Degree of Sum of Polynomials

Theorem
Let $(R, +, \circ)$ be a ring with unity whose zero is $0_R$.

Let $R \left[{X}\right]$ be the ring of polynomial forms over $R$ in the indeterminate $X$.

For $f \in R \left[{X}\right]$ let $\deg \left({f}\right)$ be the degree of $f$.

Then:
 * $\forall f, g \in R \left[{X}\right]: \deg \left({f + g}\right) \le \max \left\{{\deg \left({f}\right), \deg \left({g}\right)}\right\}$

Proof
First we associate to $f=a_0+a_1X+\cdots+a_nX^n\in R \left[{X}\right]$ a formal vector $x_f=(a_0,a_1,\ldots,a_n,0,\ldots)\in R^\infty$, and let $x_f^i\in R$ denote the element at the $i^\text{th}$ position.

Then $\displaystyle \deg(f)=\sup\{i\in\N:x_f^i\neq 0\}$.

The sum $+$ and product $\circ$ in the polynomial ring $R[X]$ induce operations $+',\ \circ'$ on the subset $S=\{x\in R^\infty:x=x_f\text{ for some }f\in R \left[{X}\right]\}$. These are given by


 * $x_{f+g}^i=x_f^i+x_g^i$


 * $\displaystyle x_{f\circ g}^i=\sum_{j+k=i}x_f^jx_g^k$

Let $f,g\in R \left[{X}\right]$


 * 1. Let $d=\max\{\deg(f),\deg(g)\}$ Then $x_f^i=0=x_g^i$ for all $i>d$, so


 * $\displaystyle \sup\{i\in\N:x_{f+g}^i\neq 0\}=\sup\{i\in\N:x_f^i+x_g^i\neq 0\}\leq d$