Here's the response I gave to
@hartnn , some time ago.
Post 1:
Sorry, but, you should not define
\[
\deg(0)=0
\]
Typically, it's a good idea to define it as:
\[
\deg(0)=-\infty
\]
But, it is mainly used for well-ordering principles in rings \(R\), such that it is \(R[x]\). It can also be left undefined, but either definition causes problems at some point or another in a proof.
Post 2:
And I say "well-ordering principles" with a grain of salt. I mean it to be something similar to the applied well-ordering of \(\mathbb{Z}\), or absolute ordering of \(\mathbb{Q}\) or \(\mathbb{R}\).
Post 3:
Last random, separated post... hopefully. So, we take, for example:
\[P(x), Q(x)\in R[x]\]
Where \(R[x]\) is a polynomial ring, then, we wish to maintain the following desirable properties (required for any absolute measure):
\[
\deg(P(x)+Q(x))\leq\max(\deg(P(x)), \deg(Q(x)))\\
\deg(P(x)Q(x))\leq \deg(P(x))+\deg(Q(x))
\]
That's why it leads us to the conclusion of \(\deg(0)=-\infty\) (where \(0\in R\) is the additive identity). The point is, though, it *could* screw you over, either way, in a proof (as it has me, more than once in number theory), unless you define that specific case as something different, or treat it as a separate case.