Nullpolünoom

Nullpolünoom, known in English as the zero polynomial, is the polynomial in a polynomial ring whose all coefficients are zero. In the polynomial ring R[x] over a commutative ring R with unity, it serves as the additive identity: for any polynomial P, P plus the zero polynomial equals P. The zero polynomial is unique, and a polynomial is the zero polynomial if and only if every coefficient is zero.

The degree of the zero polynomial is typically undefined or assigned a special value such as negative infinity, since it has no nonzero leading term. It is identically zero as a function on any ring: evaluating it at any element gives zero.

Core properties include that multiplying any polynomial by the zero polynomial yields the zero polynomial, and adding the zero polynomial to any polynomial leaves that polynomial unchanged. It is not a unit in the polynomial ring (except in the degenerate case where the ring itself is trivial, i.e., 0 = 1).

In algebraic terms, the set containing only the zero polynomial forms the zero ideal of the polynomial ring. It also plays the role of the zero vector in the vector space of polynomials when coefficients come from a field or a fixed ring.

Examples of the zero polynomial are simply 0, or any expression with all coefficients equal to zero. The concept is fundamental in polynomial arithmetic, providing the identity element for addition and behaving predictably under multiplication and evaluation.