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\chapter{IDEALS: Arithmetic for polynomial ideals}
\label{IDEALS}
\typeout{{IDEALS: Arithmetic for polynomial ideals}}

{\footnotesize
\begin{center}
Herbert Melenk \\
Konrad--Zuse--Zentrum f\"ur Informationstechnik Berlin \\
Takustra\"se 7 \\
D--14195 Berlin--Dahlem, Germany \\[0.05in]
e--mail: melenk@zib.de
\end{center}
}

\ttindex{IDEALS}

This package implements the basic arithmetic for polynomial ideals
by exploiting the Gr\"obner bases package of \REDUCE.
In order to save computing time all intermediate Gr\"obner bases
are stored internally such that time consuming repetitions
are inhibited. A uniform setting facilitates the access.

\section{Initialization}

Prior to any computation the set of variables has to be declared
by calling the operator $I\_setting$ .  For example in order to initiate
computations in the polynomial ring $Q[x,y,z]$ call
\begin{verbatim}
    I_setting(x,y,z);
\end{verbatim}
A subsequent call to $I\_setting$ allows one to select another set
of variables; at the same time the internal data structures
are cleared in order to free memory resources.

\section{Bases}

An ideal is represented by a basis (set of polynomials) tagged
with the symbol $I$, {\em e.g.\ }
\begin{verbatim}
   u := I(x*z-y**2, x**3-y*z);
\end{verbatim}
Alternatively a list of polynomials can be used as input basis; however,
all arithmetic results will be presented in the above form. The
operator $ideal2list$ allows one to convert an ideal basis into a
conventional \REDUCE\ list.

\subsection{Operators}

Because of syntactical restrictions in \REDUCE, special operators
have to be used for ideal arithmetic:

\begin{verbatim}
         .+            ideal sum (infix)
         .*            ideal product (infix)
         .:            ideal quotient (infix)
         ./            ideal quotient (infix)
         .=            ideal equality test (infix)
         subset        ideal inclusion test (infix)
         intersection  ideal intersection (prefix,binary)
         member        test for membership in an ideal
                         (infix: polynomial and ideal)
         gb            Groebner basis of an ideal (prefix, unary)
         ideal2list    convert ideal basis to polynomial list
                         (prefix,unary)
\end{verbatim}

Example:

\begin{verbatim}
    I(x+y,x^2) .* I(x-z);

      2                      2    2
   I(X  + X*Y - X*Z - Y*Z,X*Y  - Y *Z)
\end{verbatim}

Note that ideal equality cannot be tested with the \REDUCE\ equal sign:
\begin{verbatim}
   I(x,y)  = I(y,x)       is false
   I(x,y) .= I(y,x)       is true
\end{verbatim}