Given a succession of nonsingular points which are on a nonhyperelliptic curve of curve genus 
, but are not a group of the canonical series, the number of
groups of the first 
which cannot constitute the group of simple poles
of a rational function is 
. If points next to each other are taken, then the theorem
becomes: Given a nonsingular point of a nonhyperelliptic curve of curve
genus 
,
then the orders which it cannot possess as the single pole of a rational
function are 
in number.
Weierstrass's Gap Theorem
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References
Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 290, 1959.Referenced on Wolfram|Alpha
Weierstrass's Gap TheoremCite this as:
Weisstein, Eric W. "Weierstrass's Gap Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/WeierstrasssGapTheorem.html