Convert Laurent polynomial to polynomial?

[oskarhenriksson]

Suppose that I have a Laurent polynomial $f\in k[x^\pm]$ where all exponents happen to be nonnegative, say

R, t = LaurentPolynomialRing(QQ, "t")
f = 5*t^3 - 7*t + 1

and that I want to view it as a polynomial $g\in k[x]$. Is there a nice way to do this in Oscar?

Naively, I would have hoped to be able to form the inclusion $\varphi\colon k[x]\hookrightarrow k[x^\pm]$ and use the preimage command, like this:

S, t = polynomial_ring(QQ,["t"])
phi = hom(S,R,[t])
g = preimage(phi, f)

but R doesn't seem to have the correct datatype for it to be possible to set up algebra homomorphisms like this.

(Sidenote: Part of the reason I want to do this is because I want to be able to find the roots of $f$ with the roots command. If there is a smooth way to do this without passing via $k[t]$, that would also be of interest!)

[fieker]

On Mon, Jan 08, 2024 at 04:21:34AM -0800, Oskar Henriksson wrote: Suppose that I have a Laurent polynomial $f\in k[x^\pm]=R$ where all exponents happen to be nonnegative, say ``` R, t = LaurentPolynomialRing(QQ, "t") f = 5*t^3 - 7*t + 1 ```` and that I want to view it as a polynomial in the subring of regular polynomials, $S=k[x]$. Is there a nice way to do this in Oscar? Naively, I would have hoped that this would work: ``` S, t = polynomial_ring(QQ,["t"]) phi = hom(S,R,[t]) g = preimage(phi, f) ```` but `R` doesn't seem to have the correct datatype for it to be possible to set up algebra homomorphisms like this. (_Sidenote:_ Part of the reason I want to do this is because I want to be able to find the roots of $f$ with the `roots` command. If there is a smooth way to do this without passing via $k[t]$, that would also be of interest!)

In your case, for a quick answer: f.poly is what you want - provided f.mindeg is, as here, 0. We need to provide a better interface for that

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[thofma]

1. I agree, that we need to improve the univariate experience and bring it on par with the multivariate one.
2. Maybe it makes sense to extend the roots function to Laurent polynomials. Then the only question is what to do about zero. Always exclude it? E.g. should we have

   R, t = LaurentPolynomialRing(QQ, "t")
   roots(t) == [] # or
   roots(t) == [zero(R)]