The right type for the right job

On 8/21/06, Ralf Hemmecke <[email protected]> wrote:

> >> Looking at this thing I would say that if you take
> >>
> >> R = Q[s,c]       -- polynomial ring in two variables over rationals
> >> I = (s^2+c^2-1)R -- ideal in R
> >> A = R/I          -- factor structure
> >> S = A[[x]]       -- formal power series
> >>
> >> then S would be a perfect candidate for the result type of the above
> >> expression. And there is no "Expression Integer".
> >> While constructing the result of "series", Axiom should try hard to get
> >> a reasonable (in some sense minimal) type for the result.

The above is a very nice example of a domain for a restricted use,
where only the algebraic relation s^2+c^2=1 is important in the
coefficients of the power series we wish to work with.

I have a very different purpose in mind, and I wonder if anyone can
come up with the right type for it. Say I have a few symbols
[x,y,z,...] and a binary operagor g. I'm only interested in
polynomials or rational functions with, say, integer coefficients in
the formal expressions g(x,x), g(x,y), g(y,z), etc. Is there a type
that restricts to just this sort of expressions? Expression Integer
can handle them, but it will also allow other symbols like a,b,c,...
and different operators f(a), h(x,c), etc. None of the Polynomial or
similar domains in Axiom can handle this situation because non-symbols
like g(x,y).

What's the solution?

Igor