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Thus when defining formal Laurent series, one requires Laurent series with only finitely many negative terms. Similarly, the sum of two convergent Laurent series need not converge, though it is always defined formally, but the sum of two bounded below Laurent series (or any Laurent series on a punctured disk) has a non-empty annulus of convergence.
A formal power series can be loosely thought of as an object that is like a polynomial, but with infinitely many terms.Alternatively, for those familiar with power series (or Taylor series), one may think of a formal power series as a power series in which we ignore questions of convergence by not assuming that the variable X denotes any numerical value (not even an unknown value).
The formal Laurent series over a finite field: the ring of integers of F q ((T)) is the ring of formal power series F q [[T]]. Its maximal ideal is (T) (i.e. the set of power series whose constant terms are zero) and its residue field is F q. Its normalized valuation is related to the (lower) degree of a formal Laurent series as follows:
In mathematics, local class field theory, introduced by Helmut Hasse, [1] is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension of ...
Then the valuation ring R consists of rational functions with no pole at x = a, and the completion is the formal Laurent series ring F((x−a)). This can be generalized to the field of Puiseux series K {{ t }} (fractional powers), the Levi-Civita field (its Cauchy completion), and the field of Hahn series , with valuation in all cases returning ...
A Laurent polynomial with coefficients in a field is an expression of the form p = ∑ k p k X k , p k ∈ F {\displaystyle p=\sum _{k}p_{k}X^{k},\quad p_{k}\in \mathbb {F} } where X {\displaystyle X} is a formal variable, the summation index k {\displaystyle k} is an integer (not necessarily positive) and only finitely many coefficients p k ...
Formal power series are used in combinatorics to describe and study sequences that are otherwise difficult to handle, for example, using the method of generating functions. The Hilbert–Poincaré series is a formal power series used to study graded algebras.
The definition of a Puiseux series includes that the denominators of the exponents must be bounded. So, by reducing exponents to a common denominator n, a Puiseux series becomes a Laurent series in an n th root of the indeterminate. For example, the example above is a Laurent series in /.