generalised rational function fields (rabbits in rabbit fields)
galathaea
my posting newsserver is unreachable and google remains}
a while back
i came up with a construction to analyse composition
in generalised polynomial fields
it didn't seem to have a lot of interesting structure at first
but some recent messing around
has shown that they may be useful
for some structural analysis on the generalised polynomials
so this post is to see if anyone has seen these before
because i have no idea what others would call them
i call them rabbits
start with generalised polynomials
g_0, g_1, ..., g_{n-1}
and form the product
n-1 j
--- / \ w
| | | g | n
| | \ j /
j=0
for n=2
this is just the definition of a rational function
for larger n
this generalises the idea
and yet carries over a lot of the basic structure
polynomials may be composed
p1(p2(x))
over the ring of polynomials
laurent polynomials may be composed
over the field of rational functions
and there is a sense
in which composing generalised polynomials
is done in a field generated by these rabbits
notice first
that rabbits composed with generalised monomials
produce other rabbits
this is obvious for the generating monomials
j
w
n
x
and
of course
any product of rabbits is rabbit
(the law of lagomorphic multiplicity)
n-1 j n-1 k
/ --- / \ w \ / --- / \ w
| | | | g (x) | n | | | | | g (x) | n
\ | | \1 j / / \ | | \2 k /
j=0 k=0
n-1 j
--- / \ w
= | | | g (x) g (x) | n
| | \ 1 j 2 j /
j=0
unlike in the rational function case
in general
sums add more structure
but the law is obvious
and the result has many easily provable properties
this is easy to see if you look at why rational functions work
in rational functions
sums look like
-1 -1
(f (x)) (f (x)) + (g (x)) (g (x))
0 1 0 1
now
in rational function theory
there is the almost trivial relation
-1
(f(x)) (f(x)) = 1
so one can multiply the left hand term by
-1
(g (x)) (g (x))
1 1
and the right term by
-1
(f (x)) (f (x))
1 1
-1 -1 -1
all terms have (f (x)) (g (x)) = (f (x) g (x))
1 1 1 1
which can be distributed out
-1
(f (x) g (x) + f (x) g (x)) (f (x) g (x))
0 1 1 0 1 1
and since the generalised polynomials are a ring
and products and sums of polynomials are again polynomials
this again a 2-rabbit!
for generalised n-rabbits
there is a similar lemma like rationals
where
(lemma of identity in cartwheel symmetry)
n-1 j
--- w
| | n
| | (f(x)) = 1
j=0
proof:
use a^j a^k = a^(j + k)
and 1 + w_n + (w_n)^2 + ... + (w_n)^(n-1) = 0
now n-rabbit sums can use the same approach
and it shows an interesting property
but it doesn't reduce to another n-rabbit
the point is that these cartwheels
only remove one foot of the rabbit at a time
n-1 j n-1 k
/ --- w \ / --- w \
| | | n | | | | n |
\ | | (f (x)) / + \ | | (g (x)) / =
j=0 j k=0 k
n-2 j n-2
k n-1
/ / --- w \ / --- w \
\ w
| | | | n | | | | n |
| n
\ \ | | (f (x) g (x)) / + \ | | (f (x) g (x)) / / (f (x)
g (x))
j=0 j n-1 k=0 n-1 k n-1
n-1
but if we continue and multiply the right inner term
with the cartwheel of g_(n-2)
and the left by the cartwheel of f_(n-2)
we introduce new terms of the form
n-1
w
n
(?)
which are different for each terms
and could not distribute out
still
the ability to grab a foot
is a useful structural property
that helps explain what the additive completion of n-rabbits looks
like
this is how the n-rabbit fields are generated from their rabbits
and since the sum doesn't collapse to rabbit
the construction may be iterated
providing a grading of ever deeper fields
i.e. starting from a generalised polynomial field G[x]
we can form it's first rabbit field R(G[x])
and the rabbit field R(R(G[x])) of that
and so on
each field contains every previous
(f) (1)^(w_n) ... (1)^((w_n)^(n-1))
is a rabbit that introduces the embedding
and there are many relations between rabbit constructions
and composition rules
one reason these types of fields are interesting
is because they give meaning to periodicities in compositions like
w
3
if f(x) = x
then f(f(f(x))) = x
so
anyway
i'm sure these have a simple enough representation in familiar
algebraics
what do i look up?
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
galathaea: prankster, fablist, magician, liar
galathaea
let me give my guess of what's going on
the generalised polynomial rings
are integral domains
the rabbit extensions
are generalisations
of taking their field of fractions
the particular application in composition
helps provide an algebraic description
which is applicable over the generalised polynomials
but it seems the construction can be done
over any integral domain
as a generalisation of the field of fractions
for instance
this can be applied to Z
David Bernier
[...]
