Removal of 5 Terms from a Degree 21 Polynomial

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.