### Folded polynomial systems, Part 4: Some properties of the underlying algebraic geometry

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

### Readers are encouraged to read Fixed points of folded polynomial systems, Parts 1,2 and 3 first and reveiw the nomenclature notes below.

We continue to study the fixed points of a particular folded polynomial system \begin{align*} F^{((n))}(z,w)&=\underbrace{F(F(F(\cdots F))\cdots)}_{\text{n times}}\\ &=\{g(z,w),h(z,w)\} \end{align*} with $$$$F(z,w)=\left\{\begin{array}{l} 1-1/4 w+z^2 \\ z \end{array} \right\}$$,$$ or equivalently the roots of \begin{align*} V_n(z,w)&=F^{((n))}-\{z,w\} \\ \\ &= \left\{\begin{array}{l} g(z,w)-z \\ h(z,w)-w \end{array} \right\}\\ \\ &= \left\{\begin{array}{l} u_n(z,w) \\ v_n(z,w) \end{array} \right\} \end{align*}. Setting $V_n=0$ produces two algebraic expressions: $$\begin{array}{c} u_n(z,w)=0 \\ v_n(z,w)=0. \end{array}$$ We next write $u_n$ and $v_n$ in standard form: $$\begin{array}{ll} u_n(z,w)=a_0(z)+a_1(z)w+a_2 w^2+\cdots+a_r(z)w^{r}; & r=2^{n-1} \\ v_n(z,w)=b_0(z)+b_1(z)w+b_2 w^2+\cdots+b_s(z)w^{s}; & s=2^{n-2} \end{array}$$ that implicitly define algebraic curves $w_{u}^n(z)$ and $w_{v}^n(z)$.

We now focus on $w_u$. The analysis is similar for $w_v$.

1. ### Some notes about nomenclature:

1. A central property of algebraic curves like $w_u$ is singular points. These are the points in $\mathbb{C}^2$ such that either $\displaystyle \frac{\partial u}{\partial w}=0$ or the values of $z$ such that the coefficient on the highest powers of $w$ is zero. That is, the roots to $a_r(z)=0$. Since singular points are in terms of $\{z,w\}$, singular points often do not affect all branching of the curves and the principle of extending a branch beyond a singular point is the process of extending a branch beyond its ring of definition over a singular point that is not affecting the analycitity of the branch. It's simpler also in practice to simply reference the $z$-value of the singular point as a singular-$z$ point keeping in mind there will be some branches which are analytic at this value of $z$ and other branches of the curve which are singular.

2. This discussion centers on the branching geometry of an algebraic curve. This geometry is often expressed in terms of "monodromy" which describes how the sheets of a multi-valued function wrap around one or more singular points. For example, as a circular path around the origin travels around $\sqrt{z}$, the path moves from one sheet of the function to the next then back to the first sheet over a $4\pi$ route. In this way, the path is cycling over the two single-valued sheets of $\sqrt{z}$ and we say the branching is "ramified" since a $2\pi$ route does not return to the same point. If the branching returns to the same point after a $2\pi$ route, the branch is "unramified". The branching can be described in two ways: The first way is simply to describe the number of single-valued sheets of the branch. This is the cycle size. For $\sqrt{z}$, the cycle size is $2$ or the function is $2$-cycle. A list of cycle sizes is given in parentheses such as $(1,1,1,2,2,4)$ which describes a set of cycles containing three $1$-cycles, two $2$-cycles and one $4$-cycle.

The second way to describe the cyclic geometry is by the monodromy or sheet sequence the path travels based on a pre-defined sequence, usually a sorted order at a specific value of $z$. In the case of $\sqrt{z}$, the monodromy sequence is simply the first sheet to the second and this is given with french brackets or $\{1,2\}$. Another example is the $8$-cycle branch of $w_u^5$ below with monodromy $\{3,16,15,13,8,11,12,9\}$. This monodromy indicates that a path around this branch first begins on sheet $3$, then after $2\pi$, continues (analytically) to sheet $16$, then $15$ and so on cycling back to sheet $3$ again after $16\pi$. An algebraic curve can have multiple branches surrounding a set of singular points such as the set $\{1,2,4,8,6,5,7,3\},\{9,11,12,10\}$ and $\{13,14,16,15\}$ of $w_u^5$. See Section 10 and reference On the branching geometry of algebraic functions. for more information.

