reduce-historical/r38/packages/groebner/new1.tex

the following is new in july 2001:

\subsubsection{$greduce$\_$orders$: Reduction with several term orders}
The shortest polynomial with different polynomial term orders is computed
with the operator $greduce$\_$orders$:
 
\begin{description}
\ttindex{$greduce$\_$orders$}
\item[{\it greduce\_orders}]($exp$, \{$exp1$, $exp2$, \ldots , $expm$\}
[,\{$v_1$,$v_2$ \ldots $v_n$\}]);
 
where {\it exp} is an expression and $\{exp1, exp2,\ldots , expm\}$ is
a list of any number of expressions or equations. The list of variables
$v_1,v_2 \ldots v_n$ may be omitted; if set, the variables must be a list.
\end{description}
 
The expression {\it exp} is reduced by {\it greduce} with the orders
in the shared variable {\it gorders}, which must be a list of term
orders (if set). By default it is set to
 
\begin{center}
$\{revgradlex,gradlex,lex\}$
\end{center}
 
The shortest polynomial is the result.
The order with the shortest polynomial is set to the shared variable
{\it gorder}. A Groebner basis of the system \{$exp1$, $exp2$, \ldots ,
$expm$\} is computed for each element of $orders$.
With the default setting {\it gorder} in most cases will be set
to {\it revgradlex}.
If the variable set is given, these variables are taken; otherwise all
variables of the system \{$exp1$, $exp2$, \ldots , $expm$\} are
extracted.
 
The Groebner basis computations can take some time; if interrupted, the
intermediate result of the reduction is set to the shared variable
$greduce$\_$result$, if one is done already. However, this is not
nesessarily the minimal form.
 
If the variable {\it gorders} should be set to orders with a parameter,
the term oder has to be replaced by a list; the first element is the
term oder selected, followed by its parameter(s), e.g.
 
\begin{center}
$orders:=\{\{gradlexgradlex,2\},\{lexgradlex,2\}\}$
\end{center}