I was looking into this issue
<https://github.com/sagemath/sage/issues/36201#issue-1885203561>
This throws error:
-----------------------------------------------------------------------------------------
sage: R1.<x,y,z> = QQ[]
sage: p = (x^2 - y^2) * (x + y + z)
sage: S = QQ['z']['x,y']
sage: try:
....: S(p)
....: except Exception as e:
....: print(e)
x^3 + x^2*y - x*y^2 - y^3 is not a constant polynomial
But this passes
--------------------------------------------------------------------------------------------
sage: R2.<z,x,y> = QQ[]
sage: p = (x^2 - y^2) * (x + y + z)
sage: S(p)
x^3 + x^2*y - x*y^2 - y^3 + z*x^2 + (-z)*y^2
sage:
So the output of
p = R1.random_element()
S(p) and S(R2(p))
should they be same or different ?
One reason to this behavior is the ordering of monomial. Ref1
<https://github.com/sagemath/sage/blob/79c047c0a22a98bea4567d182c694fd4df1aea81/src/sage/rings/polynomial/multi_polynomial.pyx#L487C1-L487C38>
The 1st one hits the Ref1 but 2nd one don't. That leads to
ConstantPolynomial
<https://doc.sagemath.org/html/en/reference/polynomial_rings/sage/rings/polynomial/polynomial_element.html#sage.rings.polynomial.polynomial_element.ConstantPolynomialSection>
Error.
On Sunday, July 28, 2024 at 3:42:28 AM UTC+5:30 Dima Pasechnik wrote:
> On Sat, Jul 27, 2024 at 9:59 PM 'Animesh Shree' via sage-support
> <sage-s...@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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>
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