Re: -- explorer le fourier --

On Apr 29, 4:16 am, Hagen <k...@xxxxxxxxxxx> wrote:
it is possible to define
<S, R>[x] as the generalised polynomial ring
with S the semigroup of exponents
and R the ring of coefficients

What exactly do you mean here? In algebra there is a
notion of a >semi-group ring<: take a commutative
semi-group (S,+) and a commutative ring R and consider
the set

R[S]:={ f:S-->R : {s : f(s)<>0} is finite}

Addition in R[S] is pointwise.
Multiplication:

(f*g)(s):=f(u_1)g(v_1)+f(u_2)g(v_2)+...

where the sum is taken over all pairs (u,v)
with u+v=s.

Assume that R has a multiplicative neutral 1_R.

Setting x^s to be the map that sends s to 1_R and everything
else to 0, we have:

x^s*x^t=x^(s+t).

An element f of R[S] can be expressed as a generalized
polynomial:

f=(sum over s in S) f(s) x^s

yes!

in fact
this is the algebraic setting needed
if one wants to generalise certain famous properties of polynomials

the most natural first question for me
was a generalisation of serre's question:
are all projective modules over W[x] free?

quillen-suslin generalises to laurent polynomials
using only the fact that the semigroup (monoid) of exponents
is toric
(torsion free, cancellative, seminormal, ..)

it seems to extend to W[x]
but i have not worked out all the details with units..

and these general constructs have a number of
properties
more generally derivable

however
the cyclotomic generalisation has a number of
of special properties
that make it very natural for study

just as a quick example
even though f(x) e W[x] does not guarantee f(x)^w e
e W[x]

Algebraically f^w is not defined in the semi-group ring.
But of course you see its elements as functions in the
same way polynomials can be interpreted as functions.
If you chose S and R within the complex numbers, then
the elements of R[S] can indeed be seen as functions
C-->C. Nevertheless it is unclear that then the function
f^w "comes" from an element of R[S].

it becomes true for the monomials M of R[S]
if R is such that r e R, s e S --> "r^s" e R

in particular
if R = Z_2 with the obvious definitions
then for any m e M there is m^s e M

this iteration
(interpretation as a function
with a binary "application" or "composition" operator)
has become important in the rest of the theory
so i've been looking to what structures
preserve it's interesting properties

-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
.

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