(maxima.info)Introduction to Polynomials

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14.1 Introduction to Polynomials
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Polynomials are stored in Maxima either in General Form or as Canonical
Rational Expressions (CRE) form.  The latter is a standard form, and is
used internally by operations such as factor, ratsimp, and so on.

   Canonical Rational Expressions constitute a kind of representation
which is especially suitable for expanded polynomials and rational
functions (as well as for partially factored polynomials and rational
functions when RATFAC is set to 'true').  In this CRE form an ordering
of variables (from most to least main) is assumed for each expression.
Polynomials are represented recursively by a list consisting of the main
variable followed by a series of pairs of expressions, one for each term
of the polynomial.  The first member of each pair is the exponent of the
main variable in that term and the second member is the coefficient of
that term which could be a number or a polynomial in another variable
again represented in this form.  Thus the principal part of the CRE form
of 3*X^2-1 is (X 2 3 0 -1) and that of 2*X*Y+X-3 is (Y 1 (X 1 2) 0 (X 1
1 0 -3)) assuming Y is the main variable, and is (X 1 (Y 1 2 0 1) 0 -3)
assuming X is the main variable.  "Main"-ness is usually determined by
reverse alphabetical order.  The "variables" of a CRE expression needn't
be atomic.  In fact any subexpression whose main operator is not + - * /
or ^ with integer power will be considered a "variable" of the
expression (in CRE form) in which it occurs.  For example the CRE
variables of the expression X+SIN(X+1)+2*SQRT(X)+1 are X, SQRT(X), and
SIN(X+1).  If the user does not specify an ordering of variables by
using the RATVARS function Maxima will choose an alphabetic one.  In
general, CRE's represent rational expressions, that is, ratios of
polynomials, where the numerator and denominator have no common factors,
and the denominator is positive.  The internal form is essentially a
pair of polynomials (the numerator and denominator) preceded by the
variable ordering list.  If an expression to be displayed is in CRE form
or if it contains any subexpressions in CRE form, the symbol /R/ will
follow the line label.  See the RAT function for converting an
expression to CRE form.  An extended CRE form is used for the
representation of Taylor series.  The notion of a rational expression is
extended so that the exponents of the variables can be positive or
negative rational numbers rather than just positive integers and the
coefficients can themselves be rational expressions as described above
rather than just polynomials.  These are represented internally by a
recursive polynomial form which is similar to and is a generalization of
CRE form, but carries additional information such as the degree of
truncation.  As with CRE form, the symbol /T/ follows the line label of
such expressions.