What is the degree of the zero polynomial and why is it so? [duplicate]

Solution 1:

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.

Solution 2:

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 $\dim(0)=0$, 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 $\deg(P)\geq 2$.