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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).
Every field is a Henselian local ring. (But not every field with valuation is "Henselian" in the sense of the fourth definition above.) Complete Hausdorff local rings, such as the ring of p-adic integers and rings of formal power series over a field, are Henselian. The rings of convergent power series over the real or complex numbers are Henselian.
Quotient rings of the ring are used in the study of a formal algebraic space as well as rigid analysis, the latter over non-archimedean complete fields. Over a discrete topological ring, the ring of restricted power series coincides with a polynomial ring; thus, in this sense, the notion of "restricted power series" is a generalization of a ...
In the complex case, algebraic geometry begins by defining the complex affine space to be . For each , we define , the ring of analytic functions on to be the ring of holomorphic functions, i.e. functions on that can be written as a convergent power series in a neighborhood of each point.
3. An analytic ring is a quotient of a ring of convergent power series in a finite number of variables over a field with a valuation. analytically This often refers to properties of the completion of a local ring; cf. #formally 1. A local ring is called analytically normal if its completion is an integrally closed domain. 2.
The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynomial with infinitely many terms. Conversely, every polynomial is a power ...
Every field is a G-ring; Every complete Noetherian local ring is a G-ring; Every ring of convergent power series in a finite number of variables over R or C is a G-ring.; Every Dedekind domain in characteristic 0, and in particular the ring of integers, is a G-ring, but in positive characteristic there are Dedekind domains (and even discrete valuation rings) that are not G-rings.
The Weierstrass preparation theorem can be used to show that the ring of convergent power series over the complex numbers in a finite number of variables is a Wierestrass ring. The same is true if the complex numbers are replaced by a perfect field with a valuation.