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.
","abstract_html":"<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>","abstract_has_math":true,"creators":["Forst, Maxwell"],"institution":null,"degree_name":"Mathematics, PhD","degree_level":"Open Access Dissertation","degree_discipline":"Institute of Mathematical Sciences","degree_department":null,"school":null,"contributors":["Michael Orrison","Allon Percus","Jeffrey D. Vaaler"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-01-01T08:00:00Z","date_published":"2023-01-01T08:00:00Z","updated_at":"2026-07-24T01:40:22Z","subjects":["Diophantine approximation","geometry of numbers","height functions","lattices","number theory","polynomials","Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarship.claremont.edu/cgu_etd/586","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Michael Orrison","Allon Percus","Jeffrey D. Vaaler"]},{"key":"dc:creator","label":"Author","values":["Forst, Maxwell"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2023-11-08T08:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Institute of Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Open Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Mathematics, PhD"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Diophantine approximation","geometry of numbers","height functions","lattices","number theory","polynomials","Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarship.claremont.edu/cgu_etd/586"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Lattice Extensions and Zeros of Multilinear Polynomials"]}]}],"canonical_facts":{"dc:contributor":["Michael Orrison","Allon Percus","Jeffrey D. Vaaler"],"dc:creator":["Forst, Maxwell"],"dc:date.available":["2023-11-08T08:00:00Z"],"dc:description.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.
"],"dc:identifier":["https://scholarship.claremont.edu/cgu_etd/586"],"dc:subject":["Diophantine approximation","geometry of numbers","height functions","lattices","number theory","polynomials","Mathematics"],"dc:title":["Lattice Extensions and Zeros of Multilinear Polynomials"],"thesis:degree_discipline":["Institute of Mathematical Sciences"],"thesis:degree_level":["Open Access Dissertation"],"thesis:degree_name":["Mathematics, PhD"]},"updated_at":"2026-07-24T01:40:22Z"}