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,…,αJ distinct, nonzero algebraic numbers (which are not necesarily in k), and m1,m2,…,mJ 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 αj with multiplicity at least mj 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 αj and the remaining data in the problem: m1,m2,…mJ, N and the field constants associated with k.KeywordsGalois GroupAlgebraic NumberProduct FormulaGalois ExtensionIndex ColumnThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.