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The rings of convergent power series over the real or complex numbers are Henselian. Rings of algebraic power series over a field are Henselian. A local ring that is integral over a Henselian ring is Henselian. The Henselization of a local ring is a Henselian local ring. Every quotient of a Henselian ring is Henselian.
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).
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 ...
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.
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.
The order of the power series f is defined to be the least value such that there is a α ≠ 0 with = | | = + + +, or if f ≡ 0. In particular, for a power series f(x) in a single variable x, the order of f is the smallest power of x with a nonzero coefficient.
There are some other related constructions. A formal power series ring [[]] consists of formal power series , together with multiplication and addition that mimic those for convergent series. It contains R[t] as a subring. A formal power series ring does not have the universal property of a polynomial ring; a series may not converge after a ...
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.