2 - Puiseux' Theorem

Summary

The theorem of Puiseux states that a polynomial equation f(x, y) = 0 has a solution in which y is expressed as a power series in fractional powers of x. In this chapter we will give several versions of this theorem, of increasing sharpness. In the first section we present the classical algorithm for calculating the successive terms in the power series, and show that this does yield a solution. However, to obtain a convergent power series requires more work, and in the second section we give a different approach giving an introduction to the geometry of the situation and an existence proof for convergent power series solutions.

The next short section collects the results describing the relations between curves, their branches, tangents and multiplicities, which are basic for later chapters.

The fourth section establishes some basic properties of the rings of power series, in particular that they are unique factorisation domains, and deduces that the solutions obtained in the preceding sections must all be the same.

Solution in power series

We want to solve a polynomial equation f(x, y) = 0. There are several ways to find a solution for y in terms of x, but we begin with one which gives an effective method of calculation. For this, it will make no difference if we allow f to be a formal power series. The basis of the method of proof goes back to Newton [142].