zero polynomial

The zero polynomial in a ring $R[X]$ of polynomials over a ring $R$ is the identity element 0 of this polynomial ring:

$$f+\text{𝟎}=\text{𝟎}+f=f\mathit{\hspace{1em}}\forall f\in R[X]$$

So the zero polynomial is also the absorbing element for the multiplication of polynomials.

All coefficients of the zero polynomial are equal to 0, i.e.

$$\text{𝟎}:=(0,\mathrm{\hspace{0.17em}0},\mathrm{\hspace{0.17em}0},\mathrm{\dots }).$$

Because always

$$f\cdot \text{𝟎}=\text{𝟎}$$

and because in general  $\mathrm{deg}(fg)=\mathrm{deg}(f)+\mathrm{deg}(g)$  when $R$ has no zero divisors, one may define that that the zero polynomial has no degree (http://planetmath.org/Polynomial) at all, or alternatively that

$$\mathrm{deg}(\text{𝟎})=-\mathrm{\infty }$$

(see the extended real numbers).