To What Degree Can We Use Degree?

Guest post by PhD student Ben Rapone.

Introduction

The $\textbf{degree}$ of a function $f$ defined on a set $\Omega$ and its image $f\left(\Omega\right)$ is an extension of the winding number, which counts the number of times a closed curve travels counterclockwise around a given point. Intuitively, the degree counts the number of times $f$, in some sense, ''wraps'' $\Omega$ around a point in $f\left(\Omega\right)$. So what additional information does the degree give, and why would you care about it?

The degree was developed as a way to measure, or keep careful count of, the number of solutions to a system of nonlinear equations. By ''careful'' we mean consistent with respect to some types of perturbation of the system. Degree theory, as we shall see shortly, provides some very nice tools for verifying the existence of solutions to nonlinear systems of equations.

We will keep this conversation light, general and mostly geared towards imparting an intuition concerning what information the degree imparts, and in what ways an application oriented person not necessarily familiar with higher level mathematics could use the degree to their general advantage. For the curious mathematician wishing for a more in depth and general derivation and application there are many summaries, blogs, lecture notes, and theses (check this one out, for instance) awaiting you on the web (Google is always your friend). Here we will not provide any proofs, but will instead refer the reader to those already given in different venues (why recreate the wheel I ask?). Let's proceed, then.

To keep the discussion simple, in this post we will concern ourselves with degrees of continuous functions over bounded, "nice" manifolds. In particular we will narrow our focus at the moment to continuous functions over bounded regions in $\mathbb{R}^n$. With that in mind, let us define the following setting over which we will define the degree:
\begin{equation} \label{eq:Vars} \Omega\subset\mathbb{R}^n \text{ open and bounded } \end{equation} \begin{equation} \label{eq:cont} f:\bar{\Omega}\rightarrow \mathbb{R}^n \text{ continuous } \end{equation} \begin{equation} \label{eq:Jdef} y \in f \left(\bar{\Omega}\right) \setminus f\left(\partial\Omega\right) \text{ s.t. the Jacobian, $J_f(x)$, is defined $\forall x\in\Omega$ with $f(x)=y$} \end{equation}
In other words we will assume $y$ is in the range of $f$ over the closure of $\Omega$ but not in the image of $f$ over the boundary of $\Omega$, and if $x\in\Omega$ such that $f(x)=y$ then $J_f(x)$, the Jacobian of $f$ at $x$, is defined.

Let's look at two examples to illustrate the setting described in Equations \eqref{eq:Vars}, \eqref{eq:cont}, and \eqref{eq:Jdef}.

Figure 1

Figure 2

Figure 1 represents a continuous mapping from $\mathbb{R}$ to $\mathbb{R}$. In particular, its domain and range are specified as $\Omega \approx (-0.2,5.2)$ and $F(\Omega)\approx(-1.3,1.45)$ (these intervals are shaded in the corresponding colors on the $x$- and $y$-axis, respectively). The values where the degree is not defined are indicated by red x's and correspond to the image of the boundary of $\Omega$, $\{-1.3,1.45\}$, and points where the derivative doesn't exist, i.e., where $y = \pm 1$. All other points of $F(\Omega)$ are fair game including the value $y=0.5$, which we will refer back to for an illustration of how we calculate the degree.

Figure 2 represents a continuous mapping from $\mathbb{R}^2$ to $\mathbb{R}^2$, with the domain $\Omega$ the filled circle, and range $F(\Omega)$ the filled triangle. Here $F$ maps the boundary of the circle to the boundary of the triangle and creates a folding of $\Omega$ along the red and green curves in such a way as to make the Jacobian undefined at each point along them. This folding is highlighted using the embedded arrows to show the movement of space and illustrated by the transformation of the pink curve. It follows then that the degree is not defined along the boundary of the triangle and at any point along the red or green curve. Every other point is a viable option if we assume $F$ is smooth everywhere else.

So how do we calculate the degree? The next definition for the degree under our restrictions will answer this question.

Definition:

\begin{equation}\label{ddef} \operatorname{deg}\left(f,\Omega,y\right)=\sum\limits_{x\in f^{-1}(y)}\operatorname{sign}\left(J_f(x)\right) \end{equation}
where $\operatorname{sign}\left(J_f(x)\right)$ denotes the sign of the Jacobian of $f$ at $x$, i.e., $$ \operatorname{sign}\left(J_f(x)\right)= \left\{ \begin{array}{ll} -1 & \mbox{if } J_f(x)< 0, \\ \ \ \ 0 & \mbox{if } J_f(x)= 0,~\mbox{ and } \\ \ \ \ 1 & \mbox{if } J_f(x)> 0. \\ \end{array} \right. $$

By restricting ourselves to the settings specified in Equations \eqref{eq:Vars}, \eqref{eq:cont}, and \eqref{eq:Jdef}, we ensure that $\operatorname{sign}\left(J_f(x)\right)$ exists for each $x\in f^{-1}(y)$. Hence, if we can guarantee that the sum $\sum_{x\in f^{-1}(y)}\operatorname{sign}\left(J_f(x)\right)$ converges, then we can guarantee that $\operatorname{deg}\left(f,\Omega,y\right)$ is defined. One way to provide this guarantee is to require the sum to be finite, i.e., require that only finitely many $x\in\Omega$ exist such that $f(x)=y$. In some sense this restriction limits the number of "foldings" that $F$ can do. Okay now, say we have this additional restriction and the degree is defined for the values we wish to check. How can this setting benefit us? Let's illustrate this with our simple example in Figure 1.

As we observed earlier, the degree is defined at $y=0.5$ and so we can compute $\operatorname{deg}\left(f,\Omega,y\right)$ using definition \eqref{ddef} to be $1+(-1)+1+(-1)+1=1$. So how does this conform with our intuition? We can think of $f$ as taking the set $\Omega$, stretching it out and laying it down on $\Omega$ the way you would lay a long sheet down on a short surface, folding back and forth so it fits on the surface. Any value lying between the image of the boundary points has the beginning of the sheet below it and the end above it, so $f$ lays over it at least one time, with potentially a bunch of additional folds that must come in pairs (something very reminiscent of the intermediate value theorem). This might seem trivial, and it is here, but when $\Omega$ and $f$ are not so nice the degree becomes an invaluable tool with the help of some well meaning theorems.

Theory for Applications

Alright, so what's the use of all this stuff about degree any way? For some of you, it might all seem like a waste of time because, by definition, in order to calculate the degree we must have knowledge of solutions to the very equation we are trying to verify solutions to. Despair not, however, for the following theorem will put your concerns to rest. This theorem is quoted directly from this book on Theorems of Leray-Schauder Type And Applications, where the details of a proof are also available.

Theorem:

Let $\Omega\subset\mathbb{R}^b$ be an open bounded subset and $f:\bar{\Omega}\rightarrow\mathbb{R}^b$ be a continuous mapping. If $p\not\in f\left(\partial\Omega\right)$, then there exists an integer $\operatorname{deg}\left(f, \Omega,p\right)$ satisfying the following properties: $ \quad \text{[i] (Normality) $\operatorname{deg}\left(I, \Omega,p\right)=1$ if and only if $p\in\Omega$, where $I$ denotes the identity mapping. } \\ \quad \text{[ii] (Solvability) If $\operatorname{deg}\left(f, \Omega,p\right)\not= 0$, then $f(x)=p$ has a solution in $\Omega$. } \\ \quad \text{[iii] (Homotopy) If $f_t(x):[0,1]\times\bar{\Omega}\rightarrow\mathbb{R}^n$ is continuous and $p\not\in \bigcup\limits_{t\in[0,1]}f_t\left(\partial\Omega\right)$, then} \\ \quad \quad \quad \quad \text{$\operatorname{deg}\left(f, \Omega,p\right)$ does not depend on $t\in[0,1]$. } \\ \quad \text{[iv] (Additivity) Suppose that $\Omega_1, \Omega_2$ are two disjoint open subsets of $\Omega$ and} \\ \quad \quad \quad \quad \text{$p\not\in f\left(\bar{\Omega}-\Omega_1\cup\Omega_2\right)$}. \\ \quad \quad \quad \quad \text{Then $\operatorname{deg}\left(f, \Omega,p\right)=\operatorname{deg}\left(f, \Omega_1,p\right)+\operatorname{deg}\left(f, \Omega_2,p\right)$. }\\ \quad \text{[v] $\operatorname{deg}\left(f, \Omega,p\right)$ is a constant on any connected component of $\mathbb{R}^n\setminus f(\partial\Omega)$. } $

One nice advantage this theorem has given us is seen in the homotopy invariance of the degree, property [iii]. As a consequence, we could equate the verification of solutions to one system with that of solutions to another, potentially much simpler, system. For instance, Frommer, Hoxha, and Lang (also see here) were able to develop a test involving interval arithmetic to prove existence of zeros of functions using interval arithmetic, which depends on the homotopy invariance property of the degree. Of course there is a wide range of theory concerning the computation of the degree outside of utilizing just the properties found in the theorem, which I invite you to explore. Here are some quick suggestions to take a look at:

1. On the complexity of isolating real roots and computing with certainty the topological degree by Mourrain, Vrahatis, and Yakoubsohn;
2. The calculation of the topological degree by quadrature by O'Neil and Thomas (also see here);
3. and, of course, the book from which we took the theorem.

Additionally though, keep an eye out for my own collaborative research with Dr. Dvijotham and Dr. Krishnamoorthy (the author of this blog) where we take full advantage of the homotopy invariance property to construct optimization based techniques for calculating robustness of solutions to systems of quadratic equations. Check my own website to see more about myself, and new blog posts about my research.