Working with univariate polynomial ideals?

[oskarhenriksson]

Some of my applied algebra students are a bit confused about how to deal with the univariate polynomial ring $\mathbb{Q}[x]$ in Oscar. The issue is that there are two different ways to think about them, corresponding to two different data types in Oscar:

• truly univariate polynomial rings of type QQPolyRing
  (obtained by writing R,x=polynomial_ring(QQ,"x") or R,x=QQ["x"])
• multivariate polynomial rings of type QQMPolyRing that happen to just have a single generator
  (obtained via R,(x,)=polynomial_ring(QQ,["x"])).

You need QQPolyRing to be able to do tasks that are specific to the univariate case (like computing roots), but you need QQMPolyRing to do general computations for polynomial rings (like groebner_basis or primary_decomposition).

If you need both points of view, you need to create two ideals in two rings, which easily gets confusing in a teaching situation...

Is there any way to make this more beginner-friendly? Naively, it would of course be nice if as much as possible of the QQMPolyRing functionality just automatically worked on QQPolyRing objects, but perhaps QQPolyRing plays roles in other parts of Oscar that would make this undesirable? (For instance, ideal(R,[x^2+2]) gives an ideal of type Hecke.PIDIdeal{QQPolyRingElem}, so this is related to the Hecke algebra parts.)

[fieker]

There are several things, but I guess some would be (can be) highly confusing. The most important difference, from a proactical point of view, is the different interface to the internals (coefficients, ...). Univariate are always dense, multivariate types are sparse.
Thus for f = x^3-2, a call to coeff(f, 0) will give -2 in the univariate case and an error in the multivariate. In the multivariate case, coeff(f, 1) -> 1, coeff(f, 2) -> -2 and all others error. This allows to implement different algorithms for arithmetic

• univariate multiplication of 2 degree n polynomials can be done in O(n log(n) operations
• multivariate of 2 polynomials with n terms is O(n^2)

However that means in the univariate case (x^100-1) *(x^101-1) will take > 100 operations, while its 4 in the multivariate case.

There is no reason why we cannot add roots to multivariates - after all we can already factor... that would be trivial.
As for ideals: univariate ideals were only added as people wanted them, not due to any (serious) application. Having said that, univariate rings (over fields) have a simple structure that we want to exploit Also, my vague recollection: the gcd(f, g) is a Groebner basis to the ideal generated by f and g...

So maybe the more intersting question: what are your applications? What would help you? (roots is easy:

Oscar.roots(f::MPolyRingElem) = [-constant_coefficient(t)//leading_coefficient(t) for (t, e) = factor(f) if total_degree(t) == 1]

OK, this is not optimal...)

[thofma]

I think it would be great, if the ideal functionality available for the *MPoly rings would also work for the univariate rings. There is unfortunately nothing automatic we could do. We add them as they are needed/requested. Apart from groebner_basis and primary_decomposition, are there any other functions on your list?