Bezouts Theorem generalization to n dimensions proof reference

Question

For my bachelors thesis in the context of total degree homotopies I am looking for a reference for a Bezout type theorem like this: Let $V\subseteq \mathbb{C}^n$ be a complex algebraic set of dimension $0$ defined by $n$ polynomials $p_1,...,p_n$ then the cardinality of $V$ is bounded by the product of the degrees of the polynomials $p_i$. For a generic choice of the Polynomials the cardinality of $V$ equals the product of the degrees.

Reference to a similar result for projective algebraic sets would work aswell, as I think I could get to the affine case from there myself

I have found reference for the first part, can't find any for the second part. Pretty sure it is true anyways as I have seen it mentioned briefly here and there.

Thanks in advance.

Answer 1

I don't know a reference where this is stated as a single theorem, but you can obtain the projective version as a combination of the following, all of which are in section 8.4 of Fulton's Intersection Theory:

• $\deg V = \deg p_1 \times \deg p_2 \times \dots \times \deg p_n $ (Proposition 8.4)
• A repeated application of a Bertini theorem (Example 8.4.12(b), though not stated precisely in this form): If $X \subset \mathbf{P}^n$ is smooth then for generic $p$ the intersection (with multiplicity) $X \cdot V(p)$ is smooth.
• If $V \subset \mathbf{P}^n$ is $0$-dimensional and smooth then $\deg V = \# V$, this follows from the definition of $\deg$ in the beginning of the section.

As you alluded to, the affine version follows by choosing a hyperplane in $\mathbf{P}^n$ that avoids $V$ to be the hyperplane at infinity.