Summary: AN EFFECTIVE WEIERSTRASS DIVISION THEOREM
ABSTRACT. We prove an effective Weierstrass Division Theorem for algebraic restricted
power series with p-adic 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 number-theoretic
criterion for membership in ideals of polynomial rings with integer coefficients.
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
Let f1, . . . , fn Z[X], where X = (X1, . . . , XN ) is an N-tuple of indeterminates.
The starting point for this paper was the following criterion for membership in the ideal of