functions and polynomials

21 Mar 2007

      How widely applicable is the following idea?

Let  f: Z x Z -> Z  be a binary FUNCTION (in the sense of sets)
on the integers, with the property that

 - for each x:Z,  f(x,-) : Z -> Z is a (agrees with a unique)
   POLYNOMIAL, whose coefficients are functions of x; and similarly

 - for each y:Z,  f(-,y) : Z -> Z is also a polynomial.

Then  f(x,y)  was itself a polynomial in two variables.

This generalises to a disjoint union of sets of variables,
ie to functions   Z^X x Z^Y -> Z   that are polynomials in one
set of variables for each value of the other, and vice versa.

The (possible) categorical generalisation is this:

Let T be a strong monad on a topos, CCC or even a symmetric monoidal
closed category, and K=T0 its free algebra.   Then there is a natural
transformation
    r_X:  T X --->  K^(K^X),
which we suppose to be mono.  (How widely is this the case?)

Then the above result states that (in that case) the square

                            (K^Y)
    T (X + Y) >---------> TX
       V      |              V
       |      |              |
       | -----               |     K^Y
       |                     | r_X
       |           K^X       |
       V        r_Y          V
        (K^X) >-------->  (K^X x K^Y)
      TY                 K

is a pullback (in fact, an intersection).

I am primarily concerned with the case where T encodes the theory
of frames over either Set or Dcpo, though if the result extends to
commutative rings or other algebraic theories, that would be very
interesting too.

Paul Taylor