Section 10: Radius of convergence of algebraic power series

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

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 singular points of an algebraic function are the zeros of $a_n(z)$ and the resultant of $f(z,w)$ with its derivative $f_w$. And a distinguishing property of algebraic power series is that they often have radii of convergences extending across many singular points. This contrasts sharply with single-valued analytic functions with power expansions that never extend across the nearest singular point. We will using the following outline to accomplish this task:

  1. Compute a distinct set of singular points for the function,
  2. Identify the annular rings between singular points,
  3. Compute the monodromy (branching) of each annular ring,
  4. Compute the monodromy around each singular point including if necessary, around the point at infinity.
  5. Design a robust method of illustrating the branching geometry of each annular ring,
  6. Design a graphics interface showing these regions and monodromies and using this information, infer information about the radius of convergence of algebraic power series for the function.

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. Our first objective is to design a very precise method for identifying and comparing numerically-determined singular points. Recall, the singular points of an algebraic function are the zeros to the resultant of $f(z,w)$ with its partial $f_w$, or $R\big[f,f_w\big]$. 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: $$ \begin{align*} f_s&=\{-0.33719-0.426404 i,\\ &-0.33719+0.426404 i,\\ &-0.033241+0.617736 i,\\ &-0.033241-0.617736 i,\\ &0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,0.,\\ &0.684072,\\ & +0.373365 i,0.684072,\\ & -0.373365 i,1.79847,\\ & +0.982974 i,1.79847,\\ & -0.982974 i,3.10911\} \}\\ \end{align*} $$

Once we have the set of singular points $\{s_i\}$, we wish to compute the values of the function at these singular points. In the case of poles, the function approaches infinity. In the case of ramified points, the function has roots with multiplicities greater than one. Thus, for each singular point, we solve for the roots using the Mathematica command NSolve[theFunction==0/.z->s_i] and then determine the number and type of multiple roots. For a $2$-cycle branch, we would obtain a double root. For a $3$-cycle branch, a triple root and so on. For this function, we have $2$-cycle branches around each singular point so for each, we would obtain a double root representing the singular value at each singular point.

A plot of the singular points in the z-plane is shown in Figure 1: 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 $f(z,w)$

RingColor code Monodromy
1tan{{1}, {2}, {3, 4}, {5}}
2blue{{1,3,4,2}, {5}}
3blue{{1,3,4,2}, {5}}
4blue{{1,2,4,3}, {5}}
5tan{{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)Multiple root (w)Color code Monodromy
00tan{{1}, {2}, {3, 4}, {5}}
-0.3371-0.4264i-0.0179-0.5103itan{{1}, {2,3}, {4}, {5}}
-0.3371+0.4264i-0.0179+0.5103itan{{1}, {2,4}, {3}, {5}}
-0.03324+0.6177i0.7322+0.057itan{{1}, {2}, {3}, {4,5}}
-0.03324-0.6177i0.7322-0.057itan{{1}, {2}, {3}, {4,5}}
0.6841+0.3734i-0.5197-0.6496itan{{1,4}, {2}, {3}, {5}}
0.6841-0.3734i-0.5197+0.6496itan{{1,4}, {2}, {3}, {5}}
1.7985+0.9830i-1.6227+0.597itan{{1,2}, {3}, {4}, {5}}
1.7985-0.9830i-1.6227-0.597itan{{1,2}, {3}, {4}, {5}}
When we do this, we can create the following monodromy plot for this function where we have color-coded the ring outside of the singular points, green:

And then using the multiple roots of the function at the singular points in Table 2, we can generate a 3-D plot showing the location of the singular points (red points) of the function. It is these points which determine the radius of convergence of algebraic power series.

Using the tables and plots above, we can obtain valuable information regarding the radius of convergence of power series for each annular ring of the function. First, let's consider the first and second ring separated by the first two singular points $s_1=-0.3371-0.4264i$ and $s_2=-0.3371+0.4264i$. The first ring has a single $2$-cycle branch and three single cycle branches and the next ring has a single $4$-cycle branch and one single-cycle branch. From this data alone we can conclude the radius of convergence for the power series of the $2$-cycle branch is $\big|s_1\big|$. And the reason is simple: the second ring does not support a $2$-cycle branch.

