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 in algebra. It claims that the number of non-zero complex solutions of a system of Laurent polynomial equations f₁ = 0, ..., f_n = 0 is equal to the mixed volume of the Newton polytopes of f₁, ..., f_n, assuming that all non-zero coefficients of f_n are generic. More precise statement is as follows:

Theorem statement

Let $A$ be a finite subset of ${\mathbb {Z}}^{n}$ . Consider the subspace $L_{A}$ of the Laurent polynomial algebra ${\mathbb {C}}[x_{1}^{{\pm 1}},\ldots ,x_{n}^{{\pm 1}}]$ consisting of Laurent polynomials whose exponents are in $A$ . That is:

$L_{A}=\{f\mid f(x)=\sum _{{\alpha \in A}}c_{\alpha }x^{\alpha }\},$ where $c_{\alpha }\in {\mathbb {C}}$ and for each $\alpha =(a_{1},\ldots ,a_{n})\in {\mathbb {Z}}^{n}$ we have used the shorthand notation $x^{\alpha }$ to write the monomial $x_{1}^{{a_{1}}}\cdots x_{n}^{{a_{n}}}$ .

Now take $n$ finite subsets $A_{1},\ldots ,A_{n}$ with the corresponding subspaces of Laurent polynomials $L_{{A_{1}}},\ldots ,L_{{A_{n}}}$ . Consider a generic system of equations from these subspaces, that is:

$f_{1}(x)=\ldots =f_{n}(x)=0,$

where each $f_{i}$ is a generic element in the (finite dimensional vector space) $L_{{A_{i}}}$ .

The Bernstein–Kushnirenko theorem states that the number of solutions $x\in ({\mathbb {C}}\setminus 0)^{n}$ of such a system is equal to $n!V(\Delta _{1},\ldots ,\Delta _{n})$ , where $V$ denotes the Minkowski mixed volume and for each $i$ , $\Delta _{i}$ is the convex hull of the finite set of points $A_{i}$ . Clearly $A_{i}$ is a convex lattice polytope. It can be interpreted as the Newton polytope of a generic element of generic element of the subspace $L_{{A_{i}}}$ .

In particular, if all the sets $A_{i}$ are the same $A=A_{1}=\cdots =A_{n}$ , then the number of solutions of a generic system of Laurent polynomials from $L_{A}$ is equal to $n!vol(\Delta )$ where $\Delta$ is the convex hull of $A$ and vol is the usual $n$ -dimensional Euclidean volume. Note that even though the volume of a lattice polytope is not necessarily an integer but it is an integer after multiplying by $n!$ .

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

↑
- David A. Cox; J. Little; D. O'Shea Using algebraic geometry.
Second edition. Graduate Texts in Mathematics, 185. Springer, 2005. xii+572 pp. ISBN 0-387-20706-6
↑ D. N. Bernstein, "The number of roots of a system of equations", Funct. Anal. Appl. 9 (1975), 183–185
↑ A. G. Kouchnirenko, "Polyhedres de Newton et nombres de Milnor", Invent. Math. 32 (1976), 1–31
↑ Moscow Mathematical Journal volume in honor of Askold Khovanskii (Mosc. Math. J., 7:2 (2007), 169–171)

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