Hamburger-Noether expansions and approximate roots of polynomials

Abstract

Let k be a field. To any pair (x,y) ~ (0,0) of elements in tk [[t]], and hence to any irreducible and residually rational power series f e k[[X,Y]], we associate a matrix, the Hamburger-Noether tableau of (x,y), which, in essence, is a description of the algorithm used to compute an element of minimal order in the integral closure of k [[x,y]] by successive quadratic transformations.Our main aim then is to reprove basic results of Abhyankar and Moh on approximate roots of polynomials and their use in the study of branches of plane curves, and to extend these results to certain situations over fields of positive characteristic where one important technical tool of Abhyankar and Moh, the NewtonPuiseux expansion of a branch, is not available.

Plane Frobenius Sandwiches

“…By [7,Theorem 5.3], we conclude that there exists ay G tf with one place at infinity, of degree cx/cs, such that the projective curve defined by y meets C at infinity with intersection multiplicity rh. Therefore the intersection of u andy on Spectf is c\/cs -rh = 1.…”

Plane Frobenius sandwiches

“…The key step in the proof of the polynomial case of the Theorem rests on recent work of Peter Russell [7,Theorem 5.3] on approximate roots of polynomials, work which may be regarded as extending earlier results of Abhyankar and Moh [1,2]. The proof of the Theorem for power series is comparatively trivial.…”

“…Since Noether's Palermo paper of 1890, a vast amount has been published about HN and about the less generally applicable, but much older, NewtonPuiseux expansion. We refer only to [7] for what is needed here; in the writer's opinion it represents the latest word on the subject, both in its content and in the particular way in which it proceeds from the basics to some of the deepest results in the theory of plane, and algebroid plane, curves. The reader is directed to the references given in [7] and to the introduction to that paper, where among other things is indicated the rôle played by the work of Moh and of Abhyankar, over the past fifteen years or so, in the revival of interest in the subject.…”

“…Defining it either way, we have (c) Given coprime irreducible/, g G F, the intersection multiplicity ,(/, g) = dimr/(/T + gT) = 2 PA + min{Psc's, p'scs), k i<s where p¡, c¡, a¡ and p\, c'¡, a-are the entries of HN(x, y;f) and HN(x, y;g) respectively, and the integer s > 1 is determined by p¡c¡ -p'ici and a, = a'¡ for i < s, Psc's ^ Pscs or as ^ a's-(^ee Ü ' 3-3]. Note that our s is s + 1, in [7].) (d) Keep the notation of (c).…”

Relative Frobenius of Plane Singularities

“…Define x^, yoo£ tk [[t\] by ' (1/x, y/x), if ordx < ordy, (l/y,x/y), ifordx>ord>', (1/x, y/x -a), if ordx = ordy, and a = constant term of y/x. Then a description of the singularity of e at infinity is given by the characteristic sequence [xx , y^] of xx and yx , as defined in [3] or [6]. On the other hand let (x ,y) = (x,yp) and consider (x'^, j/j .…”