Now consider the three single-cycle branches in ring 1. Since ring $2$ supports one single-cycle branch, there is the possibility one of the single-cycle branches in the first ring can be analytically continued across the first set of (congugate) singularities. This however is a necessary condition and not a sufficient one. And there is another necessary condition: the two singular points separating the first and second ring must support continuing the branch across the singular point. And in order to do this, the monodromy around each of these singular points must contain single-cycle branches which are not poles and this function does not contain poles and according to Table 2, these points do support single-cycle branches. But again, this is only a necessary condition. A sufficient condition for analytic continuation across a singular point that is not a pole is that we should begin at a point in one ring, traverse an analytically continuous path around the singular points and return to the starting point on the same branch sheet we started the path. We can accomplish this task easily through numerical integration.

However, there is a visual means of checking this albeit not a rigorous proof of analytic continuation: We simply plot the branch sheets along with the singular points. If the branch sheet in one ring is analytically continuous with a branch sheet in another ring with no singular points impinging upon the surfaces, then the two branches are analytic continuations of one another and the radius of convergence for the power series representing the inner branch is at least as large as the radius of the outer ring. We will demonstrate this with the single-cycle branches in the first and second ring.

Let's consider the first two non-zero singular points on the first ring and the (real-part of the) single-cycle branch identified by it's root, $\{5\}$. This plot is shown in Figure 4. Note how these singular points do not intersect this branch. Compare this with the $\{1\}$ cycle shown in Figure 5 where we see one of the singular points intersecting the branch. The singular point in Figure 5 is preventing the branch from being analytically continued across the second ring. However, the branch in Figure 4 is not so restricted as we can see by plotting the single-cycle branch in ring two along with the the plot in Figure 4. Note how the two branches join without intervening singular points intersecting the sheets. This shows us that we can analytically continue the $\{5\}$ branch into the second ring via the $\{1\}$ branch and that the power expansion for the $\{5\}$ branch has a radius of convergence at least equal to the radius of the second ring. However if we try to extend this branch into the third ring with the single-cycle branch there, we find the singular points in the second ring impinging upon the branch sheets and so preventing analytic continuation into the next ring. This is shown in Figure 7. Using this graphic argument, we can conclude, non-rigorously, that the radius of convergence of the power expansion for the $\{5\}$ branch in the first ring extends to the second ring or $|s_4|\approx|-0.03324+0.6177i|\approx 0.6186$.

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.

Analyzing the continuations for all branches, we can then create the continuation graph of Figure 8 showing which branches continue into other rings. The designation "2-2" between "ring 1" and "ring 2" represents the ramification of the two singular points between these rings: Each singular points has a single 2-cycle branch and three single-cycle branchs. The "2" above Ring 1 represents the ramification at the origin. These help us decide which branches are capable of continuations. In the case of continuations between ring 1 and 2, the two singular points between these rings have three single-cycles (non-poles) that can support continuation and in fact one does continue the {5} branch. Now consider ring 2 and ring 3: Ring 2 has a 4-cycle branch. In order for this branch to continue into ring 3, the singular points beween the rings would have to support four single cycle branches. We see from the ramification of "2-2" that we do not have this support so can conclude the 4-cycle branch in ring 2 does not continue into ring 3. The same reasoning can be used to conclude the 4-cycle branches in ring 3 and 4 also do not have continuations. The reader may wish to experimentally derive the continuations for the fifth degree function in Section 3 with 21 rings using AFRender.

One added feature of this plot is that we can easily compute the genus of the function from the list of singular point ramifications using the Riemann-Hurwitz formula: $$ g=1/2 \sum_p (r-1)-n+1 $$ where $n$ is the degree of $f(z,w)$ in $w$, and the sum is over all the ramifications of all the singular points including infinity (in the formula we subtract one from each ramification). Using this formula and Figure 8, we see the sum is 5, the degree is 5 so that the genus of this function is 1 or its normal Riemann surface is a torus.

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