p-adic analytic functions

[Kannappan Sampath]

Hello everyone -- 

I am trying to do the following computation --- 

Suppose that h(z) is a polynomial defined over an extension of the 23-adic integers with unit constant term h(0) and that the uniformizer divides h(z)-h(0). "By Newton's binomial theorem", it admits a square root. I want to be able to compute this square root. 

Set aside precision issues for a moment; short of coding this by hand, are there methods in Sage that I should be looking at? I haven't kept in touch with the developments on Sage in the past three years and so I am not sure what has been done. 

Thank you for any pointers you might have!     

                                       Best, 

                                       Kannappan 

---------

Kannappan Sampath

Department of Mathematics | University of Michigan

EH Rm 3080 | kn...@umich.edu

[David Roe]

On Thu, Jun 20, 2019 at 1:55 PM Kannappan Sampath <kn...@umich.edu> wrote:

Hello everyone -- 

I am trying to do the following computation --- 

Suppose that h(z) is a polynomial defined over an extension of the 23-adic integers with unit constant term h(0) and that the uniformizer divides h(z)-h(0). "By Newton's binomial theorem", it admits a square root. I want to be able to compute this square root. 

I'm not quite sure what the setup is; what is "it" that admits a square root?  Perhaps you could provide a sequence of Sage commands illustrating what you're looking for?

Have you tried just calling sqrt() on the element?

David

Set aside precision issues for a moment; short of coding this by hand, are there methods in Sage that I should be looking at? I haven't kept in touch with the developments on Sage in the past three years and so I am not sure what has been done. 

Thank you for any pointers you might have!     

                                       Best, 

                                       Kannappan 

---------

Kannappan Sampath

Department of Mathematics | University of Michigan

EH Rm 3080 | kn...@umich.edu

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[Kannappan Sampath]

On Thu, Jun 20, 2019 at 3:10 PM David Roe <roed...@gmail.com> wrote:

On Thu, Jun 20, 2019 at 1:55 PM Kannappan Sampath <kn...@umich.edu> wrote:

Hello everyone -- 

I am trying to do the following computation --- 

Suppose that h(z) is a polynomial defined over an extension of the 23-adic integers with unit constant term h(0) and that the uniformizer divides h(z)-h(0). "By Newton's binomial theorem", it admits a square root. I want to be able to compute this square root. 

I'm not quite sure what the setup is; what is "it" that admits a square root?  Perhaps you could provide a sequence of Sage commands illustrating what you're looking for?

Sorry, "it" was supposed to be h(z). 

Have you tried just calling sqrt() on the element?

Admittedly, after coercing the element into a formal power series ring, I was able to do compute "the" formal square root of h. 

For now, I think this would suffice. I have a hacky way to finish the rest of the computation I have in mind, I think. 

I will perhaps write a trac ticket with more precise description of the functionality that would be nice to have. 

Thank you for your help! 

David

Set aside precision issues for a moment; short of coding this by hand, are there methods in Sage that I should be looking at? I haven't kept in touch with the developments on Sage in the past three years and so I am not sure what has been done. 

Thank you for any pointers you might have!     

                                       Best, 

                                       Kannappan 

---------

Kannappan Sampath

Department of Mathematics | University of Michigan

EH Rm 3080 | kn...@umich.edu

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