Annie Carter's profile - activity

Is there any hope if the ring of coefficients has more than one variable? The application I'm interested in is when the coefficient ring has the form $\mathbf{Z}[a_1, \ldots, a_n]$ and I reduce modulo some power of a maximal ideal of the form $(p, a_1, \ldots, a_n)$ with $p$ a prime.

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?

I have a polynomial in two variables $t_1$ and $t_2$ (say $2t_1 + at_2$) defined over a ring which is itself a polynomial ring (say $\mathbf{Z}[a]$). I'd like to reduce the coefficients of the polynomial modulo an ideal of the latter ring (say $(2)$ or $(a)$ or $(2,a)$). When I execute

M.<a> = PolynomialRing(ZZ)
R.<t1,t2> = PolynomialRing(M)
m = M.ideal(a)
(2*t1 + a*t2).change_ring(QuotientRing(M,m))

I get 2*t1, as I would expect. On the other hand, the code

M.<a> = PolynomialRing(ZZ)
R.<t1,t2> = PolynomialRing(M)
m = M.ideal(2)
(2*t1 + a*t2).change_ring(QuotientRing(M,m))

gives me a type error ("polynomial must have unit leading coefficient"). And the input

M.<a> = PolynomialRing(ZZ)
R.<t1,t2> = PolynomialRing(M)
m = M.ideal(2,a)
(2*t1 + a*t2).change_ring(QuotientRing(M,m))

gives the output 2*t1 + abar*t2 rather than the 0 I would have expected. What should I do to get the outputs I would expect (namely 2*t1, abar*t2, and 0, respectively)?

It is not difficult to define a multivariate power series; for example, the following works:

R.<a,x,y> = PowerSeriesRing(ZZ,3)
f = a*x + y + x*y + O(a,x,y)^3

But what I would really like is to treat the series f as a series in x and y, with a acting as a variable coefficient, so that, for example, I could instead write f = a*x + y + x*y + O(x,y)^3. I tried the following:

M = PolynomialRing(ZZ,'a')
R.<x,y> = PowerSeriesRing(M,2)
f = a*x + y + x*y + O(x,y)^3
f

This generates the error "name a is not defined." What can I do to get a into the coefficient ring?