UC Irvine

In this dissertation, we determine best-to-date upper bounds on the resolvent degree of solving general polynomials. Chapters 1 and 2 provide the necessary background information and Chapters 3 and 4 establish new results.

In Chapter 1, we provide a high-level introduction of the dissertation and provide a history of the literature on resolvent degree. We also establish standard mathematical notation and terminology.

In Chapter 2, Section 1, we recall the definitions of essential dimension and resolvent degree, as well as some standard results. In Section 2, we give an introduction to the theory of Tschirnhaus transformations and explain how we will use this theory to obtain upper bounds on resolvent degree by determining special points on Tschirnhaus complete intersections.

In Chapter 3, Section 1, we recover the classical notion of the polars of a hypersurface at a point, so that we can introduce the polar cone of a hypersurface at a point and the connection between the polar cone at a point and lines on the hypersurface through that point. We then extend these notions to intersections of hypersurfaces and introduce iterated polar cones and their connections to k-planes in Section 2. Finally, we recover the obliteration algorithm of Sylvester and present it in a modern geometric context in Section 3.

In Chapter 4, we establish our bounds on resolvent degree and Sections 1, 2, and 3 each highlight different constructions. In Section 4.4, we indicate obstructions to further bounds on resolvent degree via iterated polar cone constructions. We then proceed to establish approximations via elementary functions and compare the bounds we obtain with previous bounds in Sections 5 and 6. Finally, we posit several questions for future research in Section 7.

Appendix A contains numerical information regarding our bounding function G'(m) and the previous best bounding function F(m), as well as implementations of the geometric obliteration algorithm. Appendix B contains three translations [Sut2019, Sut2021A, Sut2021B] of papers in the classical resolvent degree literature [Che1954, Kle1905, Wim1927].