polynomial_ring cached=false for tropical geometry

[ederc]

See #2455.

Tests need to be adjusted, maybe @YueRen can provide some help.

[YueRen]

@ederc Thanks for the changes, much appreciated! I will look over them tomorrow.

[YueRen]

(This is more a memo to myself, though comments are welcome) Okay, here is why the change is problematic and what I plan to do to fix it:

Consider a polynomial ring K[x1,..,xn] over a valued field K. In addition to the original polynomial ring K[x1,..,xn], computations in tropical geometry require

• a polynomial ring R[x1,...,xn] over the valuation ring R (for tropical Groebner bases)
• a polynomial ring k[x1,...,xn] over the residue field field k (this is where initial ideals live)

Currently, I just (re-)create R[x1,...,xn] and k[x1,...,xn] from K[x1,...,xn] and blindly trust that it always ends up being the same polynomial ring. This is probably not very smart.

What should happen is that K[x1,..,xn] should store R[x1,..,xn] and k[x1,..,xn] as attributes. This will require lots of changes, so it might take a bit of time (ETA August).

[fingolfin]

Why should K[x1,..,xn] store R[x1,..,xn] and k[x1,..,xn] as attributes? That would be a bit unusual. Perhaps it is unavoidable for some reason, but normally I would expect that there is some kind of parent object (perhaps a "tropical variety" or something like that) which contains these rings.

I had a quick look at the changes in this PR and the tropical code in general but it wasn't apparent to me at all how and when these

[YueRen]

Why should K[x1,..,xn] store R[x1,..,xn] and k[x1,..,xn] as attributes? That would be a bit unusual. Perhaps it is unavoidable for some reason, but normally I would expect that there is some kind of parent object (perhaps a "tropical variety" or something like that) which contains these rings.

I had a quick look at the changes in this PR and the tropical code in general but it wasn't apparent to me at all how and when these

I should probably save it in TropicalSemiringMap then. TropicalSemiringMap's main job is to encode a valuation on K and a choice between min-convention and max-convention, and it is always passed along in tropical (algebraic) computations. It could also store a dictionary, which maps K[x] to R[x] and k[x] (in case multiple polynomial rings have the same K).