Algebraic Geometry and its Applications pp 19-84 | Cite as

# Square-root Parametrization of Plane Curves

Chapter

## Abstract

By calculating the genus of an irreducible algebraic plane curve of degree *n* in terms of its singularities, we see that, counted properly, the curve can have at most \(\frac{{(n - 1)(n - 2)}}
{2}\) double points, and it can be rationally parametrized iff this maximum is reached. If the maximum falls short by one or two, then the curve can still be parametrized by square-roots of rational functions. Such a square-root parametrization is used for factoring certain bivariate polynomials over a finite field.

### Keywords

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### References

- [1]S. S. Abhyankar,
*Coverings of algebraic curves*, American Journal of Mathematics, vol 79, 1957, pages 825–856.MathSciNetMATHCrossRefGoogle Scholar - [2]S. S. Abhyankar,
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*Galois theory on the line in nonzero characteristic*, Bulletin of the American Mathematical Society, (to appear).Google Scholar - [5]S. S. Abhyankar and C. Bajaj,
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*Springer Lecture Notes In Computer Science*, vol 358, 1989, pages 274–284.MathSciNetGoogle Scholar - [7]J-P. Serre,
*A letter as an appendix to the square-root paper of Abhyankar*, These Proceedings.Google Scholar

## Copyright information

© Springer-Verlag New York, Inc. 1994