This is the standard fact that the image of a connected space under a continuous map is connected. Apply this to the restriction of \(f\) to the connected set \(s\).

The subspace topology on \(s\subset \mathbb Z\) is discrete, because \(\mathbb Z\) itself is discrete. A connected discrete space cannot contain two distinct points: otherwise one of those points would be a nontrivial clopen subset, contradicting connectedness. Hence \(s\) has at most one point. Since \(s\) is assumed nonempty, it must be of the form \(\{ k\} \) for some \(k\in \mathbb Z\).

Compose the map \(x\mapsto f(x)\in \mathbb C\) with the real-part map \(\Re \colon \mathbb C\to \mathbb R\). This shows that the same function, viewed as an \(\mathbb R\)-valued map, is continuous on \(s\). Now the inclusion \(\mathbb Z\hookrightarrow \mathbb R\) is an embedding, so continuity after composing with this inclusion is equivalent to continuity of the original \(\mathbb Z\)-valued map.

In Mathlib.

In Mathlib.

In Mathlib.

In Mathlib.

This is immediate from Definition 22 and Lemma 36.

By Lemma 36 for \(x\in T\) \(Re(cos(x))\not=-1\) and hence by Lemma 39 \(PBlog(x)\in S\).

A calculation.

By Lemma 36 \(CSexp(z)\in \mathbb {R}^-\) if and only if \(CSexp({\rm Im}(z))\in \mathbb {R}^-\) if and only if \(\{ {\rm Im}(z)\in \{ \pi +(2\pi )\mathbb {Z}\} \} \). Since, by Definition 26 for \(z∈ S\), \(-\pi {\lt} Im(z) {\lt} \pi \). It follows that \(CSexp(S)\subset T\). Conversely, by Lemma 40 if \(z\in T\) then \(PBlog(z)\in S\).

By Lemma 35 \(CSexp\) is continuous and, by Lemma 40, \(PBlog\) is continuous on \(T\). Suppose that \(z,w\in S\) and \(CSexp(z)=CSexp(w)\). By Lemma 38 there is an integer \(n\) such that \(z-w =2\pi * n * I\) and \(-2\pi {\lt} Im(z)-Im(w){\lt}2\pi \). It follows that \(n=0\) and hence that \(z=w\). This shows that \(CSexp|_S\) is one-to-one. Since \(CSexp|_S\) is one-to-one and \(CSexp({\rm PBlog}(z))=z\) for all \(z\in T\), it follows that \(CSexp\colon S\to T\) and \({PBlog}\colon T\to S\) are inverse functions. Since each is continuous, they are inverse homeomorphisms.

If \(\frac{\Im (w)+\pi }{2\pi }\) were an integer, say equal to \(k\), then

But this says that \(\Im (w)\) is an odd multiple of \(\pi \), contradicting the definition of \(\tilde S\).

Fix \(n\in \mathbb Z\). The fiber where

is exactly the strip

inside \(\tilde S\). The point is that equality at one of the endpoints would make \(\frac{\Im (w)+\pi }{2\pi }\) an integer, which is excluded by Lemma 42. Hence each fiber is open in \(\tilde S\), so the map is locally constant, and therefore continuous.

Let \(z\in S\), so \(-\pi {\lt}\Im (z){\lt}\pi \). For \(\varphi (z,n)=z+2n\pi i\), the imaginary part is

If this were equal to an odd multiple \((2k+1)\pi \), then

which is impossible because \(\Im (z)\) lies strictly between \(-\pi \) and \(\pi \). Therefore \(\varphi (z,n)\in \tilde S\).

Set \(N(w)=\left\lfloor \frac{\Im (w)+\pi }{2\pi }\right\rfloor \). By the defining inequalities for the floor function,

Because \(w\in \tilde S\), Lemma 42 shows that the left inequality is in fact strict. Multiplying through by \(2\pi \) gives

After subtracting \(2N(w)\pi i\), the imaginary part lands in the interval \((-\pi ,\pi )\), which is exactly the condition defining \(S\).

Take \((z,n)\in S\times \mathbb Z\). Since \(z\in S\), we have \(-\pi {\lt}\Im (z){\lt}\pi \), so

Hence the floor function recovers exactly the integer \(n\). Therefore \(\tilde\varphi ^{-1}\) subtracts precisely the same shift \(2n\pi i\) that \(\varphi \) added, and it also returns the second coordinate \(n\). Thus \(\tilde\varphi ^{-1}(\varphi (z,n))=(z,n)\).

