Section 17: Radius of convergence of power expansions around singular points of algebraic functions. Some extreme examples.

$$ \newcommand{\bint}{\displaystyle{\int\hspace{-10.4pt}\Large\mathit{8}}} \newcommand{\res}{\displaystyle{\text{Res}}} \newcommand{Log}{\text{Log}} \newcommand{Arg}{\text{Arg}} \newcommand{pLog}{\text{pLog}} $$

The following are extreme radius of convergence, $(R)$, results of power expansion at the origin for $w(z)$ given by the expression $f(z,w)=0$. The term "CLSP" is the convergence-limiting singular point. Background details are described in Sections $15$ and $16$ and in paper Determining the radii of convergence of power expansions around singular points of algebraic functions.

  1. Case $P$:

  2. This $24$-degree function has $1013$ finite singular points. $$ \begin{align*} f(z,w)&= \left(\frac{3}{2} z^{17}\right)+\left(-2 z^4+2 z^{22}\right) w+\left(-\frac{1}{2} z^{21}\right) w^7+\left(3 z^{17}\right)w^8+\left(-z^3\right) w^{11}+\left(-\frac{3}{2}\right) w^{12}\\ &+\left(-3 z^8\right) w^{15}+\left(2 z^4-\frac{2}{5} z^{10}+3z^{11}-\frac{7}{2} z^{25}\right) w^{16}+\left(\frac{8}{3} z^8+4 z^{21}\right) w^{17}\\ &+\left(z^2-1 z^6\right)w^{19}+\left(-z^2\right) w^{21}+\left(z-\frac{17}{12} z^{13}\right) w^{22}+\left(-2 z^{12}-\frac{8}{3} z^{20}\right)w^{23}+\left(-\frac{13}{6} z^{24}\right) w^{24}\\ &=0 \end{align*} $$ $$ \begin{array}{c} \text{Table 1: Convergence Results for Case P power expansions at origin} \\ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline \text{Index} & \text{Series #} & \text{Type} & \text{Size} & \text{est. CLSP} & \text{act. CLSP} & \text{est. R} & \text{act. R} & \text{Dif} & \text{Series Size} \\ \hline 1 & 1 & \text{E} & 1 & 325 & 318 & 0.94792 & 0.947248 & 0.000671926 & 1863 \\ 2 & 2 & \text{L}_{-11} & 1 & 2 & 2 & 0.467857 & 0.464314 & 0.00354293 & 1024 \\ 3 & 3 & \text{L}_{-12} & 1 & 2 & 2 & 0.468067 & 0.464314 & 0.00375325 & 1024 \\ 4 & 4 & V_{11}^4 & 11 & 3 & 3 & 0.580431 & 0.576096 & 0.00433496 & 1922 \\ 5 & 15 & P_{10}^{-1} & 10 & 3 & 3 & 0.580408 & 0.576096 & 0.00431224 & 2006 \\ \hline \end{array}\\ \small \textbf{Series #: }\text{series number in table of branch series, }\textbf{Type: }\text{branch type, }\textbf{Size: }\text{Cycle size}\\ \small \textbf{est. CLSP: }\text{CLSP estimated from Root Test, }\textbf{act. CLSP: }\text{CLSP determined by analytic continuation}\\ \small\textbf{est. R, act. R, dif: }\text{corresponding estimated and actual radius of convergence and their difference}\\ \small\textbf{Series Size: }\text{Number of terms in series expansion} \end{array} $$

    It may seem to appear a discrepancy in the tables regarding the estimated value of $R$ from the root test and the nearest singular point to this estimate given in the table as est. CLSP. The estimated value of $R$ from the root test is determined by a numeric limiting process described in the cited paper above. This value will in general be slightly different than the absolute value of the nearest singular point to this value given by est. CLSP. So that even if the est. CLSP and act. CLSP are the same, the est. R and act. R may be different.

  3. Case $M$:

  4. This $33$-degree function has $1762$ finite singular points. The $3$-cycle pole at the origin has a particularly extreme geometry which makes computing the CLSP a challenge at the series size used in this study. A larger number of terms would make the analysis perhaps less so.

