# polarization by differential operators

## Primary tabs

\documentclass{article}
% 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

\begin{document}
One can construct the polars of a polynomial by means of a differential
operator.  Suppose we have a homogeneous polynomial $p (x_1, \ldots, x_n)$.
To compute the polars of $p$ we act on it with the operator
\,$\Delta = y_1 \, \partial / \partial x_1 + \cdots +y_n \, \partial / \partial x_n$;\, the $k$-th polar of $p$ equals $\Delta^k p(x_1, \ldots x_n)$.
%%%%%
%%%%%
nd{document}