Talk:AF+BG theorem

Importance of the theorem

I was surprised to learn this theorem from Wikipedia, because I have never heard of it in thirty years of research on polynomials and related algebraic geometry (I guess that I have skip this chapter when I have read Fulton a long time ago). Thinking a little about, I convince me that it is an easy consequence of a more general well known result. However this being WP:original research, I may not insert it in the article. However, as it may be interesting to the reader, I put it here.

The general result: let F₁, ..., F_k be k homogeneous polynomials in n + 1 variables such that the projective hypersurfaces they define intersect in a finite number of points (in the complex projective space P_n). This is equivalent to saying that the F_i generate an ideal of dimension one. Then a homogeneous polynomial belongs to the ideal generated by the F_i if and only if it belongs to it locally at each point of the intersection.

Sketched proof: By choosing an hyperplane "at infinity" that does not pass through any of the points of intersection, one may reduce the problem to a similar affine problem involving non-homogeneous polynomials in n variables that generate a zero-dimensional ideal I. This ideal being zero-dimensional, the algebra k[x₁, ..., x_n/I is zero-dimensional and is therefore a commutative Artinian ring. This implies that it is a direct product of local rings and the theorem follows immediately.

IMHO, this shows that this theorem is not, nowadays, a fundamental theorem, and that the rating of "low importance" is perfectly correct. D.Lazard (talk) 11:25, 22 October 2013 (UTC)