Bernstein–Kushnirenko theorem

Bernstein–Kushnirenko theorem (also known as BKK theorem or Bernstein–Khovanskii–Kushnirenko theorem ), proven by David Bernstein and Anatoli Kushnirenko () 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:

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

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