Date of Award
5-2026
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematical Sciences
Committee Chair/Advisor
Michael Burr
Committee Member
James Coykendall
Committee Member
Shuhong Gao
Committee Member
Svetlana Poznanovic
Committee Member
Matthew Macauley
Abstract
The lift of a loop in the base space of a branched cover to the cover induces a permutation of points in a fibre. The monodromy group of the branched cover is the permutation group generated by all such permutations. When loops are restricted to a particular subset of the base space, the corresponding permutation group induced by these loops is the restricted monodromy group. Monodromy groups encode structure and symmetries of many enumerative problems. We describe the relationship between the restricted monodromy group and the monodromy group of the original branched cover. Our main result is a local-to-global property: when the restricted monodromy group contains a transposition, it is the full symmetric group if and only if the unrestricted monodromy group is the full symmetric group. We show how this result may be applied to sparse polynomial systems, where the restricted monodromy group can be obtained by changing one coefficient parameter. As an application, we complete the proof of the sparse trace test. The sparse trace test verifies that a computed solution set of a sparse polynomial system is complete if and only if the trace, or sum of the solutions, is a function of the coefficient parameters.
Recommended Citation
Barnhart, Julianne, "Parameterized Polynomial Systems: Monodromy, Sparse Polynomials, and Solutions" (2026). All Dissertations. 4277.
https://open.clemson.edu/all_dissertations/4277
Author ORCID Identifier
https://orcid.org/0009-0000-9773-0058