Formal complex polynomials, evaluation, degree, leading coefficient, and monic polynomials

A formal complex polynomial is either the zero polynomial $0$, or a finite coefficient list $(a_0,\dots ,a_n)$ with $a_n\mathrm{\ne }0$; we write the latter as $p(Z)={\sum }_{k=0}^n{a}_k{Z}^k$. The list, rather than the function it induces, is the polynomial object. Define evaluation at $z\in \mathbb{C}$ by $0(z):=0,p(z):={\sum }_{k<n+1}^{\mathbb{C}}{a}_k{z}^k\text{for }p=(a_0,\dots ,a_n)\mathrm{\ne }0.$ where the latter is the initial-segment complex sum defined in Complex series, absolute convergence, complex power series, and radius of convergence. Thus evaluation is defined for the zero polynomial as well as every nonzero formal polynomial.

For nonzero $p=(a_0,\dots ,a_n)$, define $\mathrm{deg}p:=n$ and $\mathrm{lc}(p):=a_n$. The zero polynomial has no degree and no leading coefficient. A nonzero polynomial is monic when $\mathrm{lc}(p)=1$. Complex arithmetic and powers are those of $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$ and Integer powers in the complex field.