Upper Bounds on Resolvent Degree via Sylvester's Obliteration Algorithm

TL;DR

This paper revisits Sylvester's algorithm from an algebraic geometry perspective to establish improved upper bounds on the resolvent degree of general polynomials, advancing understanding of algebraic solutions.

Contribution

It introduces a modern geometric interpretation of Sylvester's algorithm and applies it to derive tighter upper bounds on RD(n).

Findings

Derived new upper bounds on RD(n) for various n.

Reformulated Sylvester's algorithm using algebraic geometry.

Enhanced understanding of algebraic solutions for polynomial equations.

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 this paper, we recover an algorithm of Sylvester for determining non-zero solutions of systems of homogeneous polynomials, which we present from a modern algebro-geometric perspective. We then use this geometric algorithm to determine improved thresholds for upper bounds on RD$(n)$.