TITLE: What is the degree of the zero polynomial and why is it so?
QUESTION [19 upvotes]: My teacher says-

The degree of the zero polynomial is undefined.

My book says-

The degree of the zero polynomial is defined to be zero.

Wikipedia says-

The degree of the zero polynomial is $-\infty$.

I am totally confused and want to know which one is true or are all true?

REPLY [7 votes]: Defining it as $-\infty$ makes the most sense.
As mentioned in Surb's answer and comments, some properties of degrees are kept intact this way, e.g.

$\deg(PQ)=\deg P+\deg Q$
If $\deg P>\deg Q$ then $\deg(P+Q)=\deg P$

It also starts making more sense if you consider expressions that can take on negative powers as well. That is, instead of $\sum_{k=0}^na_kx^k$, consider $\sum_{k=-\infty}^{n}a_kx^k$. So you could have $3x^2+2x$ and $x+1+3x^{-2}$ and $2x^{-3}-\frac45x^{-5}$. Then degree is just "the supremum of all $k$s for which $a_k\neq 0$. The degrees of these 3 expressions are 2, 1 and -3 respectively. Then it's easy to see that 0, which has no nonzero coefficients, has a degree of $-\infty$.
The same works if you consider expressions than can also have fractional degrees. Then the degree of, say, $3\sqrt{x}-x^{-3}$ is $1/2$.
Of course, this inspires the definition of a "dual" degree, which is the infimum instead of the supremum. Then the degree of $3x^4+2x^3+5x^2$ will be 2, and the degree of $0$ will be $\infty$.
Keeping the degree of $0$ undefined is understandable (not everyone wants to deal with infinities). Defining it as $-1$ has merits (if you don't consider negative powers, $0$ is one step down from nonzero constants). But there is absolutely no sense in defining the degree as $0$. The $0$ polynomial has as much similarity with constants, as constants have with linear polynomials.