Re: Converting a symbolic expression to a power series and substituting another power series

[Volker Braun]

Reusing variable names is generally a recipe for confusion:

R.<x1,x2>=PowerSeriesRing(SR)

P.<x1,x2>=PolynomialRing(QQ)

Now R and P have variables that print as "x1" and "x2", but of course they are still different variables. Now compare

sage: f   

x1*x2 + O(x1, x2)^3

sage: f[2]    # the degree-2 part in the power series variables

x1*x2

sage: f2

x1*x2 + O(x1, x2)^3

sage: f2[0]    # the degree-0 part in the power series variables

x1*x2

[marco nijmeijer]

Thank you. That is a  good point although I still do not see how it solves the issue. Suppose I define f_symb as

f_symb(t1,t2)=t1*t2

to avoid the confusion of using identical variable-names meaning different things, what would I have to do to convert f_symb to f such that I can do the substitution? Help is much appreciated.

[Volker Braun]

You can't substitute power series into the symbolic ring, since power series are not objects of the symbolic ring. It just doesn't make sense in general.

You can substitute power series into polynomials; This also makes mathematical sense:

sage: f_symb

(t1, t2) |--> t1*t2

sage: f_symb.polynomial(QQ)

t1*t2

sage: _.parent()

Multivariate Polynomial Ring in t1, t2 over Rational Field

sage: f_symb.polynomial(QQ).subs(t1=x1, t2=x2)

x1*x2

[marco nijmeijer]

Thank you, that takes me  a step further. If I formulate the solution for my self I would say: take the symbolic expression, turn it into a polynomial by the method 'polynomial', then turn it into a power series by the proper substitution. That solves my original problem. It is nice to talk to someone about these things!

Hopefully I am allowed to ask a next question regarding dealing with power series. Suppose I have a power series in x1 and x2: f=x1*x2+O(x^3). If I substitute x1=y^2 and x2=y^2 I turn this into a power series in y: g(y) =y^4+O(y^6).  Doing this in Sage in the following way:

R.<x1,x2>=PowerSeriesRing(SR)
Y.<y>=PowerSeriesRing(SR)

f= x1*x2+R.O(3)

g=f.substitute(x1=y^2,x2=y^2);g

O(y^3)

does not work. Is there a way to make Sage understand that O(x^2)  becomes O(y^6) under the substitution x=y^2 (and in general under similar substitutions)?