By definition,

Applying \(\tilde\varphi \) to this pair adds back the same quantity \(2N(w)\pi i\), so the first coordinate returns to \(w\). Therefore \(\tilde\varphi (\tilde\varphi ^{-1}(w))=w\).

The formula for the forward map is

The first term is continuous in \((z,n)\), and the second term depends only on \(n\); since \(\mathbb Z\) is discrete, every map out of it is continuous. Therefore the sum is continuous as a map into \(\mathbb C\). Lemma 44 shows that its image lies in \(\tilde S\), so it is continuous as a map into \(\tilde S\).

The second component of \(\tilde\varphi ^{-1}\) is the function

which is continuous on \(\tilde S\) by Lemma 43. The first component is

so it is obtained from the identity map and the continuous function \(N(w)\) by continuous algebraic operations. Hence both components are continuous, and therefore the product map \(\tilde\varphi ^{-1}\colon \tilde S\to S\times \mathbb Z\) is continuous.

According to Definition 26 image of \(S\) under the translation action of \((2\pi )\mathbb {Z}\) on \(\mathbb {C}\) is the union of all strips \(S(2n-1)\pi ,(2n+1)\pi )\). By Definition 28 this union is \(\tilde S\). Thus we have a map \(S\times \mathbb {Z}\to \tilde S\) defined by \((z,n)\mapsto z+2\pi *n *I\). Since translation is a homeomorphism of \(\mathbb {C}\to \mathbb {C}\), this map is a local homeomorphism onto its image \(\tilde S\). If \(n ,n'\in \mathbb {Z}\) with \(n\not=n'\) then \(S((2n-1)\pi ,(2n+1)\pi )\cap S((2n'-1)\pi ,(2n'+1)\pi )=\emptyset \). Also \(\tilde S=\coprod _{n\in \mathbb {Z}}S((2n-1)\pi ,(2n+1)\pi )\). It follows that \(\varphi \) is a bijective map and hence a homeomorphism.

By Definition 34 \(\widetilde PBlog\) is the product of \(PBlog\colon T\to S\) and \({\rm Id}_\mathbb {Z}\colon \mathbb {Z}\to \mathbb {Z}\). By Lemma 1 the first of these factors is a homeomorphism. Since \({\rm Id}_\mathbb {Z}\) is a homeomorphism. it follows from basic properties of homeomorphisms that the product \(\widetilde{PBlog}\) is a homeomorphism.

By Definition 23, \(T\) is the union of \(\{ z\mid \Re (z){\gt}0\} \) and \(\{ z\mid \Im (z)\neq 0\} \). Both are open: the first is an open strict inequality set, and the second is the complement of the closed set \(\{ \Im (z)=0\} \). Hence \(T\) is open.

By contradiction, suppose \(\Im (x)=(2k+1)\pi \) for some \(k\in \mathbb {Z}\). Then \(\Im (CSexp(x))=0\) and \(\Re (CSexp(x)){\lt}0\), so \(CSexp(x)\notin T\). This contradicts \(x\in CSexp^{-1}(T)\). Hence no odd multiple of \(\pi \) occurs as \(\Im (x)\), i.e. \(x\in \widetilde S\).

By Definition 31, \(\tilde\varphi ^{-1}_{\mathbb {C}}(x)=x-2N(x)\pi i\), where \(N(x)\) is an integer (Definition 30). By periodicity of \(CSexp\) (Lemma 38), shifting by an integer multiple of \(2\pi i\) does not change the value.

From Lemma 53, \(x\in \widetilde S\). Then Lemma 45 gives \(\tilde\varphi ^{-1}_{\mathbb {C}}(x)\in S\). Applying the left-inverse identity from Lemma 1 to \(\tilde\varphi ^{-1}_{\mathbb {C}}(x)\) gives \(PBlog(CSexp(\tilde\varphi ^{-1}_{\mathbb {C}}(x)))=\tilde\varphi ^{-1}_{\mathbb {C}}(x)\). Finally use Lemma 54.

