@@ -1,116 +1,116 @@
-\documentstyle[11pt]{article}
-
-\title{R/I\_SOLVE: Rational/Integer Polynomial Solvers}
-
-\author{Francis J. Wright \\
-School of Mathematical Sciences \\
-Queen Mary and Westfield College \\
-University of London \\
-Mile End Road, London E1 4NS, UK. \\
-E-mail: {\tt F.J.Wright@QMW.ac.uk}}
-
-\date{27 January 1995}
-
-\begin{document}
-\maketitle
-
-\begin{abstract}
-  This package provides the operators \verb|r/i_solve| that compute
-  respectively the exact rational or integer zeros of a single
-  univariate polynomial using fast modular methods.
-\end{abstract}
-
-
-\section{Introduction}
-
-This package provides operators that compute the exact rational zeros
-of a single univariate polynomial using fast modular methods.  The
-algorithm used is that described by R. Loos (1983): Computing rational
-zeros of integral polynomials by $p$-adic expansion, {\it SIAM J.
-Computing}, {\bf 12}, 286--293.  The operator \verb|r_solve| computes
-all rational zeros whereas the operator \verb|i_solve| computes only
-integer zeros in a way that is slightly more efficient than extracting
-them from the rational zeros.  The \verb|r_solve| and \verb|i_solve|
-interfaces are almost identical, and are intended to be completely
-compatible with that of the general \verb|solve| operator, although
-\verb|r_solve| and \verb|i_solve| give more convenient output when
-only rational or integer zeros respectively are required.  The current
-implementation appears to be faster than \verb|solve| by a factor that
-depends on the example, but is typically up to about 2.
-
-I plan to extend this package to compute Gaussian integer and rational
-zeros and zeros of polynomial systems.
-
-
-\section{The user interface}
-
-The first argument is required and must simplify to either a
-univariate polynomial expression or equation with integer, rational or
-rounded coefficients.  Symbolic coefficients are not allowed (and
-currently complex coefficients are not allowed either.)  The argument
-is simplified to a quotient of integer polynomials and the denominator
-is silently ignored.
-
-Subsequent arguments are optional.  If the polynomial variable is to
-be specified then it must be the first optional argument, and if the
-first optional argument is not a valid option (see below) then it is
-(mis-)interpreted as the polynomial variable.  However, since the
-variable in a non-constant univariate polynomial can be deduced from
-the polynomial it is unnecessary to specify it separately, except in
-the degenerate case that the first argument simplifies to either 0 or
-$0 = 0$.  In this case the result is returned by \verb|i_solve| in
-terms of the operator \verb|arbint| and by \verb|r_solve| in terms of
-the (new) analogous operator \verb|arbrat|.  The operator
-\verb|i_solve| will generally run slightly faster than \verb|r_solve|.
-
-The (rational or integer) zeros of the first argument are returned as
-a list and the default output format is the same as that used by
-\verb|solve|.  Each distinct zero is returned in the form of an
-equation with the variable on the left and the multiplicities of the
-zeros are assigned to the variable \verb|root_multiplicities| as a
-list.  However, if the switch \verb|multiplicities| is turned on then
-each zero is explicitly included in the solution list the appropriate
-number of times (and \verb|root_multiplicities| has no value).
-
-\begin{sloppypar}
-Optional keyword arguments acting as local switches allow other output
-formats.  They have the following meanings:
-\begin{description}
-\item[\verb|separate|:] assign the multiplicity list to the global
-  variable \verb|root_multiplicities| (the default);
-\item[\verb|expand| or \verb|multiplicities|:] expand the solution
-  list to include multiple zeros multiple times (the default if the
-  \verb|multiplicities| switch is on);
-\item[\verb|together|:] return each solution as a list whose second
-  element is the multiplicity;
-\item[\verb|nomul|:] do not compute multiplicities (thereby saving
-  some time);
-\item[\verb|noeqs|:] do not return univariate zeros as equations but
-  just as values.
-\end{description}
-\end{sloppypar}
-
-
-\section{Examples}
-
-\begin{verbatim}
-r_solve((9x^2 - 16)*(x^2 - 9), x);
-\end{verbatim}
-\[
-  \left\{x=\frac{-4}{3},x=3,x=-3,x=\frac{4}{3}\right\}
-\]
-\begin{verbatim}
-i_solve((9x^2 - 16)*(x^2 - 9), x);
-\end{verbatim}
-\[
-  \{x=3,x=-3\}
-\]
-See the test/demonstration file \verb|rsolve.tst| for more examples.
-
-
-\section{Tracing}
-
-The switch {\tt trsolve} turns on tracing of the algorithm.  It is off
-by default.
