(I asked this question on MSE, but someone suggested it would be better asked here.)

I'm a fan of the Bernstein-Khovanskii-Kushnirenko theorem (that the number of solutions in $(\mathbb{C}^*)^n$ to a generic system of Laurent polynomials is the mixed volume of the polynomials' Newton polytopes), and I was intrigued by the claim in this tribute to Askold Khovanskii that:

"Askold found various proofs of Bernstein's formula; the number of proofs that he has obtained up to now is about fifteen...and each of them can be explained to an advanced high-school student in half an hour."

Having only seen one proof of the theorem as part of an introduction to toric geometry that went somewhat over my head at the time, I'm curious how these proofs work. I'm especially interested in the "can be explained to an advanced high-school student" aspect, as I'm wondering if this might be good material for a math club talk to undergraduates or a week-long course for advanced high schoolers.

However, I have no idea where any of these proofs would be documented, if they even are documented; some poking around on Math Reviews didn't turn up much. It also seems possible that some or all of them would be in Russian, which I can't read. Does anyone have suggestions for where I might find such an elementary proof or proofs of the BKK theorem?

Fewnomials", whose English translation was published in 1991 by the American Mathematical Society: a proof of the BKK theorem appears as a corollary of Theorem 2 in §3.12. $\endgroup$ – Daniele Tampieri Feb 19 '19 at 19:55