On 12/4/09, rcs@xmission.com <rcs@xmission.com> wrote:
Addition of these forms is easy enough. Multiplication by X is a tad complicated, and multiplication of two forms is complicated. It looks like the division algorithm, where the divisor is monic, will have a wee bit of difficulty. Summation from 1...X is easier than with regular polys; integration & differentiation look doable but not neat. Shifting X -> X+1 looks easy; X -> 2X not so easy; nor X -> X^2 (or -> (X 2)). Composition P(Q(X)) is closed, and was never easy. End-for-end coefficient reversal for regular polynomials gets you X^D P(1/X); and the product of reversed polynomials is the reverse of the product; and the product of palindromic polynomials is a polyndrome. These all seem to fail for combinomials.
Might be a nice project for somebody here ...
What will you do for multivariate?
Ouch! Very good question, which I have never yet had to consider ... An obvious approach would attempt to replace a basis of monomials (homogeneous in m variables, say) by a basis of m-nomial coefficients: for m-1 = 2 variables, try somehow involving the trinomial coefficients (when i+j+k >= 0) T_ijk = 1 for i = j = k = 0; 0 for i < 0 or j < 0 or k < 0; T_{i-1,j,k} + T_{i,j-1,k} + T_{i,j,k-1} otherwise. There may be an analogy here with Bezier curves (standard curve-modelling technique for for computer graphics, etc). It's easy to interpolate 1-dimensional smooth curves, using what is essentially a glorified version of Pascal's triangle; but the direct generalisation to more dimensions appears so esoteric that the current standard (NURBS) bottles out of it, relying instead on Cartesian products of 1-dimensional interpolations. [A relevant paper from long ago lies buried somewhere in my paper files, whence it might be disinterred on request.] Should this be the case --- I'm outta there! Fred Lunnon
Hm. On the face of it, I can't see that it makes any difference to the formal properties regarding algebraic extension etc if we substitute for the traditional
P(X) = c_k X^k + . . . + c_1 X + c_0 (*)
the alternative
P(X) = c_k (x_C_k) + . . . + c_1 X + c_0 (**)
where x_C_k denotes binomial coefficient. Has anybody out there thought about this question in more detail?
On the other hand, when it comes to considering difference equations, linear recurrences and the like, we gain a completeness lacking before: the theory becomes much the same for both finite fields and rationals.
Fred Lunnon
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