Lattice Extensions and Zeros of Multilinear Polynomials

<p>We treat several problems related to the existence of lattice extensions preserving certain geometric properties and small-height zeros of various multilinear polynomials. An extension of a Euclidean lattice L₁ is a lattice L₂ of higher rank containing L₁ so that the intersection of L₂ with the subspace spanned by L₁ is equal to L₁. Our first result provides a counting estimate on the number of ways a primitive collection of vectors in a lattice can be extended to a basis for this lattice. Next, we discuss the existence of lattice extensions with controlled determinant, successive minima and covering radius. In the two-dimensional case, we also present some observations about the deep holes of a lattice as elements of the quotient torus group. Looking for basis extensions additionally connects to a search for small-height zeros of multilinear polynomials, for which we obtain several results over arbitrary number fields. These include bounds for a system of polynomials under appropriate hypotheses, as well as for a single polynomial with some additional avoidance conditions. In addition to several height inequalities that we need for these bounds, we obtain a new absolute version of Siegel's lemma which is proved using only linear algebra tools.</p>

Name thesis:degree_name

Mathematics, PhD

Level thesis:degree_level

Open Access Dissertation

Discipline thesis:degree_discipline

Institute of Mathematical Sciences

Year dc:date.available

2023

Author dc:creator

• Forst, Maxwell

Contributors dc:contributor

• Michael Orrison
• Allon Percus
• Jeffrey D. Vaaler

• Diophantine approximation
• geometry of numbers
• height functions
• lattices
• number theory
• polynomials
• Mathematics

Repository record dc:identifier

https://scholarship.claremont.edu/cgu_etd/586

OAI identifier oai:identifier

oai:scholarship.claremont.edu:cgu_etd-1608