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. > > -- > 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+unsubscr...@googlegroups.com. > To post to this group, send email to sage-support@googlegroups.com. > Visit this group at http://groups.google.com/group/sage-support?hl=en. > For more options, visit https://groups.google.com/groups/opt_out. > > -- 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+unsubscr...@googlegroups.com. To post to this group, send email to sage-support@googlegroups.com. Visit this group at http://groups.google.com/group/sage-support?hl=en. For more options, visit https://groups.google.com/groups/opt_out.
