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'zero polynomial'
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| Title of object: |
zero polynomial |
| Canonical Name: |
ZeroPolynomial2 |
| Type: |
Definition |
| Created on: |
2004-10-29 05:38:27 |
| Modified on: |
2005-03-30 16:03:10 |
| Classification: |
msc:12E05, msc:11C08, msc:13P05 |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
The {\em zero polynomial} in a ring $R[X]$ of polynomials over a ring $R$ is the additive identity element $\textbf{0}$ of this polynomial ring:
$$f+\textbf{0} = \textbf{0}+f = f \quad\forall\, f\in R[X]$$
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
$$\deg(\textbf{0}) = -\infty$$
or that the zero polynomial has no \PMlinkname{degree}{Monomial} at all. |
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