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MultivariatePolynomial examples
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The domain constructor MultivariatePolynomial is similar to Polynomial
except that it specifies the variables to be used.  Polynomial are
available for MultivariatePolynomial.  The abbreviation for
MultivariatePolynomial is MPOLY.  The type expressions

  MultivariatePolynomial([x,y],Integer)
  MPOLY([x,y],INT) 

refer to the domain of multivariate polynomials in the variables x and
y where the coefficients are restricted to be integers.  The first
variable specified is the main variable and the display of the polynomial
reflects this.

This polynomial appears with terms in descending powers of the variable x.

  m : MPOLY([x,y],INT) := (x^2 - x*y^3 +3*y)^2
     4     3 3     6       2     4      2
    x  - 2y x  + (y  + 6y)x  - 6y x + 9y
                     Type: MultivariatePolynomial([x,y],Integer)

It is easy to see a different variable ordering by doing a conversion.

  m :: MPOLY([y,x],INT)
     2 6       4     3 3     2     2     4
    x y  - 6x y  - 2x y  + 9y  + 6x y + x
                     Type: MultivariatePolynomial([y,x],Integer)

You can use other, unspecified variables, by using Polynomial in the
coefficient type of MPOLY.

  p : MPOLY([x,y],POLY INT)
                     Type: Void

Conversions can be used to re-express such polynomials in terms of
the other variables.  For example, you can first push all the
variables into a polynomial with integer coefficients.

  p :: POLY INT
    p
                     Type: Polynomial Integer

Now pull out the variables of interest.

  % :: MPOLY([a,b],POLY INT)
    p
                     Type: MultivariatePolynomial([a,b],Polynomial Integer)

Restriction:
  Axiom does not allow you to create types where MultivariatePolynomial
  is contained in the coefficient type of Polynomial. Therefore,
  MPOLY([x,y],POLY INT) is legal but POLY MPOLY([x,y],INT) is not.

Multivariate polynomials may be combined with univariate polynomials
to create types with special structures.

  q : UP(x, FRAC MPOLY([y,z],INT))
                          Type: Void

This is a polynomial in x whose coefficients are quotients of polynomials 
in y and z.

  q := (x^2 - x*(z+1)/y +2)^2 
                             2    2
         4   - 2z - 2  3   4y  + z  + 2z + 1  2   - 4z - 4
   (7)  x  + -------- x  + ----------------- x  + -------- x + 4
                 y                  2                 y
                                   y
 Type: UnivariatePolynomial(x,Fraction MultivariatePolynomial([y,z],Integer))

Use conversions for structural rearrangements. z does not appear in a
denominator and so it can be made the main variable.

  q :: UP(z, FRAC MPOLY([x,y],INT))
    2            3     2             2 4       3      2      2            2
   x   2   - 2y x  + 2x  - 4y x     y x  - 2y x  + (4y  + 1)x  - 4y x + 4y
   -- z  + -------------------- z + ---------------------------------------
    2                2                                  2
   y                y                                  y
 Type: UnivariatePolynomial(z,Fraction MultivariatePolynomial([x,y],Integer))

Or you can make a multivariate polynomial in x and z whose
coefficients are fractions in polynomials in y.

  q :: MPOLY([x,z], FRAC UP(y,INT))
     4      2     2  3     1  2    2     4y  + 1  2      4     4
    x  + (- - z - -)x  + (-- z  + -- z + -------)x  + (- - z - -)x + 4
            y     y        2       2         2           y     y
                          y       y         y
 Type: MultivariatePolynomial([x,z],Fraction UnivariatePolynomial(y,Integer))

A conversion like q :: MPOLY([x,y], FRAC UP(z,INT)) is not possible in
this example because y appears in the denominator of a fraction.  As
you can see, Axiom provides extraordinary flexibility in the
manipulation and display of expressions via its conversion facility.

See Also:
o )help DistributedMultivariatePolynomial
o )help UnivariatePolynomial
o )help Polynomial
o )show MultivariatePolynomial