3. We segregate singular-$z$ points in terms of their absolute value creating a set of rings around the origin of the $z$-plane with singular-$z$ points on these rings. The rings create a set of annular regions free of singular points. Annular branches are defined in these regions and we are concerned with computing the monodromies of these branches as well as the monodromies around each singular-$z$ point. Sometimes the annular branches are called "rings".

In the following discussion, the term "singular point" refers to the singularity at $\{z,w\}$ and "singular-$z$ point" refers to the $z$-component of the singular point.

2. ### Monodromy of $w_u$:

3. We can numerically compute $w_u$ via Puiseux series or by solving the associated differential equations. See Sections 6,7 and 8 for background on this process. Recall, $F^{((1))}=F(z,w)$ and $F^{((2))}=F(F)$ and so on. So that as we fold $F(z,w)$ into $F^{((n))}$, the degree of $w_{u}^n$ becomes $2^{(n-1)}$ and that of $w_{v}^n$ becomes $2^{n-2}$ (for $n>1$).

It's not difficult to show both $a_r$ and $b_s$ are constants. We can compute these values iteratively: If we let $a_{r}$ and $b_{s}$ represent the coefficients on the highest power of $w$ in $w_u$ and $w_v$ with $a_{r}=-1/4$ and $b_{r}=-1/4$, then: \begin{align*} a_{(n,r)}&= \left(a_{(n-1,r)}\right)^2 \\ b_{(n,s)}&= a_{(n-1,r)};\quad n>2 \end{align*} Since both $a_{r}$ and $b_{s}$ are constants, the corresponding functions $w_{u}$ and $w_{v}$ have no poles. We can easily write $V_n$ in standard form for $V_2$ through $V_6$. The leading coefficients for all $a_i(z)$ and $b_i(z)$ are constants. For example: $$\small u_3=\left(\frac{19}{4}-2 z+\frac{125 z^2}{16}-z^3+8 z^4-\frac{z^5}{2}+4 z^6+z^8\right)+\left(-\frac{31}{16}+\frac{z}{4}-4 z^2+\frac{z^3}{4}-3 z^4-z^6\right)w+\left(\frac{1}{2}-\frac{z}{32}+\frac{3 z^2}{4}+\frac{3 z^4}{8}\right)w^2+\left(-\frac{1}{16}-\frac{z^2}{16}\right)w^3+\frac{1}{256} w^4$$ This form results in a degenerate characteristic equation (see Section 7) having distinct roots and thus having a fully unramified monodromy at the origin. $V_2$ through $V_7$ are therefore fully unramified at the origin, i.e., the branch sheets at the origin are analytic.

All branch sheets at infinity have poles. This is easily seen since the coefficient on the highest power of $w$ is a constant and therefore, the expansion about infinity is an expansion about zero for the function (see Section 1): $$F^{((n))}_{\infty}=z^kF^{((n))}\bigg(\frac{1}{z},w\bigg)$$ where $k$ is the highest power of $z$ in $F^{((n))}$, produces a single coefficient in terms of $z^k$ on the highest power of $w$ in $F^{((n))}_{\infty}$. For example, making the transformation $\displaystyle F^{((4))}_{\infty}=z^{16}F^{((4))}\bigg(\frac{1}{z},w\bigg)$ produces $\displaystyle\frac{z^{16}w^8}{65536}$ as the term for the highest power of $w$.

The following tables give the ring (annular) monodromies of $w_u^2$ through $w_u^7$. Ring monodromies are given in terms of cycle sizes. For example the monodromy of ring $2$ of $w_u^3$ is $(2,2)$ meaning the ring ramifies into two $2$-cycles. Ring $14$ of $w_u^5$ ramifies into one $8$-cycle and two $4$-cycles, and ring $18$ of $w_u^6$ ramifies into two $16$-cycles. The $\infty$ ring is the monodromy about infinity. This is the monodromy about a circle enclosing all finite singular points. The tables exhibit several characteristics:

