# formal laurent series

Next Post Definition and properties of matrices with a single nonzero entry. Let Fp[T] be the polynomial ring with coeﬃcients in Fp and Fp(T) its fraction ﬁeld. Then M is a 1×1-matrix whose single entry is the smallest nonzero exponent appearing in g ( x ). This paper systematically sets up a natural definition and structure of formal Laurent series without those restrictions, including introducing a multiplication between formal Laurent series. In the multivariate case, it is not always possible to write a given f (x) ∈ K C 〚 x 〛 in the form f (x) = x e h (x) for some h (x) ∈ K C 〚 x 〛, as already illustrated in the introduction. Formal Laurent series in several variables. Continue Reading. .A. Tags: Discrete Valuation, Formal Laurent Series, Valuation Ring. Abstract. Linearity . Several kinds of formal Laurent series have been introduced with some restrictions so far. Given x ∈ Fp(T), write x as Tr g h with g;h ∈ Fp[T]; T-gh. Theorem 0.1. This paper systematically sets up a natural definition and structure of formal Laurent series without those restrictions, including introducing a multiplication between formal Laurent series. 1. Previous Post The ring of formal power series over an integral domain is an integral domain. Consider a power series f(x) ∈ 〚x −1 〛 with negative exponents (i.e., C is the negative halfline) and let g(x) ∈ ((x))∖{0} be a usual formal Laurent series. 2013;31(4):350-367. 2.2. Expo Math. Several kinds of formal Laurent series have been introduced with some restrictions so far. (2)Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, A4040 Linz, Austria. Formal series Laurent series. I would like to compute some integrals of products of Laurent series, with the goal of getting an explicit expression for the n-th coefficient of the result. We now introduce this ﬁeld. Author information: (1)Université Lille 1 Sciences et Technologies, 59655 Villeneuve d'Ascq, Cedex, France. The ring of formal Laurent series over a commutative ring A A in an indeterminate x x consists of Laurent series ∑ n ∈ ℤ f n z n \sum_{n \in \mathbb{Z}} f_n z^n, with f n ∈ A f_n \in A but where all but finitely many f n f_n for n < 0 n \lt 0 vanish. On the other hand the second series in (0.1) is a regular power series, and hence setting R 2 = (limsup n!1 jc nj1=n) 1; the second series is convergent for jz aj1=R 1. De nition 2.1. Add a comment | 1. Typically formal Laurent series have almost all negative coefficients vanishing, so I think it would be more accurate to say \$\mathbf{C} \llbracket t \rrbracket [ t^{-1} ] – Jackson Sep 15 '20 at 14:09. It gets pretty messy so I was hoping that sage could keep track of the details for me. 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