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).

Title	zero polynomial
Canonical name	ZeroPolynomial
Date of creation	2013-03-22 14:46:58
Last modified on	2013-03-22 14:46:58
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	13
Author	pahio (2872)
Entry type	Definition
Classification	msc 13P05
Classification	msc 11C08
Classification	msc 12E05
Related topic	PolynomialRingOverIntegralDomain
Related topic	OrderAndDegreeOfPolynomial
Related topic	MinimalPolynomialEndomorphism