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Weierstrass Preparation Theorem
In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point ''P''. It states that such a function is, up to multiplication by a function not zero at ''P'', a polynomial in one fixed variable ''z'', which is monic, and whose coefficients of lower degree terms are analytic functions in the remaining variables and zero at ''P''. There are also a number of variants of the theorem, that extend the idea of factorization in some ring ''R'' as ''u''·''w'', where ''u'' is a unit and ''w'' is some sort of distinguished Weierstrass polynomial. Carl Siegel has disputed the attribution of the theorem to Weierstrass, saying that it occurred under the current name in some of late nineteenth century ''Traités d'analyse'' without justification. Complex analytic functions For one variable, the local form of an analytic function ''f''(''z'') near 0 is ''z''''k''''h''(''z'') where ''h''(0) is not 0, and ''k' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Power Series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, whose terms are of the form a x^n where x^n is the nth power of a variable x (n is a non-negative integer), and a is called the coefficient. Hence, power series can be viewed as a generalization of polynomials, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the x^n are used only as position-holders for the coefficients, so that the coefficient of x^5 is the fifth ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oka Coherence Theorem
In mathematics, the Oka coherence theorem, proved by , states that the sheaf \mathcal := \mathcal_ of germs of holomorphic functions on \mathbb^n over a complex manifold is coherent.In paper it was called the idéal de domaines indéterminés. See also * Cartan's theorems A and B * Several complex variables * GAGA * Oka–Weil theorem * Weierstrass preparation theorem In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point ''P''. It states that such a function is, up to multiplication by a function not zero at ''P'', a p ... Note References * * * * * Theorems in complex analysis Theorems in complex geometry {{mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noetherian Ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence I_1\subseteq I_2 \subseteq I_3 \subseteq \cdots of left (or right) ideals has a largest element; that is, there exists an such that: I_=I_=\cdots. Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian. Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics are Noetherian (in particular the ring of integers, polynomial rings, and rings of algebraic integers in number fields), and many general theorems on rings rely heavily on Noetherian property (for example, the Laskerâ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rigid Geometry
In mathematics, a rigid analytic space is an analogue of a complex analytic space over a nonarchimedean field. Such spaces were introduced by John Tate in 1962, as an outgrowth of his work on uniformizing ''p''-adic elliptic curves with bad reduction using the multiplicative group. In contrast to the classical theory of ''p''-adic analytic manifolds, rigid analytic spaces admit meaningful notions of analytic continuation and connectedness. Definitions The basic rigid analytic object is the ''n''-dimensional unit polydisc, whose ring of functions is the Tate algebra T_n, made of power series in ''n'' variables whose coefficients approach zero in some complete nonarchimedean field ''k''. The Tate algebra is the completion of the polynomial ring in ''n'' variables under the Gauss norm (taking the supremum of coefficients), and the polydisc plays a role analogous to that of affine ''n''-space in algebraic geometry. Points on the polydisc are defined to be maximal ideals in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-archimedean Field
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields. The property, typically construed, states that given two positive numbers ''x'' and ''y'', there is an integer ''n'' such that ''nx'' > ''y''. It also means that the set of natural numbers is not bounded above. Roughly speaking, it is the property of having no ''infinitely large'' or ''infinitely small'' elements. It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ ''On the Sphere and Cylinder''. The notion arose from the theory of magnitudes of Ancient Greece; it still plays an important role in modern mathematics such as David Hilbert's axioms for geometry, and the theories of ordered groups, ordered fields, and local fields. An algebraic structure in which any two non-zero elements are ''comparable' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tate Algebra
In algebra, the ring of restricted power series is the subring of a formal power series ring that consists of power series whose coefficients approach zero as degree goes to infinity.. Over a non-archimedean complete field, the ring is also called a Tate algebra. 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 polynomial. Definition Let ''A'' be a linearly topologized ring, separated and complete and \ the fundamental system of open ideals. Then the ring of restricted power series is defined as the projective limit of the polynomial rings over A/I_: :A \langle x_1, \dots, x_n \rangle = \varprojlim_ A/I_ _1, \dots, x_n/math>. In other words, it is the completion of the polynomial ring A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iwasawa Theory
In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian varieties. More recently (early 1990s), Ralph Greenberg has proposed an Iwasawa theory for motives. Formulation Iwasawa worked with so-called \Z_p-extensions - infinite extensions of a number field F with Galois group \Gamma isomorphic to the additive group of p-adic integers for some prime ''p''. (These were called \Gamma-extensions in early papers.) Every closed subgroup of \Gamma is of the form \Gamma^, so by Galois theory, a \Z_p-extension F_\infty/F is the same thing as a tower of fields :F=F_0 \subset F_1 \subset F_2 \subset \cdots \subset F_\infty such that \operatorname(F_n/F)\cong \Z/p^n\Z. Iwasawa studied classical Galois modules over F_n by a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iwasawa Algebra
In mathematics, the Iwasawa algebra Λ(''G'') of a profinite group ''G'' is a variation of the group ring of ''G'' with ''p''-adic coefficients that take the topology of ''G'' into account. More precisely, Λ(''G'') is the inverse limit of the group rings Z''p''(''G''/''H'') as ''H'' runs through the open normal subgroups of ''G''. Commutative Iwasawa algebras were introduced by in his study of Z''p'' extensions in Iwasawa theory, and non-commutative Iwasawa algebras of compact ''p''-adic analytic groups were introduced by . Iwasawa algebra of the ''p''-adic integers In the special case when the profinite group ''G'' is isomorphic to the additive group of the ring of ''p''-adic integers Z''p'', the Iwasawa algebra Λ(''G'') is isomorphic to the ring of the formal power series Z''p'' ''T'' in one variable over Z''p''. The isomorphism is given by identifying 1 + ''T'' with a topological generator of ''G''. This ring is a 2-dimensional complete Noetherian regular lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniformizer
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions: # ''R'' is a local principal ideal domain, and not a field. # ''R'' is a valuation ring with a value group isomorphic to the integers under addition. # ''R'' is a local Dedekind domain and not a field. # ''R'' is a Noetherian local domain whose maximal ideal is principal, and not a field.https://mathoverflow.net/a/155639/114772 # ''R'' is an integrally closed Noetherian local ring with Krull dimension one. # ''R'' is a principal ideal domain with a unique non-zero prime ideal. # ''R'' is a principal ideal domain with a unique irreducible element (up to multiplication by units). # ''R'' is a unique factorization domain with a unique irreducible element (up to multiplication by units). # ''R'' is Noetherian, not a field, and every nonzero fractional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distinguished Polynomial
In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point ''P''. It states that such a function is, up to multiplication by a function not zero at ''P'', a polynomial in one fixed variable ''z'', which is monic, and whose coefficients of lower degree terms are analytic functions in the remaining variables and zero at ''P''. There are also a number of variants of the theorem, that extend the idea of factorization in some ring ''R'' as ''u''·''w'', where ''u'' is a unit and ''w'' is some sort of distinguished Weierstrass polynomial. Carl Siegel has disputed the attribution of the theorem to Weierstrass, saying that it occurred under the current name in some of late nineteenth century ''Traités d'analyse'' without justification. Complex analytic functions For one variable, the local form of an analytic function ''f''(''z'') near 0 is ''z''''k''''h''(''z'') where ''h''(0) is not 0, and ''k'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.Translator's introductionpage 35 "Indeed, with hindsight it becomes apparent that a discret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
