@@ -1,273 +1,273 @@
-\documentstyle[11pt,reduce]{article}
-
-\title{\bf XIDEAL\\
-Gr\"obner Bases for Exterior Algebra}
-
-\author{
-David Hartley\thanks{email: David.Hartley@gmd.de}       \\
-GMD, Institute I1                                       \\
-Schloss Birlinghoven                                    \\
-D--53757 St. Augustin                                   \\
-Germany                                                 \\
-\and
-Philip A.~Tuckey\thanks{email: pht@iws170.mppmu.mpg.de} \\
-Max Planck Institute for Physics                        \\
-Foehringer Ring 6                                       \\
-D--80805 Munich                                         \\
-Germany                                                 \\
-}
-
-\date{14th March 1994}
-
-\begin{document}
-\maketitle
-
-\section{Description}
-
-The method of Gr\"obner bases in commutative polynomial rings introduced by
-Buchberger (e.g.~\cite{Buchberger}) is a well-known and very important tool
-in polynomial ideal theory, for example in solving the ideal membership
-problem. XIDEAL extends the method to exterior algebras using
-algorithms from \cite{HT}.
-
-There are two main departures from the commutative polynomial case. First,
-owing to the non-commutative product in exterior algebras, ideals are no
-longer automatically two-sided, and it is necessary to distinguish between
-left and right ideals. Secondly, because there are zero divisors, confluent
-reduction relations are no longer sufficient to solve the ideal membership
-problem: a unique normal form for every polynomial does not guarantee that
-all elements in the ideal reduce to zero. This leads to two possible
-definitions of Gr\"obner bases as pointed out by Stokes \cite{Stokes}.
-
-XIDEAL constructs Gr\"obner bases for solving the left ideal membership
-problem: Gr\"obner left ideal bases or GLIBs. For graded ideals, where each
-form is homogeneous in degree, the distinction between left and right
-ideals vanishes. Furthermore, if the generating forms are all homogeneous,
-then the Gr\"obner bases for the non-graded and graded ideals are
-identical. In this case, XIDEAL is able to save time by truncating the
-Gr\"obner basis at some maximum degree if desired.
-
-XIDEAL uses the exterior calculus package EXCALC of E.~Schr\"ufer
-\cite{EXCALC} to provide the exterior algebra definitions. EXCALC must be
-loaded before XIDEAL will work. The basis 1-forms for the exterior algebra
-are automatically extracted from the input. Consequently, each expression
-must be written in terms of these 1-forms -- $p$-form kernels with $p>1$
-are not allowed. Similarly all distinct 1-forms in the input are assumed to
-be linearly independent -- if a dimension has been fixed (using the EXCALC
-\f{SPACEDIM} or \f{COFRAME} statements), then input containing more than this
-number of distinct 1-forms will generate an error. The exterior algebra can
-be based on either an abstract vector space or the cotangent space at some
-fixed point on a manifold. Any functions or 0-forms are treated as constant
-non-vanishing coefficients.
-
-The term ordering used is the graded lexicographical ordering based on the
-prevailing EXCALC kernel ordering for the basis 1-forms. This puts
-highest degree first and then sorts terms of the same degree
-lexicographically. The EXCALC kernel ordering can be changed with the
-\REDUCE{} \f{KORDER} or EXCALC \f{FORDER} statements.
-
-
-\section{Operators}
-
-\subsubsection*{XIDEAL}
-
-\f{XIDEAL} calculates a Gr\"obner left ideal basis in
-an exterior algebra. The syntax is
-\begin{verbatim}
-     XIDEAL(S:list of forms[,R:integer]):list of forms.
-\end{verbatim}
-\f{XIDEAL} calculates the Gr\"obner left ideal basis for the left ideal
-generated by \f{S} using graded lexicographical ordering based on the
-current kernel ordering. The resulting list can be used for subsequent
-reductions with \f{XMODULOP} as long as the kernel ordering is not
-changed. If the set of generators \f{S} is graded, an optional parameter
-\f{R} can be given, and \f{XIDEAL} produces a truncated basis suitable for
-reducing exterior forms of degree less than or equal to \f{R} in the left
-ideal. This can save time and space with large expressions, but the result
-cannot be used for exterior forms of degree greater than \f{R}. See also
-the switches \f{XSTATS} and \f{XFULLREDUCTION}.
