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Polynomial and rational function manipulation

MAXIMA has a polynomial and rational function package which defines many useful operations on polynomials and rational functions of one or more variables. This package uses a different data representation than the general representation. This representation known as the Canonical Rational Expression is more suitable for polynomials and rational functions. The top-level function to put an expression into internal rational form is 


rat(expr);

For example, the rational form of

is 


((mrat simp ($x) (g00018)) (g00018 3 7 2 5 0 -19) . 1)
A rational form has the following parts:
  1. The header which is the CAR part of the rational form
  2. the CRE which is the CDR of the rational form
  3. the numerator which is the CAR part of the CRE
  4. the denominator which is the CDR part of the CRE

The numerator and denominator are in polynomial representation. The header is a list whose last two items are a list of variables and a corresponding list of internally generated symbols. In the above example the variable list is ($x) and the generated symbols list is (G00018). For implementational reasons, CRE forms use internally generated symbols instead of the user-supplied variables in the data structure.

As another example, the polynomial

has the following rational form. 


((mrat simp ($x $y) (g00019 g00018)) 
	  (g00018 2 1 1 (g00019 1 2) 0 (g00019 2 1)) . 1)

Thus, a multivariate polynomial is represented as a polynomial in the first variable (g00018) with coefficients represented as polynomials in the other variables. This understanding of how polynomial and rational functions are represented does not mean that we should write code depending on their exact structure. On the contrary, it is advisable to use the system supplied manipulation functions described below.

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farrell@mcs.kent.edu