zero polynomial ZeroPolynomial2 2004-10-29 05:38:27 2009-10-10 18:09:03 Definition polynomial ring % 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 The {\em zero polynomial} in a ring $R[X]$ of polynomials over a ring $R$ is the \PMlinkescapetext{additive} identity element $\textbf{0}$ 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 \PMlinkname{degree}{Polynomial} at all, or alternatively that $$\deg(\textbf{0}) \;=\; -\infty$$ (see the extended real numbers).