-
-
-\subsubsection*{XMODULO}
-
-\f{XMODULO} reduces exterior forms to their (unique) normal forms modulo a
-left ideal. The syntax is
-\begin{verbatim}
-     XMODULO(F:form, S:list of forms):form
-\end{verbatim}
-or
-\begin{verbatim}
-     XMODULO(F:list of forms, S:list of forms):list of forms.
-\end{verbatim}
-An alternative infix syntax is also available:
-\begin{verbatim}
-     F XMODULO S.
-\end{verbatim}
-\f{XMODULO(F,S)} first calculates a Gr\"obner basis for the left ideal
-generated by \f{S}, and then reduces \f{F}. \f{F} may be either a single
-exterior form, or a list of forms, and \f{S} is a list of forms. If \f{F}
-is a list of forms, each element is reduced, and any which vanish are
-deleted from the result.  If this operator is used more than once, and
-\f{S} does not change between calls, then the Gr\"obner basis is not
-recalculated. If the set of generators \f{S} is graded, then a truncated
-Gr\"obner basis is calculated using the degree of \f{F} (or the maximal
-degree in \f{F}).
-
-\subsubsection*{XMODULOP}
-
-\f{XMODULOP} reduces exterior forms to their (not necessarily unique)
-normal forms modulo a set of exterior polynomials. The syntax is
-\begin{verbatim}
-     XMODULOP(F:form, S:list of forms):form
-\end{verbatim}
-or
-\begin{verbatim}
-     XMODULOP(F:list of forms, S:list of forms):list of forms.
-\end{verbatim}
-An alternative infix syntax is also available:
-\begin{verbatim}
-     F XMODULOP S.
-\end{verbatim}
-\f{XMODULOP(F,S)} reduces \f{F} with respect to the set of exterior
-polynomials \f{S}, which is not necessarily a Gr\"obner basis. \f{F} may be
-either a single exterior form, or a list of forms, and \f{S} is a list of
-forms. This operator can be used in conjunction with \f{XIDEAL} to produce
-the same effect as \f{XMODULO}: for a single form \f{F} in an ideal
-generated by the graded set \f{S}, \f{F XMODULO S} is equivalent to \f{F
-XMODULOP XIDEAL(S,EXDEGREE F)}.
-
-%\subsubsection*{SUBFORM}
-%
-%\f{SUBFORM} extends the \f{SUB} operator from ordinary {\REDUCE} to perform
-%substitutions for factors in exterior products. For example
-%\begin{verbatim}
-%     SUB(d X^d Y = 0, d X^d Y^d Z)         -> d X^d Y^d Z
-%     SUBFORM(d X^d Y = 0, d X^d Y^d Z)     -> 0.
-%\end{verbatim}
-%Care must be taken that the left-hand sides of substitutions are kernels,
-%eg. \verb.d X^d Y. is a kernel in the usual ordering, but  \verb.d Y^d X.
-%isn't.
-
-
-\section{Switches}
-
-\subsubsection*{XFULLREDUCE}
-
-\f{ON XFULLREDUCE} allows \f{XIDEAL} and \f{XMODULO} to calculate reduced
-(but not necessarily normed) Gr\"obner bases, which speeds up subsequent
-reductions, and guarantees a unique form (up to scaling) for the Gr\"obner
-basis. \f{OFF XFULLREDUCE} turns of this feature, which may speed up
-calculation of the Gr\"obner basis. \f{XFULLREDUCE} is \f{ON} by default.
-
-
-%\subsubsection*{XGRADED}
-%
-%\f{ON XGRADED} tells \f{XIDEAL} and \f{XMODULO} that all exterior ideals
-%under consideration are graded: consist of forms of homogeneous degree. In
-%this case, there is no distinction between left, right and two-sided
-%ideals, and it is possible to truncate the calculation of Gr\"obner bases
-%at the degree of the form or forms to be reduced. This saves considerable
-%time and space. For non-graded ideals, \f{XGRADED} must be switched
-%\f{OFF}. \f{XGRADED} is \f{ON} by default.
