Newton ’ s Method and the Convergence of Puiseux Series

Here for some natural number n, each gi ∈ C{x 1 n } with gi(0) = 0, and u(x, y) ∈ C{x 1 n , y 1 n } with u(0, 0) 6= 0. Hence the zeroes of f(x, y) are parameterized by analytic functions of one variable. (With a little more effort one can show u(x, y) ∈ C{x, y}). In the case where f is real-valued, an argument for proving a version of (1) goes back to Isaac Newton himself, as described in a 1676 letter he wrote to Oldenburg that is reproduced in [BK] p 372-375. Newton’s method produces the terms of the gi(x) through an infinite recursion, but does not show their convergence. The standard way of presenting Newton’s method entails first proving the factorization (1), and then invoking a topological argument involving Riemann surfaces to show that the resulting gi(x) are in some C{x 1 n }. We again refer to [BK] for this. (Puiseux’s original proof was somewhat different). Alternatively, one may carefully examine the algebraic properties of Newton’s algorithm as one proceeds and then, after proving the factorization (1), directly prove that the resulting gi(x) are in some C{x 1 n }; this is done in [Ca] and [Ch].