The normalized large-radius loop of a monic degree-$n$ polynomial has degree $n$

Given: A monic polynomial $p(z)=z^n+{\sum }_{j<n}{a}_j{z}^j$ of degree $n\ge 1$, the number $S={\sum }_{j<n}\mathrm{\mid }{a}_j\mathrm{\mid }$, and a real $R>\max \{1,S\}$.

For a nonzero complex polynomial, degree is the final coefficient index and monic means that its leading coefficient is $1$ (Formal complex polynomials, evaluation, degree, leading coefficient, and monic polynomials).

Complex modulus is multiplicative and satisfies the triangle inequality (Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive).

The homeomorphism $h:\mathbb{R}\mathrm{/}\mathbb{Z}\to S^1$ sends $[0]$ to $1\in S^1$ ($[t]\mapsto (\cos 2\pi t,\sin 2\pi t)$ is a homeomorphism from $\mathbb{R}\mathrm{/}\mathbb{Z}$ to the unit circle).

Maps into finite products are continuous exactly when their components are continuous; finite sums, products, and quotients of continuous real-valued maps are continuous wherever the denominator is nonzero (A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice, A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions, Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined).

Path-homotopic based circle loops have the same degree (Path-homotopic based circle loops have the same degree).

The standard loop ${\omega }_m$ has degree $m$ for every integer $m$ ($\mathrm{deg}({\omega }_n)=n$ for every integer $n$).