-
-\subsubsection*{XSTATS}
-
-\f{ON XSTATS} produces counting and timing information. As \f{XIDEAL} is
-running, a hash mark (\verb.#.) is printed for each form taken from the
-input list, followed by a sequences of carets (\verb.^.) and dollar signs
-(\verb.$.). Each caret represents a new basis element obtained by a simple
-wedge product, and each dollar sign represents a new basis element obtained
-from an S-polynomial. At the end, a table is printed summarising the
-calculation. \f{XSTATS} is \f{OFF} by default.
-
-\section{Examples}
-
-Suppose EXCALC and XIDEAL have been loaded, the switches are at their
-default settings, and the following exterior variables have been declared:
-
-\begin{verbatim}
-     pform x=0,y=0,z=0,t=0,f(i)=1,h=0,hx=0,ht=0;
-\end{verbatim}
-
-In a commutative polynomial ring, a single polynomial is its own Gr\"obner
-basis. This is no longer true for exterior algebras because of the presence
-of zero divisors, and can lead to some surprising reductions:
-\begin{verbatim}
-     xideal {d x^d y - d z^d t};
-
-          {d T^d Z + d X^d Y,
-
-           d X^d Y^d Z,
-
-           d T^d X^d Y}
-
-     f(3)^f(4)^f(5)^f(6)
-       xmodulo {f(1)^f(2) + f(3)^f(4) + f(5)^f(6)};
-
-          0
-\end{verbatim}
-
-The heat equation, $h_{xx}=h_t$ can be represented by the following
-exterior differential system.
-\begin{verbatim}
-     S := {d h - ht*d t - hx*d x,
-           d ht^d t + d hx^d x,
-           d hx^d t - ht*d x^d t};
-\end{verbatim}
-\f{XMODULO} can be used to check that the exterior differential system is
-closed under exterior differentiation.
-\begin{verbatim}
-     d S xmodulo S;
-
-          {}
-\end{verbatim}
-
-Non-graded left and right ideals are no longer the same:
-\begin{verbatim}
-     d t^(d z+d x^d y) xmodulo {d z+d x^d y};
-
-          0
-
-     (d z+d x^d y)^d t xmodulo {d z+d x^d y};
-
-           - 2*d t^d z
-\end{verbatim}
-Higher order forms can now reduce lower order ones:
-\begin{verbatim}
-     d x xmodulo {d y^d z + d x,d x^d y + d z};
-
-          0
-\end{verbatim}
-Any form containing a 0-form term generates the whole ideal:
-\begin{verbatim}
-     xideal {1 + f(1) + f(1)^f(2) + f(2)^f(3)^f(4)};
-
-          {1}
-\end{verbatim}
-
-
-\begin{thebibliography}{M}
-
-\bibitem{Buchberger}
-        B.~Buchberger, {\em
-        Gr\"obner Bases: an algorithmic method in polynomial ideal theory,}
-        in {\em Multidimensional Systems Theory\/} ed.~N.K.~Bose
-        (Reidel, Dordrecht, 1985) chapter 6.
-\bibitem{HT}
-        D.~Hartley and P.A.~Tuckey, {\em
-        A Direct Characterisation of Gr\"obner Bases in Clifford and
-        Grassmann Algebras,}
-        Preprint MPI-Ph/93--96 (1993).
-\bibitem{Stokes}
-        T.~Stokes, {\em
-        Gr\"obner bases in exterior algebra,}
-        J.~Automated Reasoning {\bf 6}(1990)233--250.
-\bibitem{EXCALC}
-        E.~Schr\"ufer, {\em
-        EXCALC, a system for doing calculations in the calculus of modern
-        differential geometry, User's manual,}
-        (The Rand Corporation, Santa Monica, 1986).