By Lemma 40, for \(z\in T\) we have \(-\pi {\lt}\Im (PBlog(z)){\lt}\pi \). Hence \(2n\pi -\pi {\lt}\Im (PBlog(z)+2n\pi i){\lt}2n\pi +\pi \), which is exactly the interval characterization of

\(\varphi \) is a homeomorphism by Lemma 50. By Lemma 51 \(\widetilde{PBlog}\colon T\times \mathbb {Z}\to S\times \mathbb {Z}\) is a homemorphism. Thus, the composition \(\varphi \circ \widetilde{PBlog}\colon T\times \mathbb {Z}\to \tilde S\) is a homeomorphism. For \((z,n)\in T\times \mathbb {Z}\),

By Lemma 38, \(CSexp(PBlog(z)+2\pi * n * I)=CSexp(PBlog(z))\), which by Lemma 51 equals \(z\). This establishes that \(\psi \) satisfies all the conditions of the Definition 19 on \(T⊆ \).

Apply the identity \(f\circ \tilde\varphi =\varphi \circ f\) to a point of the form \(\tilde\varphi ^{-1}(x)\). This gives

Now compose both sides with \(\varphi ^{-1}\) to obtain

Since this holds for every \(x\), the desired identity follows.

By Lemma 57, we also have \(f\circ \tilde\varphi ^{-1}=\varphi ^{-1}\circ f\). Since \(f\circ \tilde\varphi =\varphi \circ f\), we have \(\tilde\varphi \colon f^{-1}(U)\to f^{-1}(\varphi (U))\). Since \(\varphi \) and \(\tilde\varphi \) are homeomorphisms the induced map \(\tilde\varphi \colon f^{-1}(U) \to f^{-1}(\varphi (U))\) is a homeomorphism. Let \(\psi \colon f^{-1}(U)\to U\times I\) be a homeomorphism with \(p_1\circ \psi \) being the map \(f\colon f^{-1}U\to U\). Such a map is equivalent to a trivialization for \(f\) with base \(U\). Then

is a homeomorphism. Furthrmore, projection to the first factor is

This composition is \(f\colon f^{-1}(\varphi (U))\to \varphi (U)\), so that this homeomorphism determines a trivialization for \(f\) with base \(\varphi (U)\).

We have homeomorphism \(\mu \colon \mathbb {C}\to \mathbb {C}\) that sends \(z \to CSexp(\pi *I)z)\) and the homeomorphism \(\tilde\mu \colon \mathbb {C}\to \mathbb {C}\) defined by \(\tilde\mu (z)=z+\pi *I\) Clearly by Lemma 37 and Lemma 36 \(CSexp(\tilde\mu (z))= \mu (CSexp(z))\). By Definition 23 and Definition 35 \(\mu (T)=T'\). The result now follows from Lemma 58 and Proposition 2.

Suppose that \(x\in \mathbb {C}\) and \(x\not= 0\). Then either \(Re(x){\gt} 0\) or \(Re(x)\le 0\). If \(Re(x){\gt}0\), then by Definition 23 \(x\in T\). if \(Re(x){\lt} 0\) then by Definition 35 \(x\in T'\). Finally, if \(Re(z)=0\) and \(z\not=0\), then \(Im(z)\not= 0\) and \(z\in T\).

If \(x\in T\cup T'\), then by Definitions 23 and 35 either \(\Re (x){\gt}0\), or \(\Re (x){\lt}0\), or \(\Im (x)\neq 0\). In each case \(x\neq 0\), so \(x\in Cstar\) by Definition 24. Conversely, if \(x\in Cstar\), then \(x\neq 0\), so Lemma 59 shows that \(x\in T\) or \(x\in T'\). Hence \(x\in T\cup T'\).

By Corollary 3 \(T\cup T'= Cstar\). By Proposition 2 and Corollary 2 \(CSexp\) is a trivialization on \(T\) and on \(T'\). Hence, every point of \(Cstar\) lies in the base of a trivialization for \(CSexp\). By definition, this shows that \(CSexp\colon \mathbb {C}\to \mathbb {C}\) is a covering on \(Cstar\). Since \(CSexp(z)\not=0\) for all \(z\in \mathbb {C}\), it follows that \(CSexp^{-1}(Cstar)=\mathbb {C}\). Lastly, by Lemma 39 if \(z\in \mathbb {C}\) and \(z\not= 0\) then \(CSexp(PBlog)(z)=z\). This proves that \(CSexp\) is onto \(\{ z\in \mathbb {C}| z\not=0\} \), which by Lemma 24, is equal to \(Cstar\).

This is immediate from Definition 36.

