Talk:Regular function

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[edit] Definition

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)