Talk:Regular function

Definition

This page seems unnecessarily complicated. Example:

For example, if V is the affine line over K, the regular functions on V make up a commutative ring, under pointwise multiplication of functions, isomorphic with the polynomial ring in one variable over K. In other words, the regular functions are just polynomials in some natural parameter on the affine line.

That is ridiculous, and not helpful to anyone. —Preceding unsigned comment added by 128.173.95.192 (talk) 18:26, 6 September 2010 (UTC)

FYI, the definition given here:

In mathematics, a regular function in the sense of algebraic geometry is an everywhere-defined, polynomial function on an algebraic variety V with values in the field K over which V is defined.

seems to be in conflict with the definition given in Joe Harris, "Algebraic geometry". Harris defines a regular function on an open set of the Zariski topology, defining it as a rational function, with the denominator non-vanishing on the open set. He then presents a lemma, that the ring of functions regular at every point of the variety is just the coordinate ring.

Finally, one more lemma: that if U is the open set in the Zariski topology generated by a polynomial f, then the ring of regular functions on U is the (I hope I got this right) the polynomials over the coordinate ring in 1/f. Not clear to me if this boils down to the same thing, I'm just learning this stuff. linas 22:29, 24 December 2005 (UTC)

Besides, I thought there was a definition of regular function (complex analysis) which just says f is regular at a point p if it doesn't have a pole/singularity there, but does not otherwise require f to be free of poles, i.e. regular is a synonym for meromorphic. (if I remember correctly; I have a habit of mis-remembering.). Ah yes,

the mathworld def of regular function. linas 22:57, 24 December 2005 (UTC)