the corresponding subspaces of Laurent polynomials, <math>L_{A_1}, \ldots, L_{A_n}.</math> Consider a generic system of equations from these subspaces, that is:
- <math>f_1(x) = \cdots = f_n(x) = 0,</math>
where each <math>f_i</math> is a generic element in the (finite dimensional vector space) <math>L_{A_i}.</math>
The Bernstein–Kushnirenko theorem states that the number of solutions <math>x \in (\Complex \setminus 0)^n </math> of such a system is equal to
- <math> n!V(\Delta_1, \ldots, \Delta_n),</math>
where <math>V</math> denotes the Minkowski mixed volume and for each <math>i, \Delta_i</math> is the convex hull of the finite set of points <math>A_i</math>. Clearly, <math>\Delta_i</math> is a convex lattice polytope; it can be interpreted as the Newton polytope of a generic element of the subspace <math>L_{A_i}</math>.
In particular, if all the sets <math>A_i</math> are the same, <math>A = A_1 = \cdots = A_n,</math> then the number of solutions of a generic system of Laurent polynomials from <math>L_A</math> is equal to
- <math>n! \operatorname{vol} (\Delta),</math>
where <math>\Delta</math> is the convex hull of <math>A</math> and vol is the usual <math>n</math>-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 <math>n!</math>.
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
See also
- Bézout's theorem for another upper bound on the number of common zeros of Шаблон:Mvar polynomials in Шаблон:Mvar indeterminates.
