Suppose we have an integral (reduced and irreducible) affine curve X over an algebraically closed field k. Suppose t = f g is a rational function on X such that there exists an n ∈ N such that t n = ( f g ) n is a regular function, i.e., t n ∈ Γ ( X , O X ) (the coordinate ring or ring of regular functions on X). Does this imply that t itself is a regular function?Note that if the curve is non-singular, this is immediate, as in this case Γ ( X , O X ) is a UFD and hence b n | a n implies b | a in Γ ( X , O X ). But is this true in general for an integral affine curve?

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Suppose we have an integral (reduced and irreducible) affine curve   X over an algebraically closed field   k. Suppose   t  =      f    g   is a rational function on   X such that there exists an   n  ∈      N   such that       t    n    =  (      f    g        )    n   is a regular function, i.e.,       t    n    ∈  Γ  (  X  ,            O              X        ) (the coordinate ring or ring of regular functions on   X). Does this imply that   t itself is a regular function?Note that if the curve is non-singular, this is immediate, as in this case   Γ  (  X  ,            O              X        ) is a UFD and hence       b    n        |        a    n   implies   b      |    a in   Γ  (  X  ,            O              X        ). But is this true in general for an integral affine curve?

Ada Harrington · Accepted Answer

What about   X  =  V  (      y    2    −      x    3    )  ⊂            A        2   and   t  =  y      /    x? Then       t    2    =  x is regular, but (unless I&#039;m being dumb)   t isn&#039;t regular itself.

Suppose we have an integral (reduced and irreducible) affine curve X over an algebraically clo

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