axiom-source 20170501-3 / usr / share / axiom-20170501 / src / algebra / POLY.spad

)abbrev domain POLY Polynomial
++ Author: Dave Barton, Barry Trager
++ Description:
++ This type is the basic representation of sparse recursive multivariate
++ polynomials whose variables are arbitrary symbols. The ordering
++ is alphabetic determined by the Symbol type.
++ The coefficient ring may be non commutative,
++ but the variables are assumed to commute.

Polynomial(R) : SIG == CODE where
  R : Ring

  SIG ==> PolynomialCategory(R, IndexedExponents Symbol, Symbol) with

   if R has Algebra Fraction Integer then

     integrate : (%, Symbol) -> %
       ++ integrate(p,x) computes the integral of \spad{p*dx},
       ++ integrates the polynomial p with respect to the variable x.

  CODE ==> SparseMultivariatePolynomial(R, Symbol) add

    import UserDefinedPartialOrdering(Symbol)

    coerce(p:%):OutputForm ==
      (r:= retractIfCan(p)@Union(R,"failed")) case R => r::R::OutputForm
      a :=
        userOrdered?() => largest variables p
        mainVariable(p)::Symbol
      outputForm(univariate(p, a), a::OutputForm)

    if R has Algebra Fraction Integer then

      integrate(p, x) == (integrate univariate(p, x)) (x::%)