-
-\end{document}
+\documentstyle[11pt]{article}
+
+\title{R/I\_SOLVE: Rational/Integer Polynomial Solvers}
+
+\author{Francis J. Wright \\
+School of Mathematical Sciences \\
+Queen Mary and Westfield College \\
+University of London \\
+Mile End Road, London E1 4NS, UK. \\
+E-mail: {\tt F.J.Wright@QMW.ac.uk}}
+
+\date{27 January 1995}
+
+\begin{document}
+\maketitle
+
+\begin{abstract}
+  This package provides the operators \verb|r/i_solve| that compute
+  respectively the exact rational or integer zeros of a single
+  univariate polynomial using fast modular methods.
+\end{abstract}
+
+
+\section{Introduction}
+
+This package provides operators that compute the exact rational zeros
+of a single univariate polynomial using fast modular methods.  The
+algorithm used is that described by R. Loos (1983): Computing rational
+zeros of integral polynomials by $p$-adic expansion, {\it SIAM J.
+Computing}, {\bf 12}, 286--293.  The operator \verb|r_solve| computes
+all rational zeros whereas the operator \verb|i_solve| computes only
+integer zeros in a way that is slightly more efficient than extracting
+them from the rational zeros.  The \verb|r_solve| and \verb|i_solve|
+interfaces are almost identical, and are intended to be completely
+compatible with that of the general \verb|solve| operator, although
+\verb|r_solve| and \verb|i_solve| give more convenient output when
+only rational or integer zeros respectively are required.  The current
+implementation appears to be faster than \verb|solve| by a factor that
+depends on the example, but is typically up to about 2.
+
+I plan to extend this package to compute Gaussian integer and rational
+zeros and zeros of polynomial systems.
+
+
+\section{The user interface}
+
+The first argument is required and must simplify to either a
+univariate polynomial expression or equation with integer, rational or
+rounded coefficients.  Symbolic coefficients are not allowed (and
+currently complex coefficients are not allowed either.)  The argument
+is simplified to a quotient of integer polynomials and the denominator
+is silently ignored.
+
+Subsequent arguments are optional.  If the polynomial variable is to
+be specified then it must be the first optional argument, and if the
+first optional argument is not a valid option (see below) then it is
+(mis-)interpreted as the polynomial variable.  However, since the
+variable in a non-constant univariate polynomial can be deduced from
+the polynomial it is unnecessary to specify it separately, except in
+the degenerate case that the first argument simplifies to either 0 or
+$0 = 0$.  In this case the result is returned by \verb|i_solve| in
+terms of the operator \verb|arbint| and by \verb|r_solve| in terms of
+the (new) analogous operator \verb|arbrat|.  The operator
+\verb|i_solve| will generally run slightly faster than \verb|r_solve|.
+
+The (rational or integer) zeros of the first argument are returned as
+a list and the default output format is the same as that used by
+\verb|solve|.  Each distinct zero is returned in the form of an
+equation with the variable on the left and the multiplicities of the
+zeros are assigned to the variable \verb|root_multiplicities| as a
+list.  However, if the switch \verb|multiplicities| is turned on then
+each zero is explicitly included in the solution list the appropriate
+number of times (and \verb|root_multiplicities| has no value).
+
+\begin{sloppypar}
+Optional keyword arguments acting as local switches allow other output
+formats.  They have the following meanings:
+\begin{description}
+\item[\verb|separate|:] assign the multiplicity list to the global
+  variable \verb|root_multiplicities| (the default);
+\item[\verb|expand| or \verb|multiplicities|:] expand the solution
+  list to include multiple zeros multiple times (the default if the
+  \verb|multiplicities| switch is on);
+\item[\verb|together|:] return each solution as a list whose second
+  element is the multiplicity;
+\item[\verb|nomul|:] do not compute multiplicities (thereby saving
+  some time);
+\item[\verb|noeqs|:] do not return univariate zeros as equations but
+  just as values.
+\end{description}
+\end{sloppypar}
+
+
+\section{Examples}
+
+\begin{verbatim}
+r_solve((9x^2 - 16)*(x^2 - 9), x);
+\end{verbatim}
+\[
+  \left\{x=\frac{-4}{3},x=3,x=-3,x=\frac{4}{3}\right\}
+\]
+\begin{verbatim}
+i_solve((9x^2 - 16)*(x^2 - 9), x);
+\end{verbatim}
+\[
+  \{x=3,x=-3\}
+\]
+See the test/demonstration file \verb|rsolve.tst| for more examples.
+
+
+\section{Tracing}
+
+The switch {\tt trsolve} turns on tracing of the algorithm.  It is off
+by default.
+
+\end{document}