Complex Preliminaries

fact:

Since C is a field, C[x] is a UFD.

definition (Toy contour):

A closed Jordan curve that separates C into an exterior and interior region is referred to as a toy contour.

fact (Complex roots of a number):

Derivation of complex nth roots of a complex number z: z=reiθ=rei(θ+2kπ)z1/n=(rei(θ+2kπ))1n=r1nei(θ+2kπn) Note that one root is r^{1/n}\in {\mathbf{R}}, and the rest are separated by angles of 2\pi/n.

Complex Factoring

fact (A complex perspective on factoring):

For f a quadratic, writing \Delta \coloneqq b^2-4ac, the roots take on the form \begin{align*} z_k = {1\over 2a}\qty{-b + i\sqrt{-\Delta}} = {1\over 2a}\qty{-b + i\sqrt{4ac - b^2}} .\end{align*} For monic polynomials, this becomes slightly nicer: \begin{align*} z_k = {1\over 2}\qty{-b + i\sqrt{4c-b^2} } .\end{align*}

example (Factoring a quadratic):

There is a slightly nicer way to find roots, e.g.: \begin{align*} x^2 + 2x + 6 &= 0 \\ \implies x^2 + 2x + 1 + 4 &= 0 \\ \implies (x+1)^2 + 4 &= 0 \\ \implies x+1 &= \pm 2i \\ \implies x &= -1\pm 2i .\end{align*}

fact (Factoring z^n-1):

\begin{align*} z^n-1 &= \prod_{k=0}^{n-1} (z-\zeta_n^k) = (z-1)(z-\zeta_n)(z-\zeta_n^2)\cdots(z-\zeta_n^{n-1}) && \zeta_n \coloneqq e^{2\pi i \over n} .\end{align*}

What the roots look like:

  • n odd:

figures/2021-12-12_21-49-07.png

  • n even: \theta_0=0, increment by 2\pi/n. Always have \pm 1.

figures/2021-12-12_21-49-23.png

fact (Factoring z^n-w for w\in \CC):

Write w=Re^{i\theta}, then \begin{align*} z^n = w \implies z = R^{1\over n}e^{i(\theta + 2k\pi )\over n} = R^{1\over n}e^{i\theta\over n}e^{2\pi i k \over n} = \qty{Re^{i\theta}}^{1\over n}\zeta_n^k .\end{align*} Thus setting w_0 \coloneqq(Re^{i\theta})^{1\over n} yields \begin{align*} z^n - w = \prod_{k=0}^{n-1} (z-w_0\zeta_n^k) = (z-w_0)(z-w_0\zeta_n)\cdots (z-w_0\zeta_n^{n-1}) .\end{align*}

fact (Factoring z^n+1):

Factoring z^n+1: write 1 = e^{i\pi} to get w_0 \coloneqq e^{i\pi \over n}, then \begin{align*} z^n+1 = \prod_{k=0}^{n-1}(z-w_0\zeta_k) &= (z-e^{i\pi \over n}e^{2i\pi \over n}) (z-e^{i\pi \over n}e^{4i\pi \over n}) \cdots (z-e^{i\pi \over n}e^{2(n-1)i \pi \over n}) \\ &= (z - e^{3i\pi \over n})(z-e^{5i\pi \over n}) \cdots (z - e^{(2n-1)i\pi \over n}) .\end{align*}

What the roots look like:

  • n odd: start at \theta_0 = \pi, increment by 2\pi/n.

figures/2021-12-12_21-48-14.png

  • n even: start at \theta_0 = \pi/n, increment by 2\pi/n.

figures/2021-12-12_21-48-27.png