Date of Award

Summer 2023

Degree Type

Open Access Dissertation

Degree Name

Mathematics, PhD


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

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.

Rights Information

© 2023 Maxwell Forst


Diophantine approximation, geometry of numbers, height functions, lattices, number theory, polynomials

Subject Categories



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.



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