1. Ring $1$ of $w_u^2$ through $w_u^7$ is fully unramified, i.e., the branch sheets are analytic at the origin,
2. The ring monodromies are $2^k$-cycles for various non-negative integer $k\leq n$,
3. The global monodromy geometry reflected in the appearance of the monodromy tables below have a distinct "funnel" appearance: the rings seem to increasingly ramify away from the origin and in some cases, fully-ramifying at some point then begin to de-ramify,
4. In this example, the intrinsic monodromy of $F(z,w)$ appears to determine the global monodromy of higher folded systems. This is reflected in the similar structure, both in terms of monodromies and overall geometric shape of the tables.
$$\begin{array}{c} \text{w_u^2} \\ \begin{array}{|c|c|} \hline \text{Ring} & \text{Cycle Sizes} \\ \hline 1 & (1,1) \\ \infty & (2) \\ \hline \end{array}\\ \end{array} \hspace{20pt} \begin{array}{c} \text{w_u^3} \\ \begin{array}{|c|c|} \hline \text{Ring} & \text{Cycle Sizes} \\ \hline 1 & (1,1,1,1) \\ 2 & (2,2) \\ 3 & (4) \\ \infty & (2,2) \\ \hline \end{array}\\ \end{array} \hspace{20pt} \begin{array}{c} \text{w_u^4}\\ \begin{array}{|c|c|} \hline \text{Ring} & \text{Cycle Sizes} \\ \hline 1 & (1,1,1,1,1,1,1,1) \\ 2 & (2,2,1,1,1,1) \\ 3 & (2,2,2,2) \\ 4 & (4,4) \\ 5 & (8) \\ 6 & (8) \\ \infty & (4,4) \\ \hline \end{array}\\ \end{array} \hspace{20pt} \begin{array}{c} \text{w_u^5}\\ \begin{array}{|c|c|} \hline \text{Ring} & \text{Cycle Sizes} \\ \hline 1 & (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) \\ 2 & (2,2,1,1,1,1,1,1,1,1,1,1,1,1) \\ 3 & (2,2,1,1,1,1,1,1,1,2,2,1) \\ 4 & (2,2,2,2,1,1,1,2,2,1) \\ 5 & (2,2,2,2,2,2,2,2) \\ 6 & (4,4,2,2,2,2) \\ 7 & (4,4,4,4) \\ 8 & (8,4,4) \\ 9 & (16) \\ 10 & (4,8,4) \\ 11 & (8,8) \\ 12 & (8,8) \\ 13 & (8,8) \\ 14 & (8,4,4) \\ \infty & (4,4,4,4) \\ \hline \end{array}\\ \end{array}$$ $$\begin{array}{c} \text{w_u^6}\\ \begin{array}{|c|c|} \hline \text{Ring} & \text{Cycle Sizes} \\ \hline 1 & (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) \\ 2 & (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1) \\ 3 & (1,1,2,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1) \\ 4 & (1,1,2,1,1,1,1,2,1,1,1,1,1,1,2,2,2,1,1,1,2,1,1,1,1,1) \\ 5 & (1,1,2,1,1,1,1,2,1,2,1,2,2,2,2,1,1,1,2,1,1,1,1,1) \\ 6 & (2,2,2,1,1,2,1,2,1,2,2,2,2,1,1,1,2,1,1,1,1,1) \\ 7 & (2,2,2,1,1,2,1,2,1,2,2,2,2,1,1,2,2,1,1,2) \\ 8 & (2,2,2,2,2,2,2,2,2,2,2,1,1,2,2,1,1,2) \\ 9 & (2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2) \\ 10 & (2,2,2,2,2,2,2,2,2,2,4,2,4,2) \\ 11 & (4,2,4,2,2,2,2,2,4,2,4,2) \\ 12 & (4,4,4,4,2,2,4,2,4,2) \\ 13 & (4,4,4,4,4,4,4,4) \\ 14 & (8,4,8,4,4,4) \\ 15 & (4,4,4,4,4,4,4,4) \\ 16 & (8,4,4,8,4,4) \\ 17 & (8,8,8,8) \\ 18 & (16,16) \\ 19 & (8,8,8,8) \\ 20 & (16,16) \\ 21 & (8,8,8,8) \\ 22 & (16,16) \\ 23 & (8,8,8,8) \\ 24 & (8,16,8) \\ 25 & (16,8,8) \\ 26 & (8,8,16) \\ 27 & (8,8,8,8) \\ \infty & (8,8,8,8) \\ \hline \end{array}\\ \end{array}$$ $$\small \begin{array}{c} \text{w_u^7}\\ \begin{array}{|c|c|} \hline \text{Ring} & \text{Cycle Types} \\ \hline 1 & (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) \\ 2 & (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) \\ 3 & (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,2,2,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) \\ 4 & (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,2,2,2,1,1,2,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) \\ 5 & (1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,2,1,1,2,2,2,1,1,2,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) \\ 6 & (1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1,2,1,1,1,2,1,2,1,1,2,2,2,1,1,2,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) \\ 7 & (1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1,2,1,1,1,2,1,2,1,1,2,2,2,1,1,2,1,1,1,1,2,1,2,1,1,1,1,1,1,2,1,1,1,1,1,1,1) \\ 8 & (1,1,2,1,1,1,1,2,2,2,1,1,1,1,2,1,1,1,2,1,2,1,1,2,2,2,1,1,2,1,1,1,1,2,1,2,1,1,1,1,1,1,2,1,1,1,1,1,1,1) \\ 9 & (1,1,2,1,1,1,1,2,2,2,1,1,1,1,2,1,1,1,2,1,2,1,1,2,2,2,1,1,1,2,1,1,2,1,1,2,2,1,1,1,1,2,1,1,1,2,1,1) \\ 10 & (1,1,2,1,1,1,1,2,2,1,2,1,1,1,2,1,1,1,1,2,2,1,1,2,2,2,1,1,2,2,1,2,1,1,2,2,1,1,1,1,2,2,1,2,1,1) \\ 11 & (1,1,2,2,1,1,2,2,1,2,1,2,2,1,1,1,1,2,2,1,1,2,2,2,1,1,2,2,1,2,1,1,2,2,1,1,1,1,2,2,1,2,1,1) \\ 12 & (1,1,2,2,1,1,2,2,1,2,1,2,2,2,1,1,2,2,2,2,2,2,1,1,2,2,1,2,1,1,2,2,1,1,1,1,2,2,1,2,1,1) \\ 13 & (1,1,2,2,1,1,2,2,1,2,1,2,2,2,1,1,2,2,2,2,2,2,2,2,2,1,2,1,2,2,2,1,1,1,2,2,1,2,1,1) \\ 14 & (1,1,2,2,1,1,2,2,1,2,1,2,2,2,1,1,2,2,2,2,2,2,2,2,2,1,2,1,2,2,2,2,1,2,2,1,2,2) \\ 15 & (2,2,2,2,2,2,1,2,1,2,2,2,1,1,2,2,2,2,2,2,2,2,2,1,2,1,2,2,2,2,1,2,2,1,2,2) \\ 16 & (2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,1,2,2,2,2,1,2,2,1,2,2) \\ 17 & (2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2) \\ 18 & (2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,2,2,2,2,2,4,2,2,2,2,2,2) \\ 19 & (2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,2,2,2,2,4,4,2,2,2,2) \\ 20 & (2,2,2,2,2,2,2,4,2,2,2,2,2,4,4,4,2,2,2,2,4,4,2,2,2,2) \\ 21 & (2,2,4,2,2,2,4,2,4,2,2,4,4,4,2,2,2,2,4,4,2,2,2,2) \\ 22 & (4,4,2,2,4,4,4,2,2,4,4,4,2,2,2,2,4,4,2,2,2,2) \\ 23 & (4,4,2,2,4,4,4,2,2,4,4,4,2,4,2,4,4,2,2,4) \\ 24 & (4,4,4,4,4,4,4,4,4,4,2,4,2,4,4,2,2,4) \\ 25 & (4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4) \\ 26 & (4,8,4,4,4,4,8,4,4,4,4,4,4,4) \\ 27 & (4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4) \\ 28 & (4,8,4,4,4,4,8,4,4,4,4,4,4,4) \\ 29 & (4,12,4,4,4,4,12,4,4,4,4,4) \\ 30 & (16,4,4,16,4,4,4,4,4,4) \\ 31 & (8,8,4,4,8,4,8,4,4,4,4,4) \\ 32 & (8,8,8,8,8,8,4,4,4,4) \\ 33 & (8,8,8,8,8,8,8,8) \\ 34 & (16,8,16,8,8,8) \\ 35 & (8,8,8,8,8,8,8,8) \\ 36 & (16,8,16,8,8,8) \\ 37 & (8,8,8,8,8,8,8,8) \\ 38 & (8,8,16,8,8,16) \\ 39 & (8,8,8,8,8,8,8,8) \\ 40 & (8,8,16,8,8,16) \\ 41 & (8,8,8,8,8,8,8,8) \\ 42 & (8,8,16,8,8,16) \\ 43 & (8,8,8,8,8,8,8,8) \\ 44 & (8,8,16,8,8,16) \\ 45 & (8,8,8,8,8,8,8,8) \\ 46 & (16,8,8,16,8,8) \\ 47 & (16,16,16,16) \\ 48 & (16,16,16,16) \\ 49 & (16,32,16) \\ 50 & (16,32,16) \\ 51 & (16,32,16) \\ 52 & (16,16,16,16) \\ 53 & (16,16,16,16) \\ 54 & (16,16,16,16) \\ 55 & (16,16,16,16) \\ 56 & (16,16,16,16) \\ 57 & (16,8,16,16,8) \\ 58 & (8,8,16,8,8,8,8) \\ \infty & (8,8,8,8,8,8,8,8) \\ \hline \end{array}\\ \end{array}$$

