degree of homogeneous polynomials under addition

1. Proposition

Let \(R\) be a ring, \(R[X]\) the polynomial ring as graded ring and \(f,g \in R[X]\) homogeneous polynomials. Then

\begin{align*} \mathrm{deg}(f + g) =& \begin{cases} -\infty & \mbox{if } f = -g \\ \mathrm{max}(\mathrm{deg}(f), \mathrm{deg}(g)) & \mbox{else } \\ \end{cases} \end{align*}

see: degree

2. Proof

2.1. \(\mathrm{deg}(f) = \mathrm{deg}(g)\)

Follows from polynomial ring as graded ring. By assumption \(f + g \in R_n \setminus \{0\}\) and thus by definition \(\mathrm{deg}(f + g) = n\)

2.2. \(n \coloneqq \mathrm{deg}(f) > \mathrm{deg}(g)\)

Let \(\alpha \cdot X^{n}\) be the monomial with maximal degree. Then \(\mathrm{deg}(\alpha X^n - g) = n\), since \(\mathrm{deg}(g) < n\)

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