-
-\end{thebibliography}
-
-\end{document}
+\documentstyle[11pt,reduce]{article}
+
+\title{\bf XIDEAL\\
+Gr\"obner Bases for Exterior Algebra}
+
+\author{
+David Hartley\thanks{email: David.Hartley@gmd.de}       \\
+GMD, Institute I1                                       \\
+Schloss Birlinghoven                                    \\
+D--53757 St. Augustin                                   \\
+Germany                                                 \\
+\and
+Philip A.~Tuckey\thanks{email: pht@iws170.mppmu.mpg.de} \\
+Max Planck Institute for Physics                        \\
+Foehringer Ring 6                                       \\
+D--80805 Munich                                         \\
+Germany                                                 \\
+}
+
+\date{14th March 1994}
+
+\begin{document}
+\maketitle
+
+\section{Description}
+
+The method of Gr\"obner bases in commutative polynomial rings introduced by
+Buchberger (e.g.~\cite{Buchberger}) is a well-known and very important tool
+in polynomial ideal theory, for example in solving the ideal membership
+problem. XIDEAL extends the method to exterior algebras using
+algorithms from \cite{HT}.
+
+There are two main departures from the commutative polynomial case. First,
+owing to the non-commutative product in exterior algebras, ideals are no
+longer automatically two-sided, and it is necessary to distinguish between
+left and right ideals. Secondly, because there are zero divisors, confluent
+reduction relations are no longer sufficient to solve the ideal membership
+problem: a unique normal form for every polynomial does not guarantee that
+all elements in the ideal reduce to zero. This leads to two possible
+definitions of Gr\"obner bases as pointed out by Stokes \cite{Stokes}.
+
+XIDEAL constructs Gr\"obner bases for solving the left ideal membership
+problem: Gr\"obner left ideal bases or GLIBs. For graded ideals, where each
+form is homogeneous in degree, the distinction between left and right
+ideals vanishes. Furthermore, if the generating forms are all homogeneous,
+then the Gr\"obner bases for the non-graded and graded ideals are
+identical. In this case, XIDEAL is able to save time by truncating the
+Gr\"obner basis at some maximum degree if desired.
+
+XIDEAL uses the exterior calculus package EXCALC of E.~Schr\"ufer
+\cite{EXCALC} to provide the exterior algebra definitions. EXCALC must be
+loaded before XIDEAL will work. The basis 1-forms for the exterior algebra
+are automatically extracted from the input. Consequently, each expression
+must be written in terms of these 1-forms -- $p$-form kernels with $p>1$
+are not allowed. Similarly all distinct 1-forms in the input are assumed to
+be linearly independent -- if a dimension has been fixed (using the EXCALC
+\f{SPACEDIM} or \f{COFRAME} statements), then input containing more than this
+number of distinct 1-forms will generate an error. The exterior algebra can
+be based on either an abstract vector space or the cotangent space at some
+fixed point on a manifold. Any functions or 0-forms are treated as constant
+non-vanishing coefficients.
+
+The term ordering used is the graded lexicographical ordering based on the
+prevailing EXCALC kernel ordering for the basis 1-forms. This puts
+highest degree first and then sorts terms of the same degree
+lexicographically. The EXCALC kernel ordering can be changed with the
+\REDUCE{} \f{KORDER} or EXCALC \f{FORDER} statements.
+
+
+\section{Operators}
+
+\subsubsection*{XIDEAL}
+
+\f{XIDEAL} calculates a Gr\"obner left ideal basis in
+an exterior algebra. The syntax is
+\begin{verbatim}
+     XIDEAL(S:list of forms[,R:integer]):list of forms.
+\end{verbatim}
+\f{XIDEAL} calculates the Gr\"obner left ideal basis for the left ideal
+generated by \f{S} using graded lexicographical ordering based on the
+current kernel ordering. The resulting list can be used for subsequent
+reductions with \f{XMODULOP} as long as the kernel ordering is not
+changed. If the set of generators \f{S} is graded, an optional parameter
+\f{R} can be given, and \f{XIDEAL} produces a truncated basis suitable for
+reducing exterior forms of degree less than or equal to \f{R} in the left
+ideal. This can save time and space with large expressions, but the result
+cannot be used for exterior forms of degree greater than \f{R}. See also
+the switches \f{XSTATS} and \f{XFULLREDUCTION}.
