09/08/2022, 06:03 PM
Just to test my sage-lib:
let \(a(x)=-x+x^4+x^5\) then \(f(x)=a(a(x))=x-2x^5+O(x^6)\), \({\rm valit}[f]=4\) and we have again the 4 4th roots:
\begin{align}
f^{\circ \frac{1}{4}|1}&=x - \frac{1}{2} x^{5} - x^{7} + \frac{1}{4} x^{8} - \frac{5}{8} x^{9} + \frac{3}{2} x^{10} - \frac{17}{2} x^{11} - \frac{17}{16} x^{12} - \frac{209}{16} x^{13} + 22 x^{14} - \frac{429}{8} x^{15} + \frac{147}{8} x^{16} - \frac{31917}{128} x^{17} + \frac{2407}{16} x^{18} - \frac{4057}{8} x^{19}+O(x^20)\\
f^{\circ \frac{1}{4}|2}&=i x + 2 i x^{3} + \left(-\frac{1}{2} i + \frac{1}{2}\right) x^{4} + \frac{11}{2} i x^{5} + \left(-3 i + 4\right) x^{6} + \left(-\frac{15}{2} i - 1\right) x^{7} + \left(-16 i + \frac{101}{4}\right) x^{8} + \left(-\frac{1513}{8} i - 12\right) x^{9} + \left(\frac{43}{4} i + 101\right) x^{10} + \left(-\frac{18907}{12} i - \frac{211}{2}\right) x^{11} + \left(\frac{12329}{16} i + \frac{669}{8}\right) x^{12} + \left(-\frac{406943}{48} i - \frac{1119}{2}\right) x^{13} + \left(\frac{52139}{6} i - \frac{40595}{12}\right) x^{14} + \left(-\frac{2215209}{80} i - \frac{1901}{2}\right) x^{15} + \left(\frac{2798725}{48} i - \frac{1946047}{48}\right) x^{16} + \left(\frac{25537919}{640} i + \frac{130513}{6}\right) x^{17} + \left(\frac{7666655}{32} i - \frac{6096887}{20}\right) x^{18} + \left(\frac{3365536581}{2240} i + \frac{3905543}\\{12}\right) x^{19}+O(x^{20})\\
f^{\circ \frac{1}{4}|3}&=- x + x^{4} + \frac{1}{2} x^{5} - x^{7} + \frac{9}{4} x^{8} + \frac{5}{8} x^{9} + \frac{11}{2} x^{10} - \frac{21}{2} x^{11} + \frac{133}{16} x^{12} - \frac{367}{16} x^{13} + \frac{187}{2} x^{14} - \frac{699}{8} x^{15} + \frac{699}{4} x^{16} - \frac{82707}{128} x^{17} + \frac{17417}{16} x^{18} - \frac{11755}{8} x^{19}+O(x^{20})\\
f^{\circ \frac{1}{4}|4}&=-i x - 2 i x^{3} + \left(\frac{1}{2} i + \frac{1}{2}\right) x^{4} - \frac{11}{2} i x^{5} + \left(3 i + 4\right) x^{6} + \left(\frac{15}{2} i - 1\right) x^{7} + \left(16 i + \frac{101}{4}\right) x^{8} + \left(\frac{1513}{8} i - 12\right) x^{9} + \left(-\frac{43}{4} i + 101\right) x^{10} + \left(\frac{18907}{12} i - \frac{211}{2}\right) x^{11} + \left(-\frac{12329}{16} i + \frac{669}{8}\right) x^{12} + \left(\frac{406943}{48} i - \frac{1119}{2}\right) x^{13} + \left(-\frac{52139}{6} i - \frac{40595}{12}\right) x^{14} + \left(\frac{2215209}{80} i - \frac{1901}{2}\right) x^{15} + \left(-\frac{2798725}{48} i - \frac{1946047}{48}\right) x^{16} + \left(-\frac{25537919}{640} i + \frac{130513}{6}\right) x^{17} + \left(-\frac{7666655}{32} i - \frac{6096887}{20}\right) x^{18} + \left(-\frac{3365536581}{2240} i + \frac{3905543}{12}\right) x^{19}+O(x^{20})\\
a^{\circ \frac{1}{2}|1} &= f^{\circ \frac{1}{4}|2}\\
a^{\circ \frac{1}{2}|2} &= f^{\circ \frac{1}{4}|4}
