Section 10: Radius of convergence of algebraic power series

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

This section is current being updated

The reader is asked to review the sections in sequential order as each succeeding section builds on a previous section.

In this part we design a method to determine the radius of convergence of power expansions of algebraic functions. In this discussion, we use the term "algebraic power series" to describe these series.

If we have the general expression for the n'th term of a series, we can use the various methods of determining the radius of convergence. Unfortunately, we very often cannot formulate a general expression for algebraic power series. However, there is a way to determine the radius of convergence of these series to arbitrary precision. It is the purpose of this section to explain this method. In general, an algebraic power expansion has a radius of convergence at least equal to the distance to the nearest singular point. Recall, the $z$-values of the singular points of an algebraic function are the zeros of the Resultant $R(f,f_w)$. Once we have these, we can compute, for the finite singular points, the value of $w$ by solving with numerical accuracy, the expression $f_w(s_i)=0$.

And for our case study, we will use the function $$ f(z,w)=(z^4)+(2 z^2+z^4)w+(1+z^2+z^3)w^2+(z)w^3+(1/4-z/2)w^4+(-(1/2))w^5 $$ And this function can be seen with AFRender. And since $a_n(z)\neq 0$, this function has no poles. We have $$R\big[f,f_w\big]=\frac{1}{512} z^{11} \left(3888 z^9-28512 z^8+77400 z^7-97576 z^6+58291 z^5-36168 z^4+23488 z^3-9704 z^2+3400 z-3488\right) $$ and setting $R\big[f,f_w\big]=0$, we obtain the list of singular points: $$ \begin{align*} f_s&=\{-0.33719-0.426404 i,\\ &-0.33719+0.426404 i,\\ &-0.033241+0.617736 i,\\ &-0.033241-0.617736 i,\\ &0.,\\ &0.684072,\\ & +0.373365 i,0.684072,\\ & -0.373365 i,1.79847,\\ & +0.982974 i,1.79847,\\ & -0.982974 i,3.10911\} \}\\ \end{align*} $$

Our objective now is to compute the monodromies around each ring and around each singular point. We already know how to compute the branching geometry around each ring: we form the azimuth and radial equations and integrate around the function and using this information, deduce the monodromy. However, there are some issues with this method if the integration is not done carefully. Our method is a simple one: We select a point, $z_0$, midway between the rings, calculate the roots to $f(z_0,w)=0$. Using each of the roots, integrate $2\pi$ around the ring (or singular point), and then compare to numerical accuracy, the starting and end points of each path and from this information, determine the monodromy. Table 1 gives the monodromies for each of the rings in Figure 1. For example, the most inner ring surrounding the origin has a single $2$-cycle branch and three single-cycle branches. This configuration is color-coded tan. The next ring has a single $4$-cycle branch and one single-cycle branch color-coded blue and so forth.

Table 1: Ring monodromies for $f(z,w)$

RingMonodromy
1{{1}, {2}, {3, 4}, {5}}
2{{1,3,4,2}, {5}}
3{{1,3,4,2}, {5}}
4{{1,2,4,3}, {5}}
5{{1,2},{3},{4}, {5}}
6{{1},{2},{3},{4},{5}}
And likewise for each singular point, $s_i$, it is a simple matter to integrate over the path $z(t)=s_i+r_ie^{it}$ where each $r_i$ is half the distance from this singular point to it's nearest neighbor. Table 2 gives the monodromies around each singular point where we have likewise color-coded each branching type. For example, a monodromy consisting of a single $2$-cycle branch and three single cycle branches is color-coded tan.

Table 2: Singular point monodromies for $f(z,w)$

Singular point (z)Monodromy
0{{1}, {2}, {3, 4}, {5}}
-0.3371-0.4264i{{1}, {2,3}, {4}, {5}}
-0.3371+0.4264i{{1}, {2,4}, {3}, {5}}
-0.03324+0.6177i{{1}, {2}, {3}, {4,5}}
-0.03324-0.6177i{{1}, {2}, {3}, {4,5}}
0.6841+0.3734i{{1,4}, {2}, {3}, {5}}
0.6841-0.3734i{{1,4}, {2}, {3}, {5}}
1.7985+0.9830i{{1,2}, {3}, {4}, {5}}
1.7985-0.9830i{{1,2}, {3}, {4}, {5}}
3.1091{{1},{2,3},{4},{5}}
And our objective is to determine the following continuation diagram:

We can numerically confirm these results by using Lauren't Theorem applied to algebraic functions. Readers may wish to review the earlier sections regarding the calculations involved. Using Laurent's Theorem, we can calculate the first ten terms on the power series for the $\{5\}$ branch in ring one: $$ w_{1,5}(z)=1.45054 +0.163939 z+0.926917 z^2-0.0717314 z^3-0.602722 z^4+0.148923 z^5+0.769661 z^6-0.328224 z^7-1.21161 z^8+0.748091 z^9+2.10653 z^{10} $$ and do the same calculation for the $\{5\}$ branch in ring two: $$ w_{2,5}(z)=1.45054 +0.163939 z+0.926917 z^2-0.0717315 z^3-0.602722 z^4+0.148924 z^5+0.769661 z^6-0.328225 z^7-1.21161 z^8+0.748092 z^9+2.10653 z^{10} $$ As the reader can see, the first ten terms are the same for both branches. This gives some credence that the branches are actually analytic-continuations of each other, or the same branch. And just to check, if we compute the expansion of the $\{5\}$ branch in ring 3, we obtain the first 4 terms as: $$ w_{3,5}(z)=\frac{0.0209769}{z^3}-\frac{0.00787906}{z^2}+1.25992 z+\frac{0.223131}{z} $$ definitely not a continuation of the branches above.

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