Removal of 5 Terms from a Degree 21 Polynomial

TL;DR

This paper explores Sylvester's method for solving polynomial systems, applying it to bound the resolvent degree of degree 21 polynomials, and establishes that RD(21) is at most 15, complementing previous results.

Contribution

It applies Sylvester's method to derive a new upper bound on the resolvent degree for degree 21 polynomials, improving understanding of polynomial solvability.

Findings

Proves RD(21) ≤ 15 using Sylvester's method.

Provides an alternative proof to Sutherland’s bound using different techniques.

Enhances the theoretical framework connecting polynomial solving methods and resolvent degree.

Abstract

In 1683 Tschirnhaus claimed to have developed an algebraic method to determine the roots of any degree $n$ polynomial. His argument was flawed, but it spurred a great deal of work by mathematicians including Bring, Jerrard, Hamilton, Sylvester, and Hilbert. Many of the problems they considered can be framed in terms of the geometric notion of resolvent degree, introduced by Farb and Wolfson. Roughly speaking, we have $RD(n) \leq n-k$ if a general degree $n$ polynomial can be put into an $(n-k)$-parameter form. In the present note we discuss a method introduced by Sylvester for solving systems of polynomial equations, and apply it to finding bounds on resolvent degree. In particular, we prove the bound $RD(21) \leq 15$. This bound has been independently established by Sutherland using the classical theory of polarity.