Date of Award
Summer 2023
Degree Type
Open Access Dissertation
Degree Name
Mathematics, PhD
Program
Institute of Mathematical Sciences
Advisor/Supervisor/Committee Chair
Lenny Fukshansky
Dissertation or Thesis Committee Member
Michael Orrison
Dissertation or Thesis Committee Member
Allon Percus
Dissertation or Thesis Committee Member
Jeffrey D. Vaaler
Terms of Use & License Information
This work is licensed under a Creative Commons Attribution 4.0 License.
Rights Information
© 2023 Maxwell Forst
Keywords
Diophantine approximation, geometry of numbers, height functions, lattices, number theory, polynomials
Subject Categories
Mathematics
Abstract
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 $L_1$ is a lattice $L_2$ of higher rank containing $L_1$ so that the intersection of $L_2$ with the subspace spanned by $L_1$ is equal to $L_1$. 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.
ISBN
9798380479295
Recommended Citation
Forst, Maxwell. (2023). Lattice Extensions and Zeros of Multilinear Polynomials. CGU Theses & Dissertations, 586. https://scholarship.claremont.edu/cgu_etd/586.
