# Multivariate Laurent series

I'm trying to perform calculations with two variables $z, u$, one of which, $z$, should be invertible. Apparently Laurent series rings are not yet implemented for multivariate rings. I tried the following

R = QQ[['zed, you, zedi']]
R
Multivariate Power Series Ring in zed, you, zedi over Rational Field

x = R.gens()
x
(zed, you, zedi)

i = x[0]*x[2] - 1
i
-1 + zed*zedi

I = i*R
I
S = QuotientRing(R, I, names="z, u, xi")
S

Principal ideal (-1 + zed*zedi) of Multivariate Power Series Ring in zed, you, zedi over Rational Field
Quotient of Multivariate Power Series Ring in zed, you, zedi over Rational Field by the ideal (-1 + zed*zedi)

g = S.gens()
g
(zed, you, zedi)

g[0]*g[2]
zed*zedi


Apparently, in the quotient ring I still get that $z z^{-1}$ doesn't simplify to 1. Also, my assigned names aren't recognized. What am I doing wrong? Is this even the best way of dealing with multivariate Laurent polynomials?

Thank you. (This is my first day using sage and my first question here on ask.sagemath.org. Please let me know if I've done something wrong.)

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This is right.Multivariate Laurent series are not implemented in Sage at the moment.

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Adding them is tracked at Sage Trac ticket 19343.

( 2019-04-22 13:58:53 +0100 )edit

If only one of the variables needs to be invertible, one could work with Laurent series in one variable over a power series ring in the other variable; or conversely, power series in one variable over a Laurent series ring in the other variable:

sage: L.<z> = LaurentSeriesRing(QQ)
sage: P.<u> = PowerSeriesRing(L)


or

sage: P.<u> = PowerSeriesRing(QQ)
sage: L.<z> = LaurentSeriesRing(P)


In some cases it might be enough to use a polynomial ring or a Laurent polynomial ring for one of the variables.

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