 
Summary: AN EFFECTIVE WEIERSTRASS DIVISION THEOREM
MATTHIAS ASCHENBRENNER
ABSTRACT. We prove an effective Weierstrass Division Theorem for algebraic restricted
power series with padic coefficients. The complexity of such power series is measured
using a certain height function on the algebraic closure of the field of rational functions
over Q. The paper includes a construction of this height function, following an idea of
Kani. We apply the effective Weierstrass Division Theorem to obtain a numbertheoretic
criterion for membership in ideals of polynomial rings with integer coefficients.
CONTENTS
Introduction 1
1. Absolute values and norms of polynomials 5
2. Height functions 10
3. A height function on the algebraic closure of Q(X) 26
4. Restricted power series 32
5. Hermann's method for restricted power series 46
6. Criteria for ideal membership 56
References 60
INTRODUCTION
Let f1, . . . , fn Z[X], where X = (X1, . . . , XN ) is an Ntuple of indeterminates.
The starting point for this paper was the following criterion for membership in the ideal of
