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+% Copyright The Numerical Algorithms Group Limited 1992-94. All rights reserved.
+% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk.
+\newcommand{\XPolynomialXmpTitle}{XPolynomial}
+\newcommand{\XPolynomialXmpNumber}{9.89}
+%
+% =====================================================================
+\begin{page}{XPolynomialXmpPage}{9.89 XPolynomial}
+% =====================================================================
+\beginscroll
+The \spadtype{XPolynomial} domain constructor implements multivariate polynomials
+whose set of variables is \spadtype{Symbol}.
+These variables do not commute.
+The only parameter of this construtor is 
+the coefficient ring which may be non-commutative.
+However, coefficients and variables commute.
+The representation of the polynomials is recursive.
+The abbreviation for \spadtype{XPolynomial} is \spadtype{XPOLY}.
+
+Other constructors like \spadtype{XPolynomialRing}, \spadtype{XRecursivePolynomial} 
+\spadtype{XDistributedPolynomial},
+\spadtype{LiePolynomial} and
+\spadtype{XPBWPolynomial}
+implement multivariate polynomials
+in non-commutative variables.
+
+We illustrate now some of the facilities of the \spadtype{XPOLY} domain constructor.
+
+
+\xtc{
+Define a polynomial ring over the integers.
+}{
+\spadpaste{poly := XPolynomial(Integer) \bound{poly}}
+}
+
+\xtc{
+Define a first polynomial,
+}{
+\spadpaste{pr: poly := 2*x + 3*y-5  \free{poly} \bound{pr}}
+}
+
+
+\xtc{
+and a second one.
+}{
+\spadpaste{pr2: poly := pr*pr  \free{poly} \bound{pr2}}
+}
+
+\xtc{
+Rewrite {\bf pr} in a distributive way,
+}{
+\spadpaste{pd  := expand pr   \free{pr} \bound{pd}}
+}
+
+\xtc{
+compute its square,
+}{
+\spadpaste{pd2 := pd*pd   \free{pd} \bound{pd2}}
+}
+
+\xtc{
+and checks that:
+}{
+\spadpaste{expand(pr2) - pd2 \free{pr2} \free{pd2}}
+}
+
+
+\xtc{
+We define:
+}{
+\spadpaste{qr :=  pr**3 \free{pr} \bound{qr}}
+}
+
+\xtc{
+and:
+}{
+\spadpaste{qd :=  pd**3 \free{pd} \bound{qd}}
+}
+
+\xtc{
+We truncate {\bf qd} at degree {\bf 3}:
+}{
+\spadpaste{ trunc(qd,2) \free{qd}}
+}
+
+\xtc{
+The same for {\bf qr}:
+}{
+\spadpaste{trunc(qr,2) \free{qr}}
+}
+
+\xtc{
+We define:
+}{
+\spadpaste{Word := OrderedFreeMonoid Symbol \bound{Word}}
+}
+
+\xtc{
+and:
+}{
+\spadpaste{w: Word := x*y**2 \free{Word} \bound{w}}
+}
+
+\xtc{
+The we can compute the right-quotient of {\bf qr} by {\bf r}:
+}{
+\spadpaste{rquo(qr,w) \free{qr} \free{w}}
+}
+
+\xtc{
+and the shuffle-product of {\bf pr} by {\bf r}:
+}{
+\spadpaste{sh(pr,w::poly) \free{pr} \free{w}}
+}
+\endscroll
+\autobuttons
+\end{page}
+%