+
+
+\subsubsection*{XMODULO}
+
+\f{XMODULO} reduces exterior forms to their (unique) normal forms modulo a
+left ideal. The syntax is
+\begin{verbatim}
+     XMODULO(F:form, S:list of forms):form
+\end{verbatim}
+or
+\begin{verbatim}
+     XMODULO(F:list of forms, S:list of forms):list of forms.
+\end{verbatim}
+An alternative infix syntax is also available:
+\begin{verbatim}
+     F XMODULO S.
+\end{verbatim}
+\f{XMODULO(F,S)} first calculates a Gr\"obner basis for the left ideal
+generated by \f{S}, and then reduces \f{F}. \f{F} may be either a single
+exterior form, or a list of forms, and \f{S} is a list of forms. If \f{F}
+is a list of forms, each element is reduced, and any which vanish are
+deleted from the result.  If this operator is used more than once, and
+\f{S} does not change between calls, then the Gr\"obner basis is not
+recalculated. If the set of generators \f{S} is graded, then a truncated
+Gr\"obner basis is calculated using the degree of \f{F} (or the maximal
+degree in \f{F}).
+
+\subsubsection*{XMODULOP}
+
+\f{XMODULOP} reduces exterior forms to their (not necessarily unique)
+normal forms modulo a set of exterior polynomials. The syntax is
+\begin{verbatim}
+     XMODULOP(F:form, S:list of forms):form
+\end{verbatim}
+or
+\begin{verbatim}
+     XMODULOP(F:list of forms, S:list of forms):list of forms.
+\end{verbatim}
+An alternative infix syntax is also available:
+\begin{verbatim}
+     F XMODULOP S.
+\end{verbatim}
+\f{XMODULOP(F,S)} reduces \f{F} with respect to the set of exterior
+polynomials \f{S}, which is not necessarily a Gr\"obner basis. \f{F} may be
+either a single exterior form, or a list of forms, and \f{S} is a list of
+forms. This operator can be used in conjunction with \f{XIDEAL} to produce
+the same effect as \f{XMODULO}: for a single form \f{F} in an ideal
+generated by the graded set \f{S}, \f{F XMODULO S} is equivalent to \f{F
+XMODULOP XIDEAL(S,EXDEGREE F)}.
+
+%\subsubsection*{SUBFORM}
+%
+%\f{SUBFORM} extends the \f{SUB} operator from ordinary {\REDUCE} to perform
+%substitutions for factors in exterior products. For example
+%\begin{verbatim}
+%     SUB(d X^d Y = 0, d X^d Y^d Z)         -> d X^d Y^d Z
+%     SUBFORM(d X^d Y = 0, d X^d Y^d Z)     -> 0.
+%\end{verbatim}
+%Care must be taken that the left-hand sides of substitutions are kernels,
+%eg. \verb.d X^d Y. is a kernel in the usual ordering, but  \verb.d Y^d X.
+%isn't.
+
+
+\section{Switches}
+
+\subsubsection*{XFULLREDUCE}
+
+\f{ON XFULLREDUCE} allows \f{XIDEAL} and \f{XMODULO} to calculate reduced
+(but not necessarily normed) Gr\"obner bases, which speeds up subsequent
+reductions, and guarantees a unique form (up to scaling) for the Gr\"obner
+basis. \f{OFF XFULLREDUCE} turns of this feature, which may speed up
+calculation of the Gr\"obner basis. \f{XFULLREDUCE} is \f{ON} by default.
+
+
+%\subsubsection*{XGRADED}
+%
+%\f{ON XGRADED} tells \f{XIDEAL} and \f{XMODULO} that all exterior ideals
+%under consideration are graded: consist of forms of homogeneous degree. In
+%this case, there is no distinction between left, right and two-sided
+%ideals, and it is possible to truncate the calculation of Gr\"obner bases
+%at the degree of the form or forms to be reduced. This saves considerable
+%time and space. For non-graded ideals, \f{XGRADED} must be switched
+%\f{OFF}. \f{XGRADED} is \f{ON} by default.
