Restricting parent of polynomial to a subring

[oskarhenriksson]

Suppose I have a polynomial $f\in k[x_1,\ldots,x_n]$ such that all nonzero terms involve only the variables $x_1,\ldots,x_k$ for $k < n$.

Is there a way create a new polynomial $\hat{f}$ with the same terms, but whose parent ring instead is $k[x_1,\ldots,x_k]$?

(This would be analogous to how to_univariate function works in the $k=1$ case.)

Example:

R, (x,y,z) = polynomial_ring(QQ,["x","y","z"])
f = x^2+y^2+1

[fieker]

On Wed, Dec 06, 2023 at 11:08:27AM -0800, Oskar Henriksson wrote: Suppose I have a polynomial $f\in k[x,y,z]$ such that all nonzero terms involve only the variables $x$ and $y$.  Is there a way to modify $f$ so that its parent ring instead becomes $k[x,y]$? (This would be a bit like how the `to_univariate` function works.) **Example:** ``` R, (x,y,z) = polynomial_ring(QQ,["x","y","z"]) f = x^2+y^2+1 ```

Well.... S, (x,y) = polynomial_ring(QQ,["x","y","z"]) z = Oscar.MPolyHom_vars{typeof(S), typeof(R)}(S, R, type = :names) preimage(z, f) could be improved.... z is a map from S to R mapping vaiables to vaiables with the same name. The actual mapping is efficient.

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[oskarhenriksson]

Hm, if I run the suggested commands:

R, (x,y,z) = polynomial_ring(QQ,["x","y","z"])
f = x^2+y^2+1
S, (x,y) = polynomial_ring(QQ,["x","y","z"])
z = Oscar.MPolyHom_vars{typeof(S), typeof(R)}(S, R, type = :names)
preimage(z, f)

the resulting preimage will still have a polynomial ring in three variables as its parent. Is there a way to make the parent a polynomial ring with two variables?

Intuitively, the following would make sense to me, but it gives an error message (BoundsError: attempt to access 2-element Vector{Int64} at index [3]).

R, (x,y,z) = polynomial_ring(QQ,["x","y","z"])
f = x^2+y^2+1
S, (x,y) = polynomial_ring(QQ,["x","y"])
phi = Oscar.MPolyHom_vars{typeof(S), typeof(R)}(S, R, type = :names)
g = preimage(phi, f)

[wdecker]

R, (x,y,z) = polynomial_ring(QQ,["x","y","z"])
f = x^2+y^2+1
S, (x,y) = polynomial_ring(QQ,["x","y"])
phi = hom(R, S, [x, y, 0])
f =phi(f)