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Igusa Zeta Function
In mathematics, an Igusa zeta function is a type of generating function, counting the number of solutions of an equation, ''modulo'' ''p'', ''p''2, ''p''3, and so on. Definition For a prime number ''p'' let ''K'' be a p-adic field, i.e. : \mathbb_p\infty , ''R'' the valuation ring and ''P'' the maximal ideal. For z \in K we denote by \operatorname(z) the valuation of ''z'', \mid z \mid = q^, and ac(z)=z \pi^ for a uniformizing parameter π of ''R''. Furthermore let \phi : K^n \to \mathbb be a Schwartz–Bruhat function, i.e. a locally constant function with compact support and let \chi be a character of R^\times. In this situation one associates to a non-constant polynomial f(x_1, \ldots, x_n) \in K _1,\ldots,x_n/math> the Igusa zeta function : Z_\phi(s,\chi) = \int_ \phi(x_1,\ldots,x_n) \chi(ac(f(x_1,\ldots,x_n))) , f(x_1,\ldots,x_n), ^s \, dx where s \in \mathbb, \operatorname(s)>0, and ''dx'' is Haar measure so normalized that R^n has measure 1. Igusa's theorem ... [...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] |
Haar Measure
In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though its special case for Lie groups had been introduced by Adolf Hurwitz in 1897 under the name "invariant integral". Haar measures are used in many parts of analysis, number theory, group theory, representation theory, statistics, probability theory, and ergodic theory. Preliminaries Let (G, \cdot) be a locally compact Hausdorff topological group. The \sigma-algebra generated by all open subsets of G is called the Borel algebra. An element of the Borel algebra is called a Borel set. If g is an element of G and S is a subset of G, then we define the left and right translates of S by ''g'' as follows: * Left translate: g S = \. * Right translate: S g = \. Left and right translates map Borel sets onto Borel sets. A measure \mu on th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crelle's Journal
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by August Leopold Crelle (Berlin) in 1826 and edited by him until his death in 1855. It was one of the first major mathematical journals that was not a proceedings of an academy. It has published many notable papers, including works of Niels Henrik Abel, Georg Cantor, Gotthold Eisenstein, Carl Friedrich Gauss and Otto Hesse. It was edited by Carl Wilhelm Borchardt from 1856 to 1880, during which time it was known as ''Borchardt's Journal''. The current editor-in-chief is Rainer Weissauer (Ruprecht-Karls-Universität Heidelberg) Past editors * 1826–1856 August Leopold Crelle * 1856–1880 Carl Wilhelm Borchardt * 1881–1888 Leopold Kronecker, Karl Weierstrass * 1889–1892 Leopold Kronecker * 1892–1902 Lazarus Fuchs * 1903–1928 Kurt Hens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence Relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation. Basic example The prototypical example of a congruence relation is congruence modulo n on the set of integers. For a given positive integer n, two integers a and b are called congruent modulo n, written : a \equiv b \pmod if a - b is divisible by n (or equivalently if a and b have the same remainder when divided by n). For example, 37 and 57 are congruent modulo 10, : 37 \equiv 57 \pmod since 37 - 57 = -20 is a multiple of 10, or equivalently since both 37 and 57 have a remainder of 7 when divided by 10. Congruence modulo n (for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indicator Function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\in A, and \mathbf_(x)=0 otherwise, where \mathbf_A is a common notation for the indicator function. Other common notations are I_A, and \chi_A. The indicator function of is the Iverson bracket of the property of belonging to ; that is, :\mathbf_(x)= \in A For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers. Definition The indicator function of a subset of a set is a function \mathbf_A \colon X \to \ defined as \mathbf_A(x) := \begin 1 ~&\text~ x \in A~, \\ 0 ~&\text~ x \notin A~. \end The Iverson bracket provides the equivalent notation, \in A/math> or to be used instead of \mathbf_(x)\,. The function \mathbf_A is sometimes denoted , , , or even just . Nota ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat Varieties
