Minor remark: if you have defined x to be a polynomial ring generator then Sage will happily form rational functions without you having to define new rings:
sage: R.<x> = GF(3)[] sage: R Univariate Polynomial Ring in x over Finite Field of size 3 sage: f = x+1/x sage: f.parent() Fraction Field of Univariate Polynomial Ring in x over Finite Field of size 3 On 27 July 2014 20:26, Simon King <simon.k...@uni-jena.de> wrote: > Hi Kevin, > > On 2014-07-27, Kevin Buzzard <kevin.m.buzz...@gmail.com> wrote: >> [I've just build a degree 6 poly. Now let's build a degree 12 one] >> >> sage: h=expand((g.subs(x+2/x))*x^6) > > Let's work without the x^6 factor: > > sage: g > x^6 + 2*x^3 + x + 1 > sage: g.subs(x+2/x).expand() > 2/x + 1/x^3 + 1/x^6 > > No idea what is going wrong here. Let's try different ways to > substitute: > > sage: g(x=x+2/x).expand() > 2/x + 1/x^3 + 1/x^6 > > Same problem. Whithout expanding, it seems to be fine: > > sage: g.subs(x+2/x) > (((x + 2/x)^3 + 2)*(x + 2/x)^2 + 1)*(x + 2/x) + 1 > > I guess the above form comes from Horner's scheme. > > Now, really strange: If one copy-and-pastes the above expression and > expands it, then all is fine! > > sage: ((((x + 2/x)^3 + 2)*(x + 2/x)^2 + 1)*(x + 2/x) + 1).expand() > x^6 + 12*x^4 + 2*x^3 + 60*x^2 + 13*x + 26/x + 240/x^2 + 16/x^3 + 192/x^4 + > 64/x^6 + 161 > > That is all very puzzling to me---and is yet another reason for my creed > that one should use *proper* polynomials and not symbolic expressions > (see below for the underlying types). > >> Am I some sort of a victim of some secret property of the letter 'x'? > > Not of the letter 'x', but of the fact that the variable x is > pre-defined in Sage as a symbolic expression, which often enough causes > trouble to those who expect to work with polynomials. Note that x is *not* > the generator of a polynomial ring: > > sage: type(x) > <type 'sage.symbolic.expression.Expression'> > > whereas your polynomial g is a proper polynomial (there are various underlying > implementations of polynomials in Sage, for different purposes, and this > one relies on FLINT): > > sage: type(g) > <type 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint'> > >> I >> wanted to use x rather than another letter because that would save me from >> having to define the ring GF(3)[x,1/x] which would have involved some >> thinking on my part ;-) > > Well, the polynomial ring already is there: > > sage: g.parent() > Univariate Polynomial Ring in x over Finite Field of size 3 > > and the fraction field will automatically be created when needed. So, > you could do: > > sage: g.parent().inject_variables() > Defining x > > The preceding line works if you are in an interactive session. > Alternatively (or if you write a program), you could explicitly define > it: > > sage: x = g.parent().gen() > sage: type(x) > <type 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint'> > > Now, the substitution works out of the box, and "expansion" of symbolic > expressions is not needed, since we work with proper polynomials and > their quotients. > > sage: g.subs(x+2/x) > (x^12 + 2*x^9 + x^7 + 2*x^6 + 2*x^5 + x^3 + 1)/x^6 > > My general recommendation is: Unless you want to do calculus, get used > to define x as the generator of an appropriate polynomial ring, taking > care of the underlying range of coefficients. This is totally easy in an > interactive session: > > sage: P.<x> = QQ[] # or GF(3)['x'] in your case > > The above line both defines P and x, as polynomial ring over the > rationals, and its generator. In a program, you should write > > P = QQ['x'] # or GF(3)['x'] in your case > x = P.gen() > > instead. > > Best regards, > Simon > > > -- > 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. > For more options, visit https://groups.google.com/d/optout. -- 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. For more options, visit https://groups.google.com/d/optout.
