Order of Variables of Polynomial Matters

[Animesh Shree]

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?

[Dima Pasechnik]

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

>
> --
> You received this message because you are subscribed to the Google Groups "sage-support" group.
> To unsubscribe from this group and stop receiving emails from it, send an email to sage-support...@googlegroups.com.
> To view this discussion on the web visit https://groups.google.com/d/msgid/sage-support/30b4f982-9b1a-434f-a512-7690b6c16e6bn%40googlegroups.com.

[Animesh Shree]

I was looking into this issue

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

The 1st one hits the Ref1 but 2nd one don't. That leads to ConstantPolynomial Error.

[Dima Pasechnik]

On Sun, Jul 28, 2024 at 6:51 AM 'Animesh Shree' via sage-support
<sage-s...@googlegroups.com> wrote:
>
> I was looking into this issue
>
> 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 ?

I don't know whether S(p), i.e. coersion of p into the ring S, should work for
different monomial order, or one should explicitly use R1.change_ring

You get lucky with S and R2, as this is the same order.

I'd also be vary of using the same variable names for different rings:
if you declare
two rings, R1 amd R2, using the same variable, say, y, how does Sage
supposed to know whether
y is a member of R1, or a member of R2 (you can still use R1(y) and
R2(y) to be explicit)

HTH,
Dima

> To view this discussion on the web visit https://groups.google.com/d/msgid/sage-support/f25ed01f-2b98-47bd-bb56-52a42ed19149n%40googlegroups.com.