4. ### Branch continuations of $w_u$:

5. In this section we study analytic continuation of ring branches. For background on this concept see Section 10 and reference On the branching geometry of algebraic functions.

Our objective is to determine the analytic continuity of annular branches. These branches are the components of $w_u$ in annular regions separated by singular-$z$ points. The singular-$z$ points are sorted by absolute value and the regions between these values become the annular regions. The ring branching will often ramify into $k$-cycles. However one property of algebraic curves is that these branches can and often do extend beyond the annular region they are defined in. That is to say their analytic continuity can extend beyond the nearest singular point. This is because a singular point is actually a point in $\mathbb{C}^2$ in the form of $\{p,q\}$ and it often does not ramify sufficiently to affect the continuity of all surrounding branch sheets, and in most generic cases singular points ramify into a set of $1$-cycles and a single $2$-cycle,i.e., they minimally ramify. The $1$-cycles about a singular point then provide a possible bridge between annular branch regions.

There are two necessary conditions an annular branch may be extended beyond it's definition:

1. The singular monodromy between successive annuli must support a sufficient number of single-cycle branches to continue the branch. For example, in order to continue a $3$-cycle acrross a singular point, the singular point must have three $1$-cycles,
2. The post-annular monodromy must have the same cycle size as the branch being continued.
These however are not sufficient conditions. A sufficient condition is to analytically-continue a path from each branch sheet around the intervening singular points along single-valued branch sheets and into a supporting branch in the next annular region in such a way that the monodromy is perserved. The procedure for doing this is detailed in the link above.

Consider again the ring monodromies for $w_u^2$ and $w_u^3$: $$\begin{array}{c} \text{w_u^2} \\ \begin{array}{|c|c|} \hline \text{Ring} & \text{Cycle Types} \\ \hline 1 & (1,1) \\ \infty & (2) \\ \hline \end{array}\\ \end{array} \hspace{20pt} \begin{array}{c} \text{w_u^3} \\ \begin{array}{|c|c|} \hline \text{Ring} & \text{Cycle Types} \\ 1 & (1,1,1,1) \\ 2 & (2,2) \\ 3 & (4) \\ \infty & (2,2) \\ \hline \end{array}\\ \end{array}$$ It is obvious from these monodromies that no ring branch can be continued beyond its region of definition. In the case of $w_u^2$, ring $1$ contained $1$-cycles and the next ring contains only a $2$-cycle. Note even in the case of branches having poles as the branches at infinity, these can also be extended into since we determined above, all branchs at infinity are $1$-cycles however the continuation is not holomorphically due to the poles. And in the case of $w_u^3$, we also cannot continue the branches since each successive ring does not support the cycle type of the previous ring. However, the monodromy of $w_u^4$ does support continuations: $$\begin{array}{c} \text{w_u^4}\\ \begin{array}{|c|c|} \hline \text{Ring} & \text{Cycle Types} \\ \hline 1 & (1,1,1,1,1,1,1,1) \\ 2 & (2,2,1,1,1,1) \\ 3 & (2,2,2,2) \\ 4 & (4,4) \\ 5 & (8) \\ 6 & (8) \\ \infty & (4,4) \\ \hline \end{array}\\ \end{array}$$ Note ring $2$ contains a set of $1$-cycles that can support continuing the $1$ cycles of ring $1$, and likewise ring $3$ contains $2$-cycles capable of supporting the $2$-cycles of ring $2$. However no branches can be extended past this poin even though it may appear ring $5$ may be extended into ring $6$ but recall between these rings is at least one singular point that must have at least a minimal ramification preventing the extension of all $8$ branch sheets across it.

