Upper Bounds on Resolvent Degree and Its Growth Rate

2021/07/16 by Alexander J. Sutherland, Sutherland, Alexander J. · 3 citations
Computer Science · Mathematics · #13F20 (Secondary) #14G25 (Primary) 12E12 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Polynomial and algebraic computation #math.AC #math.AG #msc:12E12 #msc:13F20 #msc:14G25

paper · pdf · doi:10.48550/arxiv.2107.08139

40 pages. Comments welcome!

arxiv created 2021/07/16 · openalex publication_date 2021/07/16 · arxiv updated 2021/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For each n, let RD(n) denote the minimum d for which there exists a formula for the general polynomial of degree n in algebraic functions of at most d variables. In 1945, Segre called for a better understanding of the large n behavior of RD(n). In this paper, we provide improved thresholds for upper bounds on RD(n). Our techniques build upon classical algebraic geometry to provide new upper bounds for small n and, in doing so, fix gaps in the proofs of A. Wiman and G.N. Chebotarev in [Wim1927] and [Che1954].

Citations

• Anders Wiman's "On the Application of Tschirnhaus Transformations to the Reduction of Algebraic Equations''

Cited by

• Special points on intersections of hypersurfaces
• On the resolvent degree of PSU(3,q)
• Solvable points on intersections of quadrics, cubics, and quartics

Related

• Upper Bounds on Resolvent Degree via Sylvester's Obliteration Algorithm
• Removal of 5 Terms from a Degree 21 Polynomial
• Tschirnhaus transformations after Hilbert
• An upper bound for the solving degree in terms of the degree of regularity
• G.N. Chebotarev's "On the Problem of Resolvents"