Diophantine avoidance and small-height primitive elements in ideals of number fields

TL;DR

The paper develops explicit Diophantine avoidance machinery for lattices to produce small-height primitive elements in number fields and to bound generators of ideals under avoidance constraints. By blending lattice-avoidance results (Minkowski/Henk–Thiel) with polynomial-avoidance tools (Combinatorial Nullstellensatz), it proves an effective theorem guaranteeing points outside unions of sublattices that also avoid algebraic hypersurfaces, with explicit height bounds. This framework yields concrete results: (i) explicit small-height primitive generators inside ideals, including a quadratic-field bound and an avoidance theorem for primitive elements outside finite unions of ideals; (ii) analogous bounds for totally real fields on totally positive primitive elements; (iii) a Mahler-measure upper bound for non-sparse generating polynomials of a given number field; (iv) a Kornhauser-based bound for small-height generators of principal ideals in quadratic fields. Collectively, these results provide effective, invariant-dependent height controls and an alternate, constructive approach to primitive element theory with avoidance constraints, with potential applications to Ruppert-type bounds and explicit generators in algebraic number theory.

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 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 a very explicit bound like this 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 an avoidance result like this for lattice points in the positive orthant in $\mathbb{R}^d$ 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 non-sparse 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.

Theorems & Definitions (9)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 3.1
• Lemma 3.2
• Theorem 6.1
• Corollary 6.2
• Theorem 7.1