I'm trying to work with power series with coefficients in a ring consisting of elements $f/g$, where $f$ is a polynomial in several variables over the integers and g is a polynomial in one of those variables with unit constant coefficient. Ultimately, I'd like to be able to do all of the following:
- Compose power series over this ring
- Invert single-variable power series over this ring whose degree-1 coefficients are units
- Reduce the coefficients of a power series modulo some ideal of the ring
I thought I would do this by starting with the Laurent series over $\mathbf{Z}$ in the variable that occurs in the denominator, constructing a polynomial ring over this by adjoining the other variables, and letting my power series take coefficients in this ring. But I'm getting an error I don't understand when I try to compose series. The following example illustrates the problem:
M1.<b> = LaurentSeriesRing(ZZ)
M.<a,c> = PolynomialRing(M1)
R.<t1,t2> = PowerSeriesRing(M)
X = t1 + t2 + O(t1,t2)^3
R1.<t> = PowerSeriesRing(M)
f = X(t,t^2); f
This returns the type error Substitution defined only for elements of positive valuation, unless self has infinite precision despite the fact that replacing the last line by t.valuation() or (t^2).valuation() returns 1 or 2, respectively.
The problem seems to result from a combination of factors. If I replace the second line with either
M.<a,c> = PolynomialRing(ZZ)
or
M.<a> = PolynomialRing(M1)
the error vanishes and the code returns the expected t + t^2 + O(t)^3. It also gives no error if I define R and R1 to be polynomial rings, rather than power series rings, over M (and correspondingly delete the big-O notation).
Any ideas about what is triggering the error, or another way to construct these power series?