    $$ \begin{align*} f(z,w)&=\left(\frac{2}{5} z^3+\frac{1}{3} z^{10}\right)+\left(\frac{3}{4}\right) w^4+\left(\frac{2}{3} z-\frac{3}{2} z^5-\frac{1}{5} z^{13}\right) w^5+\left(-\frac{1}{2} z^2\right) w^7\\ &+\left(-z^8\right) w^8+\left(-\frac{1}{5} z^{30}\right)w^9+\left(\frac{3}{2} z^2\right) w^{11}+\left(-z^{31}-1 z^{33}\right) w^{13}+\left(7 z^{14}\right) w^{14}\\ &+\left(2z^{27}\right) w^{15}+\left(\frac{6}{5} z^{22}\right) w^{17}+\left(-\frac{7}{3} z^{22}\right) w^{21}+\left(2 z^{16}\right)w^{23}\\ &+\left(2 z^3\right) w^{25}+\left(2 z^{34}\right) w^{28}+\left(2 z^8\right) w^{29}+\left(2 z+5 z^{27}\right) w^{30}+\left(\frac{1}{5} z^{18}\right) w^{31}+\left(\frac{1}{3} z^{18}\right) w^{33}\\ &=0 \end{align*}\\ $$ $$ \begin{array}{c} \text{Table B: Convergence Results for Case M power expansions at origin} \\ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline \text{Index} & \text{Sheet} & \text{Type} & \text{Size} & \text{est. CLSP} & \text{act. CLSP} & \text{est. R} & \text{act. R} & \text{Dif} & \text{Series Size} \\ \hline 1 & 1 & V_4^3 & 4 & 4 & 2 & 0.606765 & 0.60277 & 0.00399527 & 1000 \\ 2 & 5 & P_{26}^{-1} & 26 & 6 & 2 & 0.628057 & 0.60277 & 0.0252871 & 904 \\ 3 & 31 & P_3^{-17} & 3 & 296 & 258 & 0.934127 & 0.927522 & 0.00660459 & 840 \\ \hline \end{array}\\ \end{array} $$
  5. Case $K$:

  6. This $40$-degree function has $2743$ finite singular points.

    $$ \begin{align*} f(z,w)&=\left(\frac{7}{3} z^{25}\right)+\left(-z^{21}\right) w^4+\left(-\frac{1}{4} z\right) w^5+\left(-\frac{8}{5} z^{39}\right)w^6\\ &+\left(-\frac{5}{4}-\frac{5}{4} z^{33}\right) w^8+\left(3 z^4-2 z^{27}\right) w^{13}+\left(-\frac{6}{5} z^3+\frac{1}{5}z^{27}\right) w^{20}\\ &+\left(-\frac{4}{3} z^{36}\right) w^{22}+\left(z^{17}\right) w^{23}+(-1) w^{24}+\left(\frac{1}{2} z^2\right) w^{25}+\left(z^{26}-\frac{1}{5} z^{33}\right) w^{26}\\ &+\left(-\frac{1}{3} z^{11}\right) w^{28}+\left(\frac{5}{3} z^{16}-\frac{2}{5} z^{37}\right) w^{29}+\left(-\frac{6}{5} z^{17}\right) w^{32}\\ &+\left(-2 z^{27}\right) w^{33}+\left(-5 z^{17}-4 z^{28}\right) w^{38}+\left(\frac{7}{5} z^{18}\right) w^{39}+\left(-z^{12}\right) w^{40}\\ &=0 \end{align*} $$ $$ \begin{array}{c} \text{Table C: Convergence Results for Case K power expansions at origin} \\ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline \text{Index} & \text{Sheet} & \text{Type} & \text{Size} & \text{est. CLSP} & \text{act. CLSP} & \text{est. R} & \text{act. R} & \text{Dif} & \text{Series Size} \\ \hline 1 & 1 & F_5^{24} & 5 & 76 & 14 & 0.743535 & 0.739184 & 0.00435106 & 1573 \\ 2 & 6 & V_3^1 & 3 & 83 & 14 & 0.745654 & 0.739184 & 0.00647077 & 990 \\ 3 & 9 & \text{T} & 1 & 4 & 3 & 0.713695 & 0.710527 & 0.003168 & 1024 \\ 4 & 10 & \text{T} & 1 & 4 & 2 & 0.713695 & 0.710527 & 0.003168 & 1024 \\ 5 & 11 & \text{T} & 1 & 8 & 4 & 0.716655 & 0.713423 & 0.00323201 & 1024 \\ 6 & 12 & \text{T} & 1 & 8 & 5 & 0.716655 & 0.713423 & 0.00323201 & 1024 \\ 7 & 13 & \text{T} & 1 & 10 & 11 & 0.731557 & 0.728356 & 0.00320102 & 1024 \\ 8 & 14 & \text{T} & 1 & 10 & 10 & 0.731557 & 0.728356 & 0.00320102 & 1024 \\ 9 & 15 & \text{T} & 1 & 91 & 87 & 0.753966 & 0.752053 & 0.0019128 & 1024 \\ 10 & 16 & \text{T} & 1 & 91 & 88 & 0.753966 & 0.752053 & 0.0019128 & 1024 \\ 11 & 17 & \text{T} & 1 & 78 & 29 & 0.743843 & 0.739994 & 0.00384949 & 1024 \\ 12 & 18 & \text{T} & 1 & 78 & 28 & 0.743843 & 0.739994 & 0.00384949 & 1024 \\ 13 & 19 & \text{T} & 1 & 8 & 6 & 0.717795 & 0.71466 & 0.00313488 & 1024 \\ 14 & 20 & \text{T} & 1 & 8 & 7 & 0.717795 & 0.71466 & 0.00313488 & 1024 \\ 15 & 21 & \text{T} & 1 & 14 & 13 & 0.738355 & 0.735514 & 0.00284157 & 1024 \\ 16 & 22 & \text{T} & 1 & 14 & 12 & 0.738355 & 0.735514 & 0.00284157 & 1024 \\ 17 & 23 & \text{T} & 1 & 85 & 75 & 0.748526 & 0.743153 & 0.00537252 & 1024 \\ 18 & 24 & \text{T} & 1 & 85 & 75 & 0.748526 & 0.743153 & 0.00537252 & 1024 \\ 19 & 25 & P_4^{-3} & 4 & 6 & 2 & 0.716206 & 0.710527 & 0.00567834 & 1000 \\ 20 & 29 & P_4^{-3} & 4 & 6 & 3 & 0.716206 & 0.710527 & 0.00567834 & 1000 \\ 21 & 33 & P_4^{-3} & 4 & 8 & 4 & 0.718862 & 0.713423 & 0.00543884 & 1000 \\ 22 & 37 & P_4^{-3} & 4 & 8 & 5 & 0.718862 & 0.713423 & 0.00543884 & 1000 \\ \hline \end{array}\\ \end{array} $$
  7. Case $H$:

