Year: 2026
Type: dissertation
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.
Source: TigerPrints (Clemson University)
Author: Barnhart, Julianne
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W7164635594
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