In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (''X'':''Y'':''Z'') by the Fermat equation :X^n + Y^n = Z^n.\ Therefore, in terms of the affine plane its equation is :x^n + y^n = 1.\ An integer solution to the Fermat equation would correspond to a nonzero rational number solution to the affine equation, and vice versa. But by Fermat's Last Theorem it is now known that (for ''n'' > 2) there are no nontrivial integer solutions to the Fermat equation; therefore, the Fermat curve has no nontrivial rational points. The Fermat curve is non-singular and has genus :(n - 1)(n - 2)/2.\ This means genus 0 for the case ''n'' = 2 (a conic) and genus 1 only for ''n'' = 3 (an elliptic curve). The Jacobian variety of the Fermat curve has been studied in depth. It is isogenous to a product of simple abelian varieties with complex multiplication. The Fermat curve also has gonality :n-1.\ Fermat varietie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jan Denef
Jan Denef (born 4 September 1951) is a Belgian Belgian may refer to: * Something of, or related to, Belgium * Belgians, people from Belgium or of Belgian descent * Languages of Belgium, languages spoken in Belgium, such as Dutch, French, and German *Ancient Belgian language, an extinct languag ... mathematician. He is an Emeritus Professor of Mathematics at the KU Leuven, Katholieke Universiteit Leuven (KU Leuven). Denef obtained his PhD from KU Leuven in 1975 with a thesis on Hilbert's tenth problem; his advisors were Louis Philippe Bouckaert and Willem Kuijk. He is a specialist of model theory, number theory and algebraic geometry. He is well known for his early work on Hilbert's tenth problem and for developing the theory of motivic integration in a series of papers with François Loeser. He has also worked on computational number theory. Recently he proved a conjecture of Jean-Louis Colliot-Thélène which generalizes the Ax–Kochen theorem. In 2002 Denef was an List of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resolution Of Singularities
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety ''V'' has a resolution, a non-singular variety ''W'' with a proper birational map ''W''→''V''. For varieties over fields of characteristic 0 this was proved in Hironaka (1964), while for varieties over fields of characteristic ''p'' it is an open problem in dimensions at least 4. Definitions Originally the problem of resolution of singularities was to find a nonsingular model for the function field of a variety ''X'', in other words a complete non-singular variety ''X′'' with the same function field. In practice it is more convenient to ask for a different condition as follows: a variety ''X'' has a resolution of singularities if we can find a non-singular variety ''X′'' and a proper birational map from ''X′'' to ''X''. The condition that the map is proper is needed to exclude trivial solutions, such as taking ''X′'' to be the subvariety of non- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heisuke Hironaka
is a Japanese mathematician who was awarded the Fields Medal in 1970 for his contributions to algebraic geometry. Career Hironaka entered Kyoto University in 1949. After completing his undergraduate studies at Kyoto University, he received his Ph.D. in 1960 from Harvard University while under the direction of Oscar Zariski. Hironaka held teaching positions at Brandeis University from 1960-1963, Columbia University in 1964, and Kyoto University from 1975 to 1988. He was a professor of mathematics at Harvard University from 1968 until becoming ''emeritus'' in 1992 and was a president of Yamaguchi University from 1996 to 2002. Research In 1964, Hironaka proved that singularities of algebraic varieties admit resolutions in characteristic zero. This means that any algebraic variety can be replaced by (more precisely is birationally equivalent to) a similar variety which has no singularities. He also introduced Hironaka's example showing that a deformation of Kähler manifolds need ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' join ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the ''formal'' power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence in principle has a generating function of each type (except ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Character (mathematics)
In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualified. Multiplicative character A multiplicative character (or linear character, or simply character) on a group ''G'' is a group homomorphism from ''G'' to the multiplicative group of a field , usually the field of complex numbers. If ''G'' is any group, then the set Ch(''G'') of these morphisms forms an abelian group under pointwise multiplication. This group is referred to as the character group of ''G''. Sometimes only ''unitary'' characters are considered (thus the image is in the unit circle); other such homomorphisms are then called ''quasi-characters''. Dirichlet characters can be seen as a special case of this definition. Multiplicative characters are linearly independent, i.e. if \chi_1,\chi_2, \ldots , \chi_n are different cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