If \(z\in T\), then by Definition 23 either \(\Re (z){\gt}0\) or \(\Im (z)\neq 0\). In either case \(z\neq 0\), so \(z\in Cstar\) by Definition 24.

If \(z\in T'\), then by Definition 35 either \(\Re (z){\lt}0\) or \(\Im (z)\neq 0\). Again either alternative implies \(z\neq 0\), hence \(z\in Cstar\) by Definition 24.

Because \(CSexp(z)\neq 0\) for every \(z\in \mathbb {C}\), the first coordinate of the old trivialization already lands in \(Cstar\). Thus we may keep the same source and inverse map, replace the base by \(U\cap Cstar\), and regard the target as \((U\cap Cstar)\times \mathbb {Z}\). All the axioms of a trivialization are inherited from those of \(e\).

By Proposition 2, \(CSexp\) is trivial over \(T\). Since \(T\subset Cstar\) by Lemma 61, Lemma 63 turns this into a trivialization of \(CSexpCstar\) over \(T\) viewed inside \(Cstar\).

By Corollary 2, \(CSexp\) is trivial over \(T'\). Since \(T'\subset Cstar\) by Lemma 62, Lemma 63 turns this into a trivialization of \(CSexpCstar\) over \(T'\) viewed inside \(Cstar\).

This is immediate from the construction of Corollary 5.

This is immediate from the construction of Corollary 6.

By Corollary 3, every point of \(Cstar\) lies in \(T\) or in \(T'\). By Corollaries 5 and 6, together with Lemmas 64 and 65, these two sets are bases of trivializations for \(CSexpCstar\). Hence every point of \(Cstar\) has an evenly covered neighborhood, so \(CSexpCstar\) is a covering map.

By Lemma 66, the codomain-restricted exponential \(CSexpCstar\colon \mathbb {C}\to Cstar\) is a covering map. Therefore the standard path-lifting theorem for covering maps yields a unique lift of \(\omega \) starting at \(\tilde a_0\). Finally, Lemma 60 says that forgetting the codomain restriction turns the identity \(CSexpCstar\circ \tilde\omega =\omega \) into \(CSexp\circ \tilde\omega =\omega \), which is exactly the desired conclusion.

By Lemma 66, the map \(CSexpCstar\colon \mathbb {C}\to Cstar\) is a covering map. Hence the general homotopy lifting theorem for covering maps gives a unique lift \(\tilde H\) of the homotopy \(H\) starting from the prescribed map \(f\) at time \(0\). Finally, Lemma 60 identifies the equality \(CSexpCstar\circ \tilde H=H\) with the desired equality \(CSexp\circ \tilde H=H\) after forgetting that the codomain is \(Cstar\).

By Corollary 4 \(CSexp^{-1}(\omega (a))\not=\emptyset \). Fix a point \(x\in CSexp^{-1}(\omega (a))\) and let \(\tilde\omega _x\colon [a, b]\to \mathbb {C}\) be lift of \(\omega \) through the \(CSexp\) with initial point \(x\) as guaranteed by Corollary 7.

By the Definition 25 we have \(CSexp(\tilde\omega (b))=\omega (b)\) and \(CSexp(\tilde\omega (a)=\omega (a)\). By Definition 37 \(\omega (b)=\omega (a)\). Thus, \(CSexp(\tilde\omega (b))=CSexp(\tilde\omega (a))\). By Lemma 38, there is \(k\in \mathbb {Z}\), such that \(\tilde\omega (b)-\tilde\omega (b)=2\pi *k* I\). By Definition 39, the winding number of \(\tilde\omega \) is \(k\)

Let \(\tilde\omega '\) be another lift of \(\omega \). Since \(CSexp(\tilde\omega '(t))=CSexp(\tilde\omega (t))\) for every \(t\in [ a, b]\), there is an integer \(k(t)\in \mathbb {Z}\) with \(\tilde\omega '(t)-\tilde\omega _x(t)=2\pi k(t)*I\). Since \(\tilde\omega '\) and \(\tilde\omega \) are continuous functions of \(t\) so is \(k(t)\). Since the \([ a, b]\) is connected and \(\mathbb {Z}\) is discrete, \(k(t)\) is a constant function; i.e., there is an integer \(k_0\) such that for all \(t\in [ a, b]\), we have \(\tilde\omega '(t)=\tilde\omega (t)+2\pi * k_0*I\). Thus, \(\tildeω'(b) -\tildeω'(b)=\tildeω'(a)-\tildeω(a)\). It follows from Definition 39 \(w(\tildeω')=w(\tildeω).\)

By Lemma 67, there is at least one lift \(\tilde\omega \) of \(\omega \) through \(CSexp\). Since \(\omega (a)=\omega (b)\) and \(CSexp(\tilde\omega (a))=\omega (a)\), \(CSexp(\tilde\omega (b))=\omega (b)\), Lemma 38 implies that \(\tilde\omega (b)-\tilde\omega (a)=2\pi k I\) for some \(k\in \mathbb {Z}\). Hence the winding number of this lift is \(k\).

If \(\tilde\omega '\) is any other lift, then again by Lemma 38 the values \(\tilde\omega '(a)\) and \(\tilde\omega (a)\) differ by an integral multiple of \(2\pi I\). After subtracting this multiple from \(\tilde\omega '\), we obtain another lift of \(\omega \) with the same initial value as \(\tilde\omega \). By Corollary 7, lifts with the same initial value are equal. Therefore \(\tilde\omega '\) differs from \(\tilde\omega \) by the same constant period at every point of \([a,b]\), so the endpoint difference \(\tilde\omega '(b)-\tilde\omega '(a)\) equals \(\tilde\omega (b)-\tilde\omega (a)\). Thus every lift has the same winding number \(k\).

This is one of the conclusions of Corollary 9.

By definition, \(w(\omega )\) is the integer supplied by Corollary 9. That corollary also says that every lift of \(\omega \) has winding number equal to this integer.

Choose a lift \(\tilde\omega \) of \(\omega \) using Lemma 69. Applying Corollary 8 to the homotopy \(H\) and the initial lift \(\tilde\omega \), we obtain a lift \(\tilde H\) of the whole homotopy.

Now consider the two boundary paths

By Definition 38, the projected paths agree, because \(H(a,s)=H(b,s)\) for all \(s\). At time \(s=0\), the starting points \(\tilde H(a,0)\) and \(\tilde H(b,0)\) differ by an integral multiple of \(2\pi I\) by Lemma 38, since \(\omega (a)=\omega (b)\). Shift the first boundary lift by this constant period. The shifted path is still a lift of the same projected path, and now it has the same initial value as the second boundary lift. By Corollary 7, these two lifts are equal. Therefore \(\tilde H(b,s)-\tilde H(a,s)\) is independent of \(s\).

Evaluating at \(s=0\) and \(s=1\) shows that

But the slice \(t\mapsto \tilde H(t,0)\) is a lift of \(\omega \), and the slice \(t\mapsto \tilde H(t,1)\) is a lift of \(\omega '\). Lemma 70 therefore identifies the two corresponding endpoint quotients with \(w(\omega )\) and \(w(\omega ')\), so these winding numbers are equal.

By Lemma 71, the winding number of \(\omega \) is equal to the winding number of a constant loop. Choose a point \(z_0\in \mathbb {C}\) with \(CSexp(z_0)=c\); this is possible by Corollary 4. Then the constant map \(t\mapsto z_0\) is a lift of the constant loop at \(c\) through \(CSexp\). Its endpoint difference is zero, so Lemma 70 shows that the winding number of the constant loop is zero.

By Definition 41, the endpoints of the associated path are \(\psi (CSexp(0))\) and \(\psi (CSexp(2\pi i))\). Since \(CSexp(0)=CSexp(2\pi i)=1\), these endpoints agree.

Let \(ω \colon [ 0, 2\pi ] \to \mathbb {C}\) be the loop associated to \(ρ\). Then by Definition 41, for every \(t\in [0,2\pi ]\) we have \(ω(t)=ρ(CSexp(it))\in Cstar\).

By Definition 41 the loop associated with the constant map \(f\colon S^1\to Cstar\) is a constant loop at a point of \(Cstar\). By Lemma 42 the winding number of \(f\) is equal to the winding number of the constant loop at \(f(S^1)\in Cstar\). By Lemma 10 this winding number is zero.

Let \(H\colon S^1\times I\to \mathbb {C}\) be a homotopy from \(\psi \) to \(\psi '\) whose image lies in \(Cstar\). Let \(ω\) and \(ω'\) be the loops associated to \(ψ\) and \(ψ'\) respectively. Define \(\hat H\colon [ 0, 2\pi ]\times [ 0, 1 ]\to \mathbb {C}\) by \(\hat H(t,s)=H(CSexp(it),s)\). Then by Definition 41 \(\hat H\) is a homotopy from the loop \(\omega \) to the loop \(\omega '\). The images of \(H\) and \(\hat H\) are the same, so the image of \(\hat H\) also lies in \(Cstar\). By Lemma 71, the winding numbers of \(\omega \) and \(\omega '\) are equal. By Definition 42, this means that the winding numbers of \(ψ\) and \(ψ'\) are equal.

Define a continuous map \(J\colon S^1\times [ 0,1 ]\to D^2\) by \((z,t)\mapsto (1-t)z\). Then \(\hat f\circ J(z,0)= \rho (z)\) and \(\hat f\circ J(z,1)=\hat f(0)\) for all \(z\in S^1\). This is a homotopy in \(Cstar\) from \(\rho \) to a constant map of \(S^1\to Cstar\). By Lemma 75 the winding number of \(\rho \) is equal to the winding number of a constant map \(S^1\to C\star \). By Lemma 74, the winding number of a constant map \(S^1\to \hat f(0)\in Cstar\) is zero.

By Definition‘41 and by Lemma 37 the loop \(\omega \colon [ 0, 2\pi ]\to \mathbb {C}\) associated to \(\psi _{\alpha _0,t}\) restricted to the circle of radius \(R\) is given by \(\omega (t)= \alpha _0 R^kCSexp(kt *I)\).

By Lemma 4 there is an \(\tilde\alpha _0\in \mathbb {C}\) with \(CSexp(\tilde\alpha _0)=\alpha _0 R^k\). Define \(\tilde\omega (t)=\tilde\alpha _0+kt *I\) for \(0\le t\le 2\pi \). Then by Lemma 37

By Definition 25 this means that \(\tilde\omega \) is a lift of \(\omega \) through \(CSexp\). By Definition 40 \(w(\omega )=(2\pi k*I-0)/2\pi * I = k\). By Definition 42, this means that the winding number of \(\psi _{\alpha _0,k}\) is \(k\).

Since for all \(z\in S^1\), \(|\psi (z)-\psi '(z)|{\lt}|\psi (z)|\), it follows that \(|\psi (z)|{\gt}0\) and \(|\psi '(z)|{\gt}0\) for all \(z\in S^1\). Define a homotopy \(H\colon S^1\times [ 0, 1 ]\to \mathbb {C}\) by \(H(z,t)=t\psi '(z)+(1-t)\psi (z)\). \(H(z,0)=\psi (z)\) and \(H(z,1)=\psi '(z)\), so \(H\) is a homotopy from \(\psi \) to \(\psi '\).

We establish that \(H(z,t)\not= 0\). For all \(z\in S^1\) and \(t\in [ 0, 1 ]\) we have \(|\psi (z)-H(z,t)|=|(1-t)(\psi (z)-\psi '(z))|\le |\psi (z)-\psi '(z)|{\lt}|\psi (z)|\). So \(H(z,t)\not=0\) for all \(z\in S^1\) and all \(t\in [ 0, 1 ]\).

Consequently, \(H\) is a homotopy \(S^1\times [ 0 , 1 ]\to \mathbb {C}\) from \(\psi \) to \(\psi '\) whose image lies in \(Cstar\).

By Lemma 77, there is a homotopy \(H\) from \(\psi \) to \(\psi '\) whose image lies in \(Cstar\). Thus, by Lemma 75, \(\psi \) and \(\psi '\) have the same winding number.

For each \(1\le i\le k\) set \(\beta _i=\alpha _i/\alpha _0\) Choose \(R{\gt}\sum _{i=1}^k|\beta _j|\) and \(R{\gt}1\). For any \(z\in \mathbb {C}\) with \(|z|=R\), we have

By Lemma 78 for \(R{\gt}{\rm max}(1,\sum _{i=1}^k|\beta _j|)\), and for any \(z\in \mathbb {C}\) with \(|z|=1\) \(|\alpha _0(R*z)^k-f(z)| {\lt}|\alpha _0 R^k|\). By Lemma 11 the maps defined on \(S^1\) by \(z ↦\alpha _0*(R* z)^k\) and by \(f\) have the same winding number.

But according the Lemma 76 the winding number of the map \(S^1\mapsto \mathbb {C}\) given by \(z\mapsto \alpha _0(R*z)^k=(α_0R^k)*z^k\) is \(k\). Thus, the winding number of \(f\) is also \(k\).

The proof is by contradiction. Suppose that \(p(z)=\sum _{i=0}^k\alpha _iz^{k-i} \) with \(\alpha _0\not= 0\). Suppose that \(p(z)\not= 0\) for all \(z\in \mathbb {C}\). By Theorem  8 for \(R{\gt}0\) sufficiently large the winding number of the restriction of \(p(z)\) to the circle of radius \(R\) is \(k\). Fix such an \(R\)

Let \(D^2\to \mathbb {C}\) be the map \(z\mapsto Rz\). Define \(\rho \colon D^2\to \mathbb {C}\) by \(z\mapsto p(Rz)\). The restriction of this map to the boundary circle is the restriction of \(p(z)\) to the circle of radius \(R\). Since \(p(z)\not=0 \) for all \(z\in \mathbb {C}\), the image of \(\rho \) is contained in \(Cstar\). According to Lemma 7, this implies that the winding number of of \(p\) on the circle of radius \(R\) is zero.

Since \(k{\gt}0\), this is a contradiction.

By Definition 49, the endpoints of the path are \(\psi (CSexp(0))\) and \(\psi (CSexp(2\pi i))\). Since both exponentials are equal to \(1\in S^1\), these two values coincide.

Every point on the associated loop has the form \(\rho (CSexp(it))\) by Definition 49. Since \(\rho \) already takes values in \(Cstar\), the whole image of the loop lies in \(Cstar\) as well.

The loop associated to a constant map on \(S^1\) is a constant loop by Definition 49. By Definition 50, the winding number of the map is the winding number of that loop, and Lemma 10 says the latter is zero.

Precompose the given homotopy on \(S^1\) with the parametrization \(t \mapsto CSexp(it)\) of the circle. This produces a homotopy between the associated loops in \(Cstar\). By Lemma 71 those two loops have the same winding number, hence by Definition 50 the original circle maps do as well.

Contract the disk radially to its center. Composing the extension \(F\) with this contraction gives a homotopy in \(Cstar\) from \(\rho \) to the constant boundary value \(F(0)\). Lemma 82 shows that \(\rho \) has the same winding number as this constant map, and Lemma 81 shows that constant map has winding number zero.

Use the straight-line homotopy \(H(z,t)=(1-t)\psi (z)+t\psi '(z)\). If \(H(z,t)=0\), then \(\psi (z)=t(\psi (z)-\psi '(z))\), so \(|\psi (z)| \le |\psi (z)-\psi '(z)|\), contradicting the hypothesis. Thus the entire homotopy avoids zero and stays inside \(Cstar\).

Lemma 83 supplies a homotopy through maps into \(Cstar\) between the two circle maps. Lemma 82 then implies that their winding numbers are equal.

The associated loop is \(t \mapsto \alpha _0 R^k CSexp(kit)\). Choose a logarithm of the nonzero constant \(\alpha _0 R^k\) using Lemma 4; then \(\tilde\omega (t)=\tilde\alpha _0+kit\) is a lift of this loop through \(CSexp\) by Lemma 37. The endpoint difference of the lift is exactly \(2\pi k i\), so Definition 40 gives winding number \(k\), and Definition 50 transfers this to the map on \(S^1\).

Write the polynomial as its leading term plus the sum of lower-degree terms. On the circle \(|z|=R\), each lower-degree monomial is bounded by a constant multiple of \(R^{k-1}\), so their sum is at most \(S R^{k-1}\) for a fixed constant \(S\). For \(R\) sufficiently large we have \(S R^{k-1} {\lt} |\alpha _0| R^k\), which proves that the leading term strictly dominates the remainder.

For large \(R\), Lemma 85 shows that the polynomial map \(z \mapsto p(Rz)\) is uniformly close, in the walking-dog sense, to its leading monomial \(z \mapsto \alpha _0 (Rz)^k\). Therefore Lemma 12 says these two maps have the same winding number. Lemma 84 computes the winding number of the monomial to be \(k\), so the polynomial has winding number \(k\) as well.

Assume for contradiction that the polynomial has no complex root. Then for sufficiently large \(R\), Theorem 11 shows that the restriction of \(p\) to the circle of radius \(R\) has nonzero winding number, namely its degree. But under the no-root assumption, the map \(z \mapsto p(Rz)\) extends from the boundary circle to the whole closed disk without hitting zero. Theorem 10 therefore says its winding number must be zero, a contradiction.