\end{align}
let \(a(x)=-x+x^4+x^5\) then \(f(x)=a(a(x))=x-2x^5+O(x^6)\), \({\rm valit}[f]=4\) and we have again the 4 4th roots:
\begin{align}
f^{\circ \frac{1}{4}|1}&=x - \frac{1}{2} x^{5} - x^{7} + \frac{1}{4} x^{8} - \frac{5}{8} x^{9} + \frac{3}{2} x^{10} - \frac{17}{2} x^{11} - \frac{17}{16} x^{12} - \frac{209}{16} x^{13} + 22 x^{14} - \frac{429}{8} x^{15} + \frac{147}{8} x^{16} - \frac{31917}{128} x^{17} + \frac{2407}{16} x^{18} - \frac{4057}{8} x^{19}+O(x^20)\\
f^{\circ \frac{1}{4}|2}&=i x + 2 i x^{3} + \left(-\frac{1}{2} i + \frac{1}{2}\right) x^{4} + \frac{11}{2} i x^{5} + \left(-3 i + 4\right) x^{6} + \left(-\frac{15}{2} i - 1\right) x^{7} + \left(-16 i + \frac{101}{4}\right) x^{8} + \left(-\frac{1513}{8} i - 12\right) x^{9} + \left(\frac{43}{4} i + 101\right) x^{10} + \left(-\frac{18907}{12} i - \frac{211}{2}\right) x^{11} + \left(\frac{12329}{16} i + \frac{669}{8}\right) x^{12} + \left(-\frac{406943}{48} i - \frac{1119}{2}\right) x^{13} + \left(\frac{52139}{6} i - \frac{40595}{12}\right) x^{14} + \left(-\frac{2215209}{80} i - \frac{1901}{2}\right) x^{15} + \left(\frac{2798725}{48} i - \frac{1946047}{48}\right) x^{16} + \left(\frac{25537919}{640} i + \frac{130513}{6}\right) x^{17} + \left(\frac{7666655}{32} i - \frac{6096887}{20}\right) x^{18} + \left(\frac{3365536581}{2240} i + \frac{3905543}\\{12}\right) x^{19}+O(x^{20})\\
f^{\circ \frac{1}{4}|3}&=- x + x^{4} + \frac{1}{2} x^{5} - x^{7} + \frac{9}{4} x^{8} + \frac{5}{8} x^{9} + \frac{11}{2} x^{10} - \frac{21}{2} x^{11} + \frac{133}{16} x^{12} - \frac{367}{16} x^{13} + \frac{187}{2} x^{14} - \frac{699}{8} x^{15} + \frac{699}{4} x^{16} - \frac{82707}{128} x^{17} + \frac{17417}{16} x^{18} - \frac{11755}{8} x^{19}+O(x^{20})\\
f^{\circ \frac{1}{4}|4}&=-i x - 2 i x^{3} + \left(\frac{1}{2} i + \frac{1}{2}\right) x^{4} - \frac{11}{2} i x^{5} + \left(3 i + 4\right) x^{6} + \left(\frac{15}{2} i - 1\right) x^{7} + \left(16 i + \frac{101}{4}\right) x^{8} + \left(\frac{1513}{8} i - 12\right) x^{9} + \left(-\frac{43}{4} i + 101\right) x^{10} + \left(\frac{18907}{12} i - \frac{211}{2}\right) x^{11} + \left(-\frac{12329}{16} i + \frac{669}{8}\right) x^{12} + \left(\frac{406943}{48} i - \frac{1119}{2}\right) x^{13} + \left(-\frac{52139}{6} i - \frac{40595}{12}\right) x^{14} + \left(\frac{2215209}{80} i - \frac{1901}{2}\right) x^{15} + \left(-\frac{2798725}{48} i - \frac{1946047}{48}\right) x^{16} + \left(-\frac{25537919}{640} i + \frac{130513}{6}\right) x^{17} + \left(-\frac{7666655}{32} i - \frac{6096887}{20}\right) x^{18} + \left(-\frac{3365536581}{2240} i + \frac{3905543}{12}\right) x^{19}+O(x^{20})\\
a^{\circ \frac{1}{2}|1} &= f^{\circ \frac{1}{4}|2}\\
a^{\circ \frac{1}{2}|2} &= f^{\circ \frac{1}{4}|4}
\end{align}