6. ### Branch continuation plots:

7. The following is a $w_u^4$ continuation plot. The actual cyclic monodromies are listed as $\{n_1,\cdots,n_k\}$ for a $k$-cycle branch with the values $n_i$ representing the sheet sequence of the monodromies (see reference above for more information about this notation). Singular points are the red numbers between the annular rings. The red arrows identify which cycles extend beyond their annular region of definition. For example, cycle $\{5\}$ in annulus $1$ extends over the two singular points between rings $1$ and $2$ and continues into cycle $\{5\}$ of annulus $2$. The $2$-cycle $\{1,3\}$ in annulus $2$ entends into the $2$-cycle $\{1,2\}$ of ring $3$.

$w_u^5$ has both a greater number as well as more extensive continuations traversing several rings. For example, cycle $\{4\}$ in ring 1 continues over four singular points: two between rings $1$ and $2$ and another two between rings $2$ and $3$. Cycle $\{10\}$ of ring $1$ continues over $6$ singular points into ring $4$. Notice how the fully ramified $16$-cycle branch in annulus $9$ is preventing any continuations from annulus $8$ into $10$. And two $4$-cycle branches extend from annulus $14$ across the real singular point at $1645.9$ out to infinity where they each ramify into $4$-cycle poles.

We want to focus on the continuation of branch $\{9\}$ of ring $1$ across the single singular point at $0.970241 - 0.233553 I$ between rings $1$ and $2$. The monodromy around this singular point is: $$\big\{\{1\},\{2,3\},\{4\},\{5\},\{6\},\{7\},\{8\},\{9\},\{10\},\{11\},\{12\},\{13\},\{14\},\{15\},\{16\}\big\}$$ so that $w_u^5$ ramifies into fourteen $1$-cycles and a single $2$-cycle at this singular point. Now cycle $\{9\}$ of ring $1$ continues into annulus $3$ so is not affected by the ramified branch $\{2,3\}$ at this singular point. We can illustrate this by plotting both the ramified $\{2,3\}$ singular branch, and the ring $\{9\}$ branch at the singular point. This is shown in Figure 1.

Figure 1 attempts to illustrate how a cycle can extend beyond the nearest singular point: The red sheet is a small section of the real surface of branch $\{9\}$ in the first annulus which is shown extending across the nearest singular point $z=0.970241 - 0.233553 I$ shown as the black point in the $z$-plane behind the red and green sheets of the ramified $2$-cycle $\{2,3\}$ branch (the green and yellow sheets are very close together and not easily distinguished in the plot). The blue circular arcs in the $z$-plane represent the annular ring domains for the first three rings although the first two are very close and appear as one. A vertical black line extends from the $z$-value of the singular point, into the yellow and green sheets of $\{2,3\}$. Where this line penetrates the two sheets of $\{2,3\}$ is the actual singular point $\{z,w\}\in\mathbb{C}^2$ although the figure is only showing real surfaces. The black line is shown penetrating the red sheet showing graphically how this branch is continuing across this singular point. The geometry of the imaginary sheets would be similar.

8. ### Analytic continuity of branch sheets over singular points:

9. In the previous section, we showed graphically how cycle $\{9\}$ of ring $1$ was extending beyond the first singualar point. It's important to understand this sheet was not contiguous with the ramified singular branch $\{2,3\}$. Figure 2 shows this more clearly.

