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. What will you do for multivariate? Rich -------------- Quoting Fred lunnon <fred.lunnon@gmail.com>:
On 12/4/09, Dan Asimov <dasimov@earthlink.net> wrote:
<< . . . [O]ne of the many ways in which mathematicians conventionally
complicate their lives is by defining polynomials over finite fields in an avoidably clumsy fashion. Restricting oneself to linear combinations of powers excludes a lot of functions which ought to be polynomial, but fail to do so on account of the Fermat little theorem x^p = x mod p.
I'm not aware of mathematicians who define polynomials over finite fields with respect to Fermat's Little Theorem.
Indeed they don't --- that's exactly the problem!
I thought the ring of polynomials over any field F is defined as all expressions of the form
P(X) = c_n X^n + . . . + c_1 X + c_0,
where all the c_k belong to F, and n is some nonnegative integer. With the natural addition and multiplication. (And such that two polynomials are equal precisely when all their coefficients c_k are equal.)
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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