The zero polynomial in a ring $R[X]$ of polynomials over a ring $R$ is the additive identity element ${0}$ of this polynomial ring: $$f\!+\!\textbf{0} \;=\; \textbf{0}\!+\!f \;=\; f \quad\forall\, f\in R[X]$$ 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. $$\textbf{0} \;:=\; (0,\,0,\,0,\,...).$$

Because always $$f\cdot\textbf{0} \;=\; \textbf{0}$$ and because in general $\deg(fg) = \deg(f)+\deg(g)$ when $R$ has no zero divisors, one may define that that the zero polynomial has no degree at all, or alternatively that $$\deg(\textbf{0}) \;=\; -\infty$$ (see the extended real numbers).