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 Theorem## Cite this as:

Weisstein, Eric W. "Weierstrass's Gap Theorem."
From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/WeierstrasssGapTheorem.html