John Cremona <[email protected]> writes:
> I agree: composition of power series should only be allowed when the
> "inner" one has positive valuation, i.e. zero constant term. (At
> least over an integral domain. Maybe it's ok if the constant term is
> just a zerodivisor, but I cannot think of a situation where that would
> be needed!)
>
> It would be quite fun to implement the Nottingham Group which (for
> each prime p) is the group of power series over F_p with zero constant
> term, under composition. This is of great interest to a wide range of
> number theorists and group theorists. But I am not an expert.
At least in FriCAS this is trivial -- I guess we assume also than
f'(0)=1, because otherwise we have a ring, not a group.
I'd be interested in why this could be of interest. (And why do we
restrict to finite fields...)
Martin
(sorry for the ASCII art below)
----------------------------------------------------------------------
-- notting.spad
----------------------------------------------------------------------
)abb dom NOTTING NottinghamGroup
NottinghamGroup(F: FiniteFieldCategory): Group with
retract: UnivariateFormalPowerSeries F -> %
== add
Rep := UnivariateFormalPowerSeries F
coerce f == coerce(f::Rep)$UnivariateFormalPowerSeries(F)
retract f ==
if zero? coefficient(f, 0) and one? coefficient(f, 1)
then f::Rep
else error "we expect leading term x"
1 == monomial(1,1)
f * g == f.g
inv f == revert f
----------------------------------------------------------------------
(1) )co notting.spad
(1) -> x := monomial(1,1)$UFPS PF 1783; s := retract(sin x)$NOTTING PF 1783
3 5 7 9 11
(1) x + 297x + 1679x + 427x + 316x + O(x )
Type: NottinghamGroup(PrimeField(1783))
(2) -> s^2
3 5 7 9 11
(2) x + 594x + 535x + 1166x + 1379x + O(x )
Type: NottinghamGroup(PrimeField(1783))
(3) -> s^-1
3 5 7 9 11
(3) x + 1486x + 847x + 207x + 1701x + O(x )
Type: NottinghamGroup(PrimeField(1783))
(4) -> s^-1 * s
11
(4) x + O(x )
Type: NottinghamGroup(PrimeField(1783))
(9) -> s^-1 * s
11
(9) x + O(x )
Type: NottinghamGroup(PrimeField(1783))
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