  8. This $50$-degree function has $4584$ finite singular points which took $4.5$ hours to compute via Mathematica's $\texttt{NSolve}$ function at $500$ digits of precision.

    $$ \begin{align*} f(z,w)&=\left(2 z^6+\frac{1}{2} z^7-\frac{5}{4} z^{11}+4 z^{22}+\frac{29}{10} z^{34}-1 z^{40}-\frac{13}{2} z^{43}\right)+\left(\frac{3}{5} z^{10}+\frac{7}{4} z^{24}-\frac{1}{4} z^{50}\right) w^2\\ &+\left(2 z^{17}+\frac{7}{2} z^{34}\right) w^3+\left(-\frac{3}{2} z^{30}+\frac{4}{3} z^{38}+\frac{8}{5} z^{42}\right) w^4+\left(-\frac{6}{5} z^2-\frac{1}{2} z^6+\frac{7}{3} z^{31}\right) w^9\\ &+\left(-\frac{2}{5} z^{11}-\frac{3}{2} z^{26}+1 z^{45}\right)w^{10}+\left(\frac{7}{5} z^{24}-6 z^{32}-6 z^{49}\right) w^{14}\\ &+\left(-\frac{3}{4} z^5+\frac{7}{3} z^{21}-\frac{1}{4} z^{26}+\frac{4}{5} z^{27}+\frac{4}{3} z^{32}-2 z^{36}+\frac{1}{3} z^{39}-\frac{3}{4} z^{41}-z^{43}\right)w^{16}\\ &+\left(-6 z^{14}-2 z^{31}-z^{33}\right) w^{18}+\left(-2 z^{27}-\frac{8}{3} z^{50}\right) w^{22}+\left(4z^8+\frac{4}{5} z^{25}-\frac{3}{2} z^{27}\right) w^{24}\\ &+\left(-3 z^4+\frac{8}{3} z^{22}-\frac{8}{5} z^{43}\right) w^{33}+\left(\frac{7}{3} z^{14}-\frac{3}{2} z^{18}\right) w^{34}+\left(-4+8 z^{13}-\frac{7}{4} z^{47}\right)w^{36}\\ &+\left(z^2-\frac{1}{4} z^7\right) w^{38}+\left(-\frac{1}{2} z^{20}-z^{29}+z^{46}\right) w^{40}+\left(\frac{1}{3}z^{10}+\frac{7}{4} z^{11}+\frac{8}{5} z^{21}\right) w^{47}\\ &+\left(\frac{2}{3} z^2+6 z^{26}+\frac{3}{5} z^{43}\right)w^{48}+\left(-z^9+\frac{1}{4} z^{13}+2 z^{14}+2 z^{18}+z^{36}-2 z^{44}\right) w^{49}\\ &+\left(-\frac{1}{3}z^{23}-\frac{7}{2} z^{40}+z^{42}\right) w^{50}\\ &=0 \end{align*} $$ $$ \begin{array}{c} \text{Table D: Convergence Results for Case H power expansions at origin} \\ \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline \text{Index} & \text{Sheet} & \text{Type} & \text{Size} & \text{est. CLSP} & \text{act. CLSP} & \text{est. R} & \text{act. R} & \text{Dif} & \text{Series Size} \\ \hline 1 & 1 & V_9^4 & 9 & 16 & 2 & 0.578855 & 0.564982 & 0.0138729 & 945 \\ 2 & 10 & V_{27}^2 & 27 & 20 & 2 & 0.584224 & 0.564982 & 0.0192425 & 813 \\ 3 & 37 & P_6^{-1} & 6 & 108 & 98 & 0.799442 & 0.791246 & 0.00819609 & 980 \\ 4 & 43 & P_6^{-1} & 6 & 94 & 94 & 0.787692 & 0.786365 & 0.00132686 & 980 \\ 5 & 49 & \text{L}_{-7} & 1 & 92 & 92 & 0.76072 & 0.75919 & 0.00153031 & 508 \\ 6 & 50 & \text{L}_{-14} & 1 & 92 & 92 & 0.76082 & 0.75919 & 0.00162952 & 505 \\ \hline \end{array} \\ \end{array} $$

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