Abstract

Let $K$ be a number field of degree $d$. Then every ideal $I$ in the ring of integers ${\mathcal{𝒪}}_K$ contains infinitely many primitive elements, i.e., elements of degree $d$. A bound on the smallest height of such an element in $I$ follows from some recent developments in the direction of a 1998 conjecture of W. Ruppert. We prove an explicit bound on the smallest height of such a primitive element in the case of quadratic fields. Further, we consider primitive elements in an ideal outside of a finite union of other ideals and prove a bound on the height of a smallest such element. Our main tool is a result on points of small norm in a lattice outside of an algebraic hypersurface and a finite union of sublattices of finite index, which we prove by blending two previous Diophantine avoidance results. We also obtain a bound for small-norm lattice points in the positive orthant in ${ℝ}^d$ with avoidance conditions and use it to obtain a small-height totally positive primitive element in an ideal of a totally real number field outside of a finite union of other ideals. Additionally, we use our avoidance method to prove a bound on the Mahler measure of a generating nonsparse polynomial for a given number field. Finally, we produce a bound on the height of a smallest primitive generator for a principal ideal in a quadratic number field.

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