Slope factorisation of polynomials over padics

[xcaruso]

Factorization over the p-adics is currently broken (or at least not very useful.) This implements slope factorization and adds some structure for polynomial rings over discrete valuation rings.

CC: @defeo

Component: padics

Keywords: polynomials, sd52, days71

Author: Xavier Caruso, Frédéric Chapoton

Branch/Commit: 868e84e

Reviewer: Julian Rüth, Maurizio Monge

Issue created by migration from https://trac.sagemath.org/ticket/14828

[xcaruso]

Dependencies: #14823, #14826

[roed314]

comment:2

I've basically abandoned the code at #6084, but I would like to extract pieces of it into other tickets. I'll try to take a look at this ticket once its prerequisites are fixed and see if I can extract anything useful from #6084 to add here. You have a different approach to polynomials with unknown degree here. I think I like it better than what I was doing, but I need to think about it a bit.

[xcaruso]

comment:3

Attachment: trac_14828_padic_polynomials.patch.gz

I updated my patch so that it fits with #14826 (many modifications made there).

[fchapoton]

comment:4

Hello,

maybe it would be a good idea to close first the simpler ticket #6667 ?

I have made there a patch which is rather "local" but also careless about precision issues.

I see that here you are changing the behaviour of the general Newton polygon, which I would prefer to avoid.

What would you think of trying to use for #6667 a mixture of my patch there and the part of your patch here dealing with that ?

[xcaruso]

comment:5

I agree with all your remarks.

I just do not understand the sense of "you are changing the behaviour of the general Newton polygon". Do you mean that I'm doing some changes in the class NewtonPolygon. If so, I still believe that it's better to do this way because it really makes sense to create a Newton polygon by giving the list of all valuations of the underlying polynomial (and these valuations may be infinite). Actually, I should have put this modification in #14826 but this ticket is already closed.

[fchapoton]

comment:6

It seems that the NewtonPolygon class is not doing what I would have expected. It gives

sage: NP = NewtonPolygon([ (0,0), (1,45), (3,6) ]); NP
Finite Newton polygon with 2 vertices: (0, 0), (3, 6)

So it is implicitely assuming that the polytope is rather a kind of "lower envelope". In my opinion, the doc needs to be enhanced, with more explanations and a link to the newton_polytope method below.

Compare to

sage: x,y=polygen(QQ,'x, y')
sage: p = 1 + x*y**45 + x**3*y**6
sage: p.newton_polytope()
A 2-dimensional polyhedron in ZZ^2 defined as the convex hull of 3 vertices
sage: p.newton_polytope().vertices()
(A vertex at (0, 0), A vertex at (1, 45), A vertex at (3, 6))

which is for me the expected behavior.

Anyway, I still think that this has rather to be solved in #6667.

[xcaruso]

comment:7

Replying to @fchapoton:

Anyway, I still think that this has rather to be solved in #6667.

I agree. I nevertheless prefer my patch than yours because it basically deals with precision issues.
Hence, I've extracted the relevant parts of my patch and submitted them to ticket #6667 (and I've also taken into account your remarks). If you are happy with this solution, please give a positive review to #6667 so that we can close this ticket.

[saraedum]

Changed keywords from polynomials to polynomials, sd52