Initial population.

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+\documentclass{article}
+\usepackage{axiom}
+\begin{document}
+\title{\$SPAD/src/input dpol.input}
+\author{The Axiom Team}
+\maketitle
+\begin{abstract}
+\end{abstract}
+\eject
+\tableofcontents
+\eject
+\section{License}
+<<license>>=
+--Copyright The Numerical Algorithms Group Limited 1991.
+@
+<<*>>=
+<<license>>
+
+)clear all
+odvar:=ODVAR Symbol
+-- here are the first 5 derivatives of w
+-- the i-th derivative of w is printed as w subscript 5
+[makeVariable('w,i)$odvar for i in 5..0 by -1]
+-- these are now algebraic indeterminates, ranked in an orderly way
+-- in increasing order:
+sort %
+-- we now make a general differential polynomial ring
+-- instead of ODVAR, one can also use SDVAR for sequential ordering
+dpol:=DSMP (FRAC INT, Symbol, odvar)
+-- instead of using makeVariable, it is easier to
+-- think of a differential variable w as a map, where
+-- w.n is n-th derivative of w as an algebraic indeterminate
+w := makeVariable('w)$dpol
+-- create another one called z, which is higher in rank than w
+-- since we are ordering by Symbol
+z := makeVariable('z)$dpol
+-- now define some differential polynomial
+(f,b):dpol
+f:=w.4::dpol - w.1 * w.1 * z.3
+b:=(z.1::dpol)**3 * (z.2)**2 - w.2
+-- compute the leading derivative appearing in b
+lb:=leader b
+-- the separant is the partial derivative of b with respect to its leader
+sb:=separant b
+-- of course you can differentiate these differential polynomials
+-- and try to reduce f modulo the differential ideal generated by b
+-- first eliminate z.3 using the derivative of b
+bprime:= differentiate b
+-- find its leader
+lbprime:= leader bprime
+-- differentiate f partially with respect to lbprime
+pbf:=differentiate (f, lbprime)
+-- to obtain the partial remainder of f with respect to b
+ftilde:=sb * f- pbf * bprime
+-- note high powers of lb still appears in ftilde
+-- the initial is the leading coefficient when b is written
+-- as a univariate polynomial in its leader
+ib:=initial b
+-- compute the leading coefficient of ftilde
+-- as a polynomial in its leader
+lcef:=leadingCoefficient univariate(ftilde, lb)
+-- now to continue eliminating the high powers of lb appearing in ftilde:
+-- to obtain the remainder of f modulo b and its derivatives
+ 
+f0:=ib * ftilde - lcef * b * lb
+@
+\eject
+\begin{thebibliography}{99}
+\bibitem{1} nothing
+\end{thebibliography}
+\end{document}