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Title

Computation of Puiseux Expansions for Algebraic Curves
Author

F. Hsu
Journal / Anthology

Master's thesis (Iowa State University)
Year: 1992
Description

In this paper, we give an algorithm to show how to construct y as a formal power series in x in a given equation f(x,y) = 0, where f is in C[x,y].

Since we can always translate (a,b) to (0,0) if (a,b) is on a curve, we may assume f(0,0) = 0 and it is enough to find roots y that tend to zero when x tends to zero.

Since we can always determine whether a curve is irreducible and work on the irreducible factors of a reducible curve to get all roots, we assume that the polynomial f is irreducible.

Also, multiple roots of a function can be detected by the vanishing of D(x), the discriminant of f(x, y) with respect to y, and determined, by finding the highest common factor of f and f', so the function worked in the algorithm is supposed to have no multiple roots in order to avoid any trouble.

We use the algorithm as a vehicle to obtain a proof that the equation f(x,y) = 0 has roots in the ring of formal power series, and to explain why this algorithm can be carried out. Also, some examples are given to explain how it works.

Finally, in the Appendix, we put a Mathematica program to implement the algorithm.
Subjects

*Mathematics > Algebra > Polynomials
*Mathematics > Calculus and Analysis > Series