Weierstrass's Gap Theorem

Given a succession of nonsingular points which are on a nonhyperelliptic curve of curve genus p, but are not a group of the canonical series, the number of groups of the first k which cannot constitute the group of simple poles of a rational function is p. If points next to each other are taken, then the theorem becomes: Given a nonsingular point of a nonhyperelliptic curve of curve genus p, then the orders which it cannot possess as the single pole of a rational function are p in number.

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Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 290, 1959.

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Weierstrass's Gap Theorem

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Weisstein, Eric W. "Weierstrass's Gap Theorem." From MathWorld--A Wolfram Web Resource.

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