+
+\subsubsection*{XSTATS}
+
+\f{ON XSTATS} produces counting and timing information. As \f{XIDEAL} is
+running, a hash mark (\verb.#.) is printed for each form taken from the
+input list, followed by a sequences of carets (\verb.^.) and dollar signs
+(\verb.$.). Each caret represents a new basis element obtained by a simple
+wedge product, and each dollar sign represents a new basis element obtained
+from an S-polynomial. At the end, a table is printed summarising the
+calculation. \f{XSTATS} is \f{OFF} by default.
+
+\section{Examples}
+
+Suppose EXCALC and XIDEAL have been loaded, the switches are at their
+default settings, and the following exterior variables have been declared:
+
+\begin{verbatim}
+     pform x=0,y=0,z=0,t=0,f(i)=1,h=0,hx=0,ht=0;
+\end{verbatim}
+
+In a commutative polynomial ring, a single polynomial is its own Gr\"obner
+basis. This is no longer true for exterior algebras because of the presence
+of zero divisors, and can lead to some surprising reductions:
+\begin{verbatim}
+     xideal {d x^d y - d z^d t};
+
+          {d T^d Z + d X^d Y,
+
+           d X^d Y^d Z,
+
+           d T^d X^d Y}
+
+     f(3)^f(4)^f(5)^f(6)
+       xmodulo {f(1)^f(2) + f(3)^f(4) + f(5)^f(6)};
+
+          0
+\end{verbatim}
+
+The heat equation, $h_{xx}=h_t$ can be represented by the following
+exterior differential system.
+\begin{verbatim}
+     S := {d h - ht*d t - hx*d x,
+           d ht^d t + d hx^d x,
+           d hx^d t - ht*d x^d t};
+\end{verbatim}
+\f{XMODULO} can be used to check that the exterior differential system is
+closed under exterior differentiation.
+\begin{verbatim}
+     d S xmodulo S;
+
+          {}
+\end{verbatim}
+
+Non-graded left and right ideals are no longer the same:
+\begin{verbatim}
+     d t^(d z+d x^d y) xmodulo {d z+d x^d y};
+
+          0
+
+     (d z+d x^d y)^d t xmodulo {d z+d x^d y};
+
+           - 2*d t^d z
+\end{verbatim}
+Higher order forms can now reduce lower order ones:
+\begin{verbatim}
+     d x xmodulo {d y^d z + d x,d x^d y + d z};
+
+          0
+\end{verbatim}
+Any form containing a 0-form term generates the whole ideal:
+\begin{verbatim}
+     xideal {1 + f(1) + f(1)^f(2) + f(2)^f(3)^f(4)};
+
+          {1}
+\end{verbatim}
+
+
+\begin{thebibliography}{M}
+
+\bibitem{Buchberger}
+        B.~Buchberger, {\em
+        Gr\"obner Bases: an algorithmic method in polynomial ideal theory,}
+        in {\em Multidimensional Systems Theory\/} ed.~N.K.~Bose
+        (Reidel, Dordrecht, 1985) chapter 6.
+\bibitem{HT}
+        D.~Hartley and P.A.~Tuckey, {\em
+        A Direct Characterisation of Gr\"obner Bases in Clifford and
+        Grassmann Algebras,}
+        Preprint MPI-Ph/93--96 (1993).
+\bibitem{Stokes}
+        T.~Stokes, {\em
+        Gr\"obner bases in exterior algebra,}
+        J.~Automated Reasoning {\bf 6}(1990)233--250.
+\bibitem{EXCALC}
+        E.~Schr\"ufer, {\em
+        EXCALC, a system for doing calculations in the calculus of modern
+        differential geometry, User's manual,}
+        (The Rand Corporation, Santa Monica, 1986).
+
+\end{thebibliography}
+
+\end{document}