On Sat, Jul 27, 2024 at 9:59 PM 'Animesh Shree' via sage-support
<sage-support@googlegroups.com> wrote:
>
> I saw this behavior
>
> sage: R1.<x,y,z> = QQ[]
> sage: R2.<z,x,y> = QQ[] # Rearrange variables of R1
> sage: R1
> Multivariate Polynomial Ring in x, y, z over Rational Field
> sage: R2
> Multivariate Polynomial Ring in z, x, y over Rational Field
> sage: R1 is R2
> False
> sage: print(R1.gens(), R1.variable_names())
> (x, y, z) ('x', 'y', 'z')
> sage: print(R2.gens(), R2.variable_names())
> (z, x, y) ('z', 'x', 'y')
> sage:
>
>
> It looks like the order in which we initiate the polynomials leads to
> different polynomial spaces. I thought this should not be the case.
> I got confused. Is it right or bug or intentional?
It is by design. Multivariate polynomial rings come with monomial
orders (https://en.wikipedia.org/wiki/Monomial_order)
and in this case R1 and R1 have different monomial orders.
Monomial orders are crucial for many algorithms operating on
multivariate polynomial rings, their ideals, etc.
Dima
>
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