Polynomials with Low Height and Prescribed Vanishing

Abstract

In a recent paper [2] we obtained an improved formulation of Siegel’s classical result([9],Bd. I,p. 213, Hilfssatz) on small solutions of systems of linear equations. Our purpose here is to illustrate the use of this new version of Siegel’s lemma in the problem of constructing a simple type of auxiliary polynomial. More precisely, let k be an algebraic number field, O _k its ring of integers, α₁,α₂,…,α_J distinct, nonzero algebraic numbers (which are not necesarily in k), and m₁,m₂,…,m_J positive integers. We will be interested in determining nontrivial polynomials P(X) in 0 _K [X] which have degree less than N, vanish at each α_j with multiplicity at least m_j and have low height. In particular, the height of such plynomials will be bounded from above by a simple function of the degrees and heights of the algebraic numbers α_j and the remaining data in the problem: m₁,m₂,…m_J, N and the field constants associated with k.

The research of the second author was supported by a grant from the National Science Foundation.