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The zero polynomial has every coefficient zero. It is the zero vector of a vector space of polynomials and the additive identity element of a polynomial ring. Its degree of a polynomial is often set to $-\infty$ when a formula requires one.

The zero vector of a vector space is its additive identity: $v+0=0+v=v$ for every vector $v$. It is unique, because if $0^{\prime }$ is another such identity then $0=0+0^{\prime }=0^{\prime }$. Distributivity gives $\alpha 0=\alpha (0+0)=\alpha 0+\alpha 0$, so $\alpha 0=0$ after cancellation. In a coordinate vector space it has every coordinate zero.