What is the degree of the zero polynomial and why is it so? 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?
 A: I think $-\infty $ make sense. Indeed, let $P$ a polynomial of degree $\geq 1$. Then, you have that $$\deg(PQ)=\deg(P)+\deg(Q),$$
for every polynomial $Q$. Now, if you define $\deg(0)=0$, you'll get $$\deg(0\cdot P)=0+\deg(P)>0,$$
which is not compatible with the degree formula. The only way to give a sense to this formula is to define $\deg(0)=-\infty $.
Same if you defined $\deg(0)=-1$, the formula won't be compatible if $\deg(P)\geq 2$.
A: 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.
A: Well, it depends.
Mathematical practice shows that sometimes it is useful to define the degree of the zero polynomial to be zero, sometimes to define it to be $-\infty$ and sometimes to leave is undefined. Which option one chooses depends on what one is trying to do.
This is quite different with what happens with the degree of all other polynomials, which is always defined in the same way (*) But don't think that if for the slightiest of reasons we were to fnd it useful to change the definition to do something we wanted, we would.
(*) Actually, that is not exactly true: we sometimes put degrees on polynomials which are different from the usual ones, but usually only on polynomials with more than one variable.
