exponent_vectors of universal polynomials

[SoongNoonien]

Hi, I found some inconsistencies of exponent_vectors:

julia> S=UniversalPolynomialRing(QQ)
Universal Polynomial Ring over Rational Field

julia> i,j=gens(S, ["i","j"])
(i, j)

julia> collect(exponent_vectors(i))
1-element Vector{Vector{Int64}}:
 [1]

julia> collect(exponent_vectors(j))
1-element Vector{Vector{Int64}}:
 [0, 1]

julia> ii=S(i)
i

julia> collect(exponent_vectors(ii))
1-element Vector{Vector{Int64}}:
 [1, 0]

I wondered why they have sometimes different length. First I thought that they may be zero stripped from the right but this is clearly not the case as one can see with ii. As far as I can tell this is caused by the underlying MPolys. By the time i was created S had only one variable, so i.p is contained in a ring with just one variable. But ii is now created from a ring that already has two variables, so the parent of ii.p has two variables (similarly j.p). This can be seen here:

julia> nvars(parent(i.p))
1

julia> nvars(parent(ii.p))
2

julia> nvars(parent(j.p))
2

So, is the current behavior of exponent_vectors intended?

[thofma]

Yes, I think this is intended. One "problem" is that, i,j=gens(S, ["i","j"]) is done recursively, so first _, i = QQ["i"] is constructed and then _,(_,j)) = QQ["i", "j"] and then (i, j) is returned.

Not sure if exponent_vectors is useful at all. Because gens(S) is dynamic, there is no guarantee that one can recover an element p from exponent_vectors(p) and gens(S).

What is your application for exponent_vectors? I am leaning towards deleting it.

[SoongNoonien]

Not sure if exponent_vectors is useful at all. Because gens(S) is dynamic, there is no guarantee that one can recover an element p from exponent_vectors(p) and gens(S).

If I understand the universal polynomials correctly then gens(S) should only grow. And the documentation states the exponent vectors will remain valid:

Later, when the ring has more variable, these exponent vectors will still be accepted. The exponent vectors are simply padded out to the full number of variables behind the scenes.

So, it should be no problem to recover elements from their coefficients and exponent vectors.

What is your application for exponent_vectors? I am leaning towards deleting it.

I'm using it to replace variables. For example S has variables x, y, x1, y1 and p is 3*x^2+x*y and I want to replace x with x1 and y with y1. This can clearly be done with evaluate as well, but I used exponent_vectors because of #1245. Since it is fixed now I should port to evaluate.

[thofma]

I think the documentation wants to say, that you can do the following:

julia> coeff(i, [1])
1//1

julia> coeff(i, [1, 0])
1//1

Also this works:

julia> S(collect(coefficients(i)), collect(exponent_vectors(i)))
i

So it seems everything is working as expected, including the behaviour exponent_vectors.