We can accurately compute the branch coverings around singular points of moderately-complex algebraic functions in terms of their Puiseux series. This includes the $16$-degree function $w_u^5$. Then we can trace a path from a ring branch towards the singular branch coverings to determine if the ring branch is contiguous with the singular branch. In order to do this we reproduce a diagram from the reference cited above:

Now consider a ring branch in the light blue region of the Figure 3. This branch will be analytically continuous at least up to the nearest singular point $z$. There are two such singular points on the black ring separating annular region 1 from 2 shows as the black points on the ring. In order to determine if this ring branch is analytically-continuous with the ramified singular coverings, we numerically compute an analytic path starting from point $a$ to $f$ then to $c$. The point $c$ is carefully chosen to lie in the radius of convergence of the Puiseux series for the selected ramifed singular covering. We then can compare the value of $w_u(c)$ computed by the analytically-continuous route along the branch sheet to the point $c$, to the value computed by the Puiseux series. If the values are equivalent to some numerical degree of accuracy much higher than the smallest difference between sheet values at the point $c$, then the two branches are considered contiguous. For example, if the smallest separation of sheet values at $w_u(c)$ is say $10^{-4}$, then analytic continuity is established at an accuracy of say $10^{-10}$. It is not especially difficult to achieve an accuracy of $10^{-16}$ with these calculations.

The following are numerical results testing analytic continuity of ring sheet $\{9\}$ across the first singular point. We first compute Puiseux series around the singular point. The Newton Polygon algorithm return $16$ single-valued branch functions as $w_{(s,i)}$ with $s$ being the cycle size, and $i$ the sheet index number so that when $s\neq 1$, the sheet is part of a ramified cycle. We obtain the list: $$w_{(1,1)},w_{(2,2)},w_{(2,3)},w_{(1,4)},w_{(1,5)},w_{(1,6)},w_{(1,7)},w_{(1,8)},w_{(1,9)},w_{(1,10)},w_{(1,11)},w_{(1,12)},w_{(1,13)},w_{(1,14)},w_{(1,15)},w_{(1,16)}$$ Therefore, the second and third function represent the two sheets of the ramified $2$-cycle with monodromy $\{2,3\}$. We then compute the value of $w_u(c)$ for each branch sheet returned by Newton Polygon and compare the results to the value of $w_u(c)$ computed numerically via the route from point $a$ to $c$ over branch sheet $\{9\}$ and compute the absolute differences. We obtain: $$\begin{array}{c} \text{Continuity of sheet \{9\} over first singular point sheets} \\ \\ \begin{array}{|c|c|} \hline \text{Sheet Function} & \text{Difference} \\ \hline w_{(1,1)} & 9.66664\\ w_{(2,2)} & 4.91694\\ w_{(2,3)} & 4.92184\\ w_{(1,4)} & 9.18177\\ w_{(1,5)} & 10.5229\\ w_{(1,6)} & 3.93768\\ w_{(1,7)} & 10.3929\\ w_{(1,8)} & 3.65415\\ w_{(1,9)} & 9.8421\\ w_{(1,10)}& 1.64741*10^{-16} \\ w_{(1,11)} & 10.1119\\ w_{(1,12)} & 0.147656\\ w_{(1,13)} & 1.55031\\ w_{(1,14)} & 8.48231 \\ w_{(1,15)} & 8.51386\\ w_{(1,16)} & 1.54297\\ \hline \end{array}\\ \end{array}$$ And therefore, sheet $\{9\}$ of ring $1$ is continuous with this singular point $1$-cycle sheet $w_{(1,10)}$. And since this sheet is analytic at the singular point, cycle $\{9\}$ is analytically-continuous over this singular point and we write $9\to w_{(1,10)}$. If we do this for all $16$ ring cycle sheets of ring 1, over both singular points on the first ring we obtain: $$\begin{array}{c} \text{Analytic continuity of ring 1 sheets over singular branches} \\ \\ \begin{array}{|c|c|} \hline 0.970241 - 0.233553 I & 0.970241 + 0.233553 I \\ \hline 2\to w_{(1,1)} & 1\to w_{(1,1)} \\ 3\to w_{(2,2)} & 5\to w_{(2,2)}\\ 1\to w_{(2,3)}& 2\to w_{(2,3)}\\ 5\to w_{(1,4)}& 3 \to w_{(1,4)} \\ 6\to w_{(1,5)}& 4 \to w_{(1,5)} \\ 4\to w_{(1,6)}& 6\to w_{(1,6)} \\ 8\to w_{(1,7)} & 7\to w_{(1,7)}\\ 7\to w_{(1,8)}& 8\to w_{(1,8)} \\ 10\to w_{(1,9)}& 9\to w_{(1,9)} \\ 9\to w_{(1,10)}& 10\to w_{(1,10)} \\ 14\to w_{(1,11)}& 12\to w_{(1,11)} \\ 12\to w_{(1,12)}& 14\to w_{(1,12)}\\ 11\to w_{(1,13)}& 13 \to w_{(1,13)}\\ 13\to w_{(1,14)}& 11\to w_{(1,14)} \\ 16\to w_{(1,15)}& 15\to w_{(1,15)}\\ 15\to w_{(1,16)}& 16\to w_{(1,16)}\\ \hline \end{array}\\ \end{array}$$ And from this table we can immediately see cycles $1$ and $3$ over singular point $0.970241 - 0.233553 I$ and cycles $2$ and $5$ over $0.970241 + 0.233553 I$ are not continuous over these singular points since they all continue onto ramified branches $w_{(2,2)}$ and $w_{(2,3)}$. This agrees with Figure 2 in that cycle $\{1\}$, $\{2\}$, $\{3\}$ and $\{5\}$ are the only cycles in ring $1$ not continuing into ring 2.

