I. Transformation
In this 2021 paper, one can remove four terms from an equation of degree $n$ using a Tschirnhausen transformation of degree $n-1$ (and with radical coefficients), but only if $n\geq10$. (This high bound allows enough "free" parameters.) Specifically, in the paper, given the equation with $n\geq10$,
$$x^n+c_1x^{n-1}+c_2x^{n-2}+\dots c_n=0\tag1$$
and Tschirnhausen transformation,
$$y = a_0+a_1x+\dots+a_{n-1}x^{n-1}\tag2$$
yields,
$$y^n+C_1y^{n-1}+C_2y^{n-2}+\dots+C_n = 0\tag3$$
But with appropriate radicals $a_n$, then,
$$C_1 = C_2 = C_3 = C_4 = 0$$
Of course, to reverse the degree $n-1$ transformation for $n=10$ is no longer in radicals, unlike the Bring quintic.
II. Example
Given a sample decic missing three terms,
$$x^{10} + x^6 + x^5 + x^4 + 2x^3 + x^2 + x + 1 = 0$$
and the deg-$9$ transformation,
$$x^9+ax^8+bx^7+cx^6+dx^5+ex^4+fx^3+gx^2+\color{red}mx+n-y=0$$
yields,
$$y^{10}+C_1y^9+C_2y^8+C_3y^7+\dots+C_{10} = 0$$
Since the original decic is missing three terms $(x^9, x^8, x^7)$, then,
\begin{align} C_1 &= \alpha_1\\ C_2 &= \alpha_2m+\alpha_3\\ C_3 &= \alpha_4m^2+\alpha_5m+\alpha_6 \end{align}
with the first three $C_n$ also missing their leading terms $(m, m^2, m^3)$, respectively, and the $a_k$ are in the other unknowns.
My theory is since one has 6 equations $\alpha_1 = \alpha_2 = \alpha_3 = \alpha_4 = \alpha_5 = \alpha_6 = 0$ but 8 unknowns $(a,b,c,d,e,f,g,n),$ then 2 unknowns may be used to reduce the quadratics $(\alpha_3, \alpha_5)$ to linear equations like $(\alpha_1, \alpha_2, \alpha_4)$. But this is just a theory, since my slow computer can't handle the deg-$9$ transformation. Once the system is solved, then $m$ becomes a "free" parameter to solve,
$$C_4 = \beta_1m^4+\beta_2m^3+\beta_3m^2+\beta_4m+\beta_5 = 0$$
analogous to the method used to derive the Bring-Jerrard quintic.
III. Question
- Anybody knows how to solve $C_1 = C_2 = C_3 = C_4 = 0$ in radicals? Or if you have the newer Mathematica, what are the explicit expressions for $(C_1, C_2, C_3)$ using the sample decic above? ($C_4$ is not needed.)
P.S. In Heberle's 2021 paper, "Removal of 5 Terms from a Degree 21 Polynomial", he takes the next step of five terms. There are other recent papers on Tschirnhausen transformations. With computer algebra systems like Mathematica or Maple, it makes things easier.
