Some comments of Jonathan Korman made me realize that it would have been good to include in How Many Zeroes? an “elementary” proof of the Bernstein-Kushnirenko formula via Hilbert polynomials. This approach only yields a “weak” version of the bound since it applies only to the case that the number of solutions is finite. Nevertheless, it is a rewarding approach since this shows how it can be useful to interpret polynomials, or more generally regular functions on a variety, as linear sections after an appropriate embedding, so that the number of solutions of systems of polynomials can be interpreted as the degree of a variety, and in addition, how the geometric concept of degree of a variety can be interpreted in terms of its Hilbert polynomial, an extremely fruitful algebraic tool discovered by Hilbert which by now occupies a central role in algebraic geometry.
The following posts describe this approach:
- The first one that introduces the degree of a projective variety,
- The second post describes the connection between the degree of a projective variety and its Hilbert polynomial
- The third one proves the weak version of Kushnirenko’s formula using Hilbert polynomials, and sketches the derivation of Bernstein’s and Bézout’s formulae.