10. ### Real geometry of $w_u$:

11. Next consider the real sheets of $w_u^2$ through $w_u^5$. Recall these functions have $2^{n-1}$ single-valued branches at the origin. Note the real sheets of $w_u^2$ in the first plot. This function has degree $2$ giving rise to the red and yellow sheets in the figure. In the plot for $w_u^3$ we see the red sheet of $w_u^2$ folliating into the green and blue sheets of $w_u^3$ and likewise the yellow sheet of $w_u^2$ folliating into the lower red and yellow sheets of $w_u^3$. The plot for $w_u^4$ shows a further folliation of the two pair of sheets of $w_u^3$ into two pairs of four sheets of $w_u^4$, although the sheet count is not obvious in the plot. The final plot shows the $8$ sheets of $w_u^4$ folliating into two sets of $8$ sheets of $w_u^5$. It's not certain this would continue for all $n$. These plots reflect how the algebraic folding of $F(z,w)$ gives rise to the geometric folding of the underlying algebraic curves $w_u$ and $w_v$.

12. ### Geometry of folded polynomial system roots:

13. If $\{p,q\}$ is a root of $V_n$, then $u(p,q)=0$ and $v(p,q)=0$ or $w_u(p)=q$ and $w_v(p)=q$ so that $w_u(p)=w_v(p)$. Both $w_u$ and $w_v$ are in general, multi-valued functions consisting respectively of $2^{n-1}$ and $2^{n-2}$ single-valued branch sheets so that if $\{p,q\}$ is a root of $V_n$ then there exists branch sheet $u_i$ and $v_k$ such that $w_{u(i)}(p)=w_{v(k)}(p)$. In order for this to be true, we must have $$$$\begin{array}{c} \text{Re}(w_u(p))=\text{Re}(w_v(p))\\ \text{Im}(w_u(p))=\text{Im}(w_v(p)) \end{array}$$$$

Each expression of (2) defines a set of contours along the real and imaginary sheets of $w_u$ and $w_v$ where the two functions intersect. Consider for example, the first folding of $F$ and its fixed points in terms of the roots of $V_2$: \begin{align*} V_2(z,w)&=\left\{ \begin{array}{l} 1-5z/4+(1-w/4+z^2)^2 \\ 1-5w/4+z^2 \end{array} \right\} =\left\{\begin{array}{l} 0 \\ 0 \end{array}\right\}\\ \\ &= \left\{\begin{array}{l} u(z,w) \\ v(z,w) \end{array} \right\} =\left\{\begin{array}{l} 0 \\ 0 \end{array}\right\}\\ \\ \end{align*} In this exampole, we can compute directly, functions $w_u$ and $w_v$, and in more complex cases compute these functions either by generating Puiseux series or by numerically solving the associated differential equations (see Section 1). We can then generate plots of $\text{Re}(w_u)$ and $\text{Re}(w_v)$ and illustrate their intersections showing for the set of roots $\{p,q\}$, the location of $\text{Re}(q)$ on these intersections.

In Figure 3, the two real sheets of $w_u$ are shows as the red and green surfaces and the single real sheet of $w_v$ shown as the yellow surface. The white contours are their intersections and the black points are the roots of $V_2$ along those intersections. Notice there are two roots on each branch sheet. A similar plot can be generated for the imaginary sheets.