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\title{Polynomial Input and Output}
\author{George E. Collins}
\maketitle

The primary internal representation of a polynomial is the
{\it sparse recursive} representation.  Let $A(x_1,\ldots,x_n)$
be a nonzero polynomial in $n$ variables over some domain $D$ 
for which a representation has already been established.  The
variables $x_i$ are considered to have been assigned the
particular order $x_1,x_2,\ldots,x_n$.  We express
$A(x_1,\ldots,x_n)$ uniquely as

\[\sum_{i=1}^k A_1(x_1,\ldots,x_{n-1}) x_n^{e_i}\]

with each $A_1 \neq 0$ and $e_1 > e_2 > \ldots > e_n$.  By
induction on $n$ we may assume that a representation $A_i^*$
for each $A_i$ has already been defined.  Then the sparse
recursive representation for $A$ is the list 
$(e_1,A_1^*,\ldots e_k,A_k^*)$.  Notice that names for the
variables do not occur in this internal representation.

There is a corresponding {\it external sparse recursive}
polynomial representation.  Following is an example,
for the case $n = 2$ and $D = {\bf Z}$ of this representation.

\begin{center}
$\verb@((x^2+1)y^3+(2x+8)y+(-5))@$
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