Is w_n = exp(2pi*i/n) ?
If so, the powers (w_4)^j give
1, i, -1, -i for j = 0, 1, 2, 3 resp.
So x^{i} = x^{sqrt(-1)} .
For complex numbers, x^i = exp(i*log(x))
so if x= exp(2pi), x^i = 1.
Also, | x^i | = 1 if x>0 and
arg(x^i) = log(x) (mod 2pi) .
One should have (x^i)^i = x^(-1) and starting
with exp(2pi):
exp(2pi) -> 1 -> exp(i*2pi*ik), for some k.
With k=1, we get exp(-2pi) = 1/exp(2pi) .
Also, [exp(4pi)]^i = 1. So x |-> x^i is not
injective if we allow all x>0.
(1/2)^i = exp(-i*log(2)) = cos(log(2)) - i*sin(log(2)).
So one monomial would be x^i. I don't know
about algebraic rules for x^i, but I have
a better idea of your "rabbit" objects.
David Bernier
galathaea
that's the basic construction for generalised polynomials
the rabbits are the particular construction using products
one example is the Z-generated 3-rabbits
these are expressions of the form
2
w w
1 3 3
(m ) (m ) (m )
0 1 2
with m0, m1, m2 all integers
even more concrete would be
2
w w
3 3
(6) (8) (10)
then
just like in the case of rational numbers
we see we can simplify
2
w w
3 3
because (2) (2) (2) = 1
so the above can be rewritten
2
w w
3 3
(3) (4) (5)
using these kinds of constructions
extends rings in a way i am sure has a well-known study
but i don't have a clue what it's called
Ross A. Finlayson
Yourself!
Thanks,
Ross F.
galathaea
the original formatting had been googled
so this is a repost through a more respectable means
to fix the broken formulae
nothing added)
........................
Hagen
It is a bit difficult for me to read your formulae - e.g. what
do the vertical lines surrounding g mean? So maybe you
could post the formulae in tex-code or so?
> for n=2
> this is just the definition of a rational function
Ok.
> for larger n
> this generalises the idea
> and yet carries over a lot of the basic structure
>
> polynomials may be composed
> p1(p2(x))
> over the ring of polynomials
>
> laurent polynomials may be composed
> over the field of rational functions
>
> and there is a sense
> in which composing generalised polynomials
> is done in a field generated by these rabbits
As I posted in the thread >field generated by the set of roots of
unity< the ring of generalized polynomials is the group ring
Q[(R,+)], where are R is the integral closure of Z in the nth
cyclotomic field over Q. Hence the rabbits are elements of that
ring.
Moreover this ring is isomorphic (not naturally as it seems) to the
subring Q[x_1,...,x_n,x_1^(-1),...,x_n^(-1)] of the rational function
field in n variables.
For fixed n the field generated by all rabbits is thus a subfield of
Q(x_1,...,x_n). Is it proper for n>2?
galathaea
\prod_{j=0}^{n-1} \left( g_{j} \right)^{\omega_{n}^{j}}
or
the product of the polynomials
raised to powers of an nth root of unity
but i see now that you are posting through the math forum
does the "Plain Text" option
(on the upper right hand side of the post window)
help clear up the formulas?
i think some of my questions below
come from a misunderstanding of the formulae
> > for n=2
> > this is just the definition of a rational function
>
> Ok.
note that this means
the laurent-generated 2-rabbit field
is the same as the rational function field
it seems later you may imply it is the laurent polynomial field
> > for larger n
> > this generalises the idea
> > and yet carries over a lot of the basic structure
>
> > polynomials may be composed
> > p1(p2(x))
> > over the ring of polynomials
>
> > laurent polynomials may be composed
> > over the field of rational functions
>
> > and there is a sense
> > in which composing generalised polynomials
> > is done in a field generated by these rabbits
>
> As I posted in the thread >field generated by the set of roots of
> unity< the ring of generalized polynomials is the group ring
> Q[(R,+)], where are R is the integral closure of Z in the nth
> cyclotomic field over Q. Hence the rabbits are elements of that
> ring.
these rabbits won't be elements of the same ring
as the generalised polynomials
these particular rabbits are generated from the generalised
polynomials
in text-friendly form
(1 + x) (1 + x^(w_3))^(w_3) (1 - x)^((w_3)^2)
is an example rabbit
but (1 + x^(w_3))^(w_3) is outside the ring of generalised polynomials
> Moreover this ring is isomorphic (not naturally as it seems) to the
> subring Q[x_1,...,x_n,x_1^(-1),...,x_n^(-1)] of the rational function
> field in n variables.
see my questions in the previous thread
i'm still confused about this point
> For fixed n the field generated by all rabbits is thus a subfield of
> Q(x_1,...,x_n). Is it proper for n>2?
i expected a different type of question
is this thing a field at all?
for n = 2 it is clear it is
but for n > 2
it is not so obvious
i've been assuming there is
because i think you can choose another element
with enough settable variables to find an inverse
it's clear that the rabbits themselves all have inverses
and are thus units
but the sums of rabbits that are not themselves rabbits
it's less clear...
Hagen
Ok, I see that now.
Hagen
Some remarks / comments:
Do you want to view the 'generalized polynomials' and
derived structures from an algebraic point of view?
I ask this because in your definition of a 'rabbit' you raise a
generalized polynomial to a non-integer power. To make this
working one has to introduce another algebraic structure -
namely exponentiation with certain elements.
Eventually this leads to a kind of generalization of the concept
of a semigroup-ring, where the semigroup of exponents is
replaced by a ring of exponents. Even doing so I was not able
to define things in a way that fits with the needs.
On the other hand one could view 'generalized polynomials'
as 'generalized polynomial functions' (on which set?). Then
the problem of uniqeness appears: two algebraically different
generalized polynomials could give rise to the same function.
H
galathaea
> Some remarks / comments:
>
> Do you want to view the 'generalized polynomials' and
> derived structures from an algebraic point of view?
yes
actually my secret focus
is on endomorphisms
which comes from the analysis of generalised tchebyshefs
and the polynomials of generalised trigonometry
and i think you've made many things clearer here
to give an idea of the algebraic approach i want
the (0,3)-trigonometric function
has the generalised polynomial form
2
w w
1 / 3 3 \
- | x + x + x |
3 \ /
then there is a generalised 3-tchebyshef
2
w w
/ 1 / 3 3 \\
T | - | x + x + x || =
3 n \ 3 \ //
2
n w n w
1 / n 3 3 \
- | x + x + x |
3 \ /
it's basically the generalisation of the power map
n
x |-> x
which is the 1-trigonometric form
but in 1-trigonometry
the study is over the standard Z[x]
and the endomorphism rule of composition
says every polynomial is an endomorphism
composition just works in that ring
in the 2-trigonometry (or the classical case)
the tchebyshef functions are still polynomials
and so corresponds to an endomorphism
i want to characterise the polynomials as rings
and characterise the endomorphism group
every generalised polynomial ring
has standard polynomials as endomorphisms
by the basic ring rules
however
the n-tchebyshef are no longer polynomial n > 3
so i have a number of questions
on the algebraic meaning of tchebyshefs
i've posted many of the formulae
that i think characterise the algebra of tchebyshefs
( the sum
the product
the cyclotomic walk
... )
and i want to see if they are endomorphisms
the same for the horizontal functions
which _are_ endomorphisms
(in fact automorphisms of order n)
there is a lot of structural similarity
between horizontals and tchebyshefs
and they may be combined into a family
(a semigroup even)
of generalised trigonometric transformations
that's what started it
studying the laurent polynomial case
there were some indications that
the endomorphisms' relationship to compositions
actually could derive structure
from understanding the greater composition structure
general composition takes us outside the ring
so not all compositions are endomorphisms
but these extensions
seem to have a lot of useful structure
this is where the rabbits come from
> I ask this because in your definition of a 'rabbit' you raise a
> generalized polynomial to a non-integer power. To make this
> working one has to introduce another algebraic structure -
> namely exponentiation with certain elements.
> Eventually this leads to a kind of generalization of the concept
> of a semigroup-ring, where the semigroup of exponents is
> replaced by a ring of exponents. Even doing so I was not able
> to define things in a way that fits with the needs.
given a ring R
look at it as a formal tuple of elements of R
(r0, r1, r2, ..., r_(n-1))
it has the projection property
(r0, r1, r2, ..., r_(n-1)) = (k r0, k r1, k r2, ..., k r_(n-1))
and it has the multiplication property
(r0, r1, r2, ..., r_(n-1)) (s0, s1, s2, ..., s_(n-1)) =
(r0 s0, r1 s1, r2 s2, ..., r_(n-1) s_(n-1))
additions create formal structures
---
\
/ b
---
rabbits b e B
B finite
with the rabbits foot property
this construction appears to generalise
the 2-rabbit extension
rabbit(2, R)
which is isomorphic to the field of fractions frac(R)
> On the other hand one could view 'generalized polynomials'
> as 'generalized polynomial functions' (on which set?). Then
> the problem of uniqeness appears: two algebraically different
> generalized polynomials could give rise to the same function.
i have been curious about that
particularly since each function
has a countably infinite collection of branches
and possibly an amazingly rich monodromy
in a standard interpretation of exponentiation
there does appear to be a zero structure
that follows some rules of analysis
much like the fundamental theorem of algebra
but then that brings me back to the algebraic structure
and a desire to look at the galois structure