Laurent polynomials in several variables should be an easy generalization of the already used model for polynomials.
For Laurent series, this can be done similarly to p-adic numbers in case of one variable; however it's not clear at the moment what kind of "error" we should allow in the multivariate case: Somewhat natural would be to give Laurent series up to terms of total degree larger than some bound, or up to O(x_0^{a0} ... x{n-1}^{a_{n-1}}) for some (a0, ..., a{n-1}). We may allow the error to lie in other arbitrary monomial ideals. But then we have two problems: 1. Notation would become cumbersome. 2. Efficient search.
So perhaps only allow Laurent polynomials (multivariate) and univariate Laurent series.
Laurent polynomials in several variables should be an easy generalization of the already used model for polynomials. For Laurent series, this can be done similarly to p-adic numbers in case of one variable; however it's not clear at the moment what kind of "error" we should allow in the multivariate case: Somewhat natural would be to give Laurent series up to terms of total degree larger than some bound, or up to O(x_0^{a0} ... x{n-1}^{a_{n-1}}) for some (a0, ..., a{n-1}). We may allow the error to lie in other arbitrary monomial ideals. But then we have two problems: 1. Notation would become cumbersome. 2. Efficient search.
So perhaps only allow Laurent polynomials (multivariate) and univariate Laurent series.