Simplification of multivariable polynomial over a polynomial ring

[Emmanuel]

Hello,

I want to work with multivariate polynomials over a multivariate polynomial ring (see below for the reason I want to do this). 

K.<a,b>=PolynomialRing(QQ, 2, order='lex')

QM.<X,Y,Z> = PolynomialRing(K, 3, order='lex')

However, I have problems when I want to simplify. Consider for example, 

F=(a*b*X^2*Y*Z + X*Y^3)/Y

This is clearly a*b*X^2*Z + X*Y^2. However, I cannot automatically simplify it since :

F.simplify()

returns AttributeError: 'sage.rings.fraction_field_element.FractionFieldElement' object has no attribute 'simplify'  whereas

F.reduce()

returns AttributeError: 'sage.rings.fraction_field_element.FractionFieldElement' object has no attribute 'simplify'.

How can I make the simplification with Sage ?

Thank you very much for your help!

NB. The reason why I want to consider multivariate polynomials over a multivariate polynomial ring, and not a multivariate polynomials with more variable is the following  I need to be able to detect monomials in a polynomial, For rexample, I want the monomials of (1+a+b)X^2Y+2bXY^2+ab+a to be (1+a+b)X^2Y, 2bXY^2 and ab+a instead of X^2Y, aX^2Y, bX^2Y, 2bXY^2, ab and a.

[Charles Bouillaguet]

On Feb 18, 2013, at 1:47 PM, Emmanuel wrote:

> Hello,
>
> I want to work with multivariate polynomials over a multivariate polynomial ring (see below for the reason I want to do this).
>
> K.<a,b>=PolynomialRing(QQ, 2, order='lex')
> QM.<X,Y,Z> = PolynomialRing(K, 3, order='lex')
>
> However, I have problems when I want to simplify. Consider for example,
>
> F=(a*b*X^2*Y*Z + X*Y^3)/Y

In fact, it is precisely the fact that you build a polynomial ring over a polynomial ring that causes problems. Look at this example :

sage: K.<a,b,X,Y,Z> = QQ[]
sage: F = (a*b*X^2*Y*Z + X*Y^3)/Y; F
a*b*X^2*Z + X*Y^2

The simplification is done automatically. But note that the result (F) no longer lives in the polynomial ring. Since it is a quotient, it lives in the **fraction field** of the ring. This is because Sage does not know in advance whether the division is exact or not. Try :

sage: parent(F)
Fraction Field of Multivariate Polynomial Ring in a, b, X, Y, Z over Rational Field

If you KNOW that the division is exact, you can do :

sage: F = (a*b*X^2*Y*Z + X*Y^3) // Y
sage: parent(F)
Multivariate Polynomial Ring in a, b, X, Y, Z over Rational Field

This division procedure is not available over your fancier ring-on-top-of-a-ring. (Btw, the error message is somewhat not-use-friendly). Your problems with monomials can be solved as follows :

sage: R.<a,b,X,Y,Z> = QQ[]
sage: f = (1+a+b)*X^2*Y+2*b*X*Y^2+a*b+a

sage: K.<a,b> = QQ[]
sage: QM.<X,Y,Z> = K[]

and you can "cast " f into QM :

sage: QM(f).monomials()
[X^2*Y, X*Y^2, 1]

sage: QM(f).coefficients()
[a + b + 1, 2*b, a*b + a]
---
Charles Bouillaguet
http://www.lifl.fr/~bouillaguet/

>
> This is clearly a*b*X^2*Z + X*Y^2. However, I cannot automatically simplify it since :
>
> F.simplify()
>
> returns AttributeError: 'sage.rings.fraction_field_element.FractionFieldElement' object has no attribute 'simplify' whereas
>
> F.reduce()
>
> returns AttributeError: 'sage.rings.fraction_field_element.FractionFieldElement' object has no attribute 'simplify'.
>
> How can I make the simplification with Sage ?
>
> Thank you very much for your help!
>
> NB. The reason why I want to consider multivariate polynomials over a multivariate polynomial ring, and not a multivariate polynomials with more variable is the following I need to be able to detect monomials in a polynomial, For rexample, I want the monomials of (1+a+b)X^2Y+2bXY^2+ab+a to be (1+a+b)X^2Y, 2bXY^2 and ab+a instead of X^2Y, aX^2Y, bX^2Y, 2bXY^2, ab and a.
>

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