Bernstein–Kushnirenko theorem

The Bernstein–Kushnirenko theorem (or Bernstein–Khovanskii–Kushnirenko (BKK) theorem), proven by David Bernstein and Anatoliy Kushnirenko in 1975, is a theorem in algebra. It states that the number of non-zero complex solutions of a system of Laurent polynomial equations ${\displaystyle f_{1}=\cdots =f_{n}=0}$ is equal to the mixed volume of the Newton polytopes of the polynomials ${\displaystyle f_{1},\ldots ,f_{n}}$, assuming that all non-zero coefficients of ${\displaystyle f_{n}}$ are generic. A more precise statement is as follows: