Insights from community regarding implementation of artin-mazur zeta function

[asutosh hota]

Hello ,

. Let’s take into consideration the zeta function by the definition:

Let’s take a function, f = x². Finding the fixed points of the function, we find the card(fix(fⁿ)) to be 2,4,6,...2n for n=1,2,3...n .Substituting the values in the above zeta function we get:

where we take the polar form of the t = a + ib, i.e., t = rexp(i ∗ θ)

We would now be able to compute the zeta function and would be able to plot the same.For the generalized implementation, we would have to create a class for artin-mazur zeta function which would have the functions to calculate the number of fixed point from each iteration and the cardinality of the sets. Then the problem can be computed with solving the formal power series.

For computing this I need to know from the community about:

1. Is there an existing model to find out fixed points in sage? Can it find the number of fixed points of the nth iterate of a function?
2.  As far other resources, I have not found any software that computes or proves rationality of an artin-mazur function. I need help from the community regarding any additional features or model that could be implemented to make this a unique addition.

Suggestions and advice would be highly appreciated from the community.

With Regards

Asutosh Hota

[Ben]

There are a few issues with your math.

For maps which are morphisms, the number of fixed points is the degree
of the map (or degree +1 in projective space), so the zeta function
should be easy to compute. In particular, your calculations for f(x) =
x^2 are wrong.

Yes, there is functionality in Sage to compute the fixed points and
iterates of maps (take a look at the projective_morphism.py file).
However, this is only feasible for the first few iterates since the
degree of the iterates grows exponentially, so the computations will
quickly be infeasible. However, with this functionality, computing the
first few coefficients of the power series should be trivial.


I'm not sure about creating a class for the zeta function. If all you're
doing is computing it (or approximating it), then it seems like it could
be a member function of the morphism class.

I can't help with (2) when you are outside of the basically trivial
situations.

Ben

On 4/6/2017 9:43 AM, asutosh hota wrote:
> Hello ,
>
> . Let’s take into consideration the zeta function by the definition:
>

> <https://lh3.googleusercontent.com/-BTX7j0GCpsI/WOZTlM_T5II/AAAAAAAAG7Y/7lqQG87iESUiHDOSuQY8ian74zSkmpJtgCLcB/s1600/1.JPG>
>
>
> Let’s take a function, /f /= /x/^2 . Finding the fixed points of the
> function, we find the /card/(/fix/(/f^n /)) to be 2,4,6,...2n for

> n=1,2,3...n .Substituting the values in the above zeta function we get:
>
>

> <https://lh3.googleusercontent.com/-By3CtwNyZSU/WOZTrbPYx2I/AAAAAAAAG7c/bglVIKJAT-IKHu3TcN8torI19uogFLhSgCLcB/s1600/22.JPG>
>
> where we take the polar form of the /t /= /a /+ /ib/, i.e., /t /=
> /r/exp(/i /∗ /θ/)

>
> We would now be able to compute the zeta function and would be able to
> plot the same.For the generalized implementation, we would have to
> create a class for artin-mazur zeta function which would have the
> functions to calculate the number of fixed point from each iteration and
> the cardinality of the sets. Then the problem can be computed with
> solving the formal power series.
>
>
> For computing this I need to know from the community about:
>

> 1. Is there an existing model to find out fixed points in sage? Can it

> find the number of fixed points of the nth iterate of a function?

> 2. As far other resources, I have not found any software that computes

> or proves rationality of an artin-mazur function. I need help from

> the community regarding any addition features or model that could be

> implemented to make this a unique addition.
>
> Suggestions and advice would be highly appreciated from the community.
>
>
> With Regards
>
> Asutosh Hota
>

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[asutosh hota]

On Thursday, 6 April 2017 21:00:07 UTC+5:30, Ben Hutz wrote:

There are a few issues with your math.

For maps which are morphisms, the number of fixed points is the degree
of the map (or degree +1 in projective space), so the zeta function
should be easy to compute. In particular, your calculations for f(x) =
x^2 are wrong.

I am afraid, but the cardinality of the fixed points functions, i.e., x^2n, is 2n. I hope that is alright? However, have I commited mistakes in summing up the power series?

 

Yes, there is functionality in Sage to compute the fixed points and
iterates of maps (take a look at the projective_morphism.py file).
However, this is only feasible for the first few iterates since the
degree of the iterates grows exponentially, so the computations will
quickly be infeasible. However, with this functionality, computing the
first few coefficients of the power series should be trivial.

That would be a great help as the problem comes down to summing up that power series and then we would be able to compute the artin-mazur function. 

I'm not sure about creating a class for the zeta function. If all you're
doing is computing it (or approximating it), then it seems like it could
be a member function of the morphism class.

I would integrate the module in the morphism class itself. 

I can't help with (2) when you are outside of the basically trivial
situations.

With regards

Asutosh Hota