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Height Zeta Function
In mathematics, the height zeta function of an algebraic variety or more generally a subset of a variety encodes the distribution of points of given height. Definition If ''S'' is a set with height function ''H'', such that there are only finitely many elements of bounded height, define a ''counting function'' :N(S,H,B) = \#\ . and a ''zeta function'' : Z(S,H;s) = \sum_ H(x)^ . Properties If ''Z'' has abscissa of convergence β and there is a constant ''c'' such that ''N'' has rate of growth : N \sim c B^a (\log B)^ then a version of the Wiener–Ikehara theorem holds: ''Z'' has a ''t''-fold pole at ''s'' = β with residue ''c''.''a''.Γ(''t''). The abscissa of convergence has similar formal properties to the Nevanlinna invariant and it is conjectured that they are essentially the same. More precisely, Batyrev–Manin conjectured the following. Let ''X'' be a projective variety over a number field ''K'' with ample divisor ''D'' giving rise to an embedding and height f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is dete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Height Function
A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typically functions from a set of points on algebraic varieties (or a set of algebraic varieties) to the real numbers. For instance, the ''classical'' or ''naive height'' over the rational numbers is typically defined to be the maximum of the numerators and denominators of the coordinates (e.g. for the coordinates ), but in a logarithmic scale. Significance Height functions allow mathematicians to count objects, such as rational points, that are otherwise infinite in quantity. For instance, the set of rational numbers of naive height (the maximum of the numerator and denominator when expressed in lowest terms) below any given constant is finite despite the set of rational numbers being infinite. In this sense, height functions can be used to prove asymptotic results such as Baker ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abscissa Of Convergence
In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of : \sum_^\infty a_n e^, where a_n, s are complex numbers and \ is a strictly increasing sequence of nonnegative real numbers that tends to infinity. A simple observation shows that an 'ordinary' Dirichlet series : \sum_^\infty \frac, is obtained by substituting \lambda_n=\ln n while a power series : \sum_^\infty a_n (e^)^n, is obtained when \lambda_n=n. Fundamental theorems If a Dirichlet series is convergent at s_0=\sigma_0+t_0i, then it is uniformly convergent in the domain : , \arg(s-s_0), \leq \theta \sigma_0. There are now three possibilities regarding the convergence of a Dirichlet series, i.e. it may converge for all, for none or for some values of ''s''. In the latter case, there exist a \sigma_c such that the series is convergent for \sigma>\sigma_c and divergent for \sigma \operatorname(s_0). A Dirichlet series may converge absolutely for all, for no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener–Ikehara Theorem
The Wiener–Ikehara theorem is a Tauberian theorem introduced by . It follows from Wiener's Tauberian theorem, and can be used to prove the prime number theorem (Chandrasekharan, 1969). Statement Let ''A''(''x'') be a non-negative, monotonic nondecreasing function of ''x'', defined for 0 ≤ ''x'' 1 to the function ''ƒ''(''s'') and that, for some non-negative number ''c'', :f(s) - \frac has an extension as a continuous function for ℜ(''s'') ≥ 1. Then the limit as ''x'' goes to infinity of ''e''−''x'' ''A''(''x'') is equal to c. One Particular Application An important number-theoretic application of the theorem is to Dirichlet series of the form :\sum_^\infty a(n) n^ where ''a''(''n'') is non-negative. If the series converges to an analytic function in :\Re(s) \ge b with a simple pole of residue ''c'' at ''s'' = ''b'', then :\sum_a(n) \sim \frac X^b. Applying this to the logarithmic derivative of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nevanlinna Invariant
In mathematics, the Nevanlinna invariant of an ample divisor ''D'' on a normal projective variety ''X'' is a real number connected with the rate of growth of the number of rational points on the variety with respect to the embedding defined by the divisor. The concept is named after Rolf Nevanlinna. Formal definition Formally, α(''D'') is the infimum of the rational numbers ''r'' such that K_X + r D is in the closed real cone of effective divisors in the Néron–Severi group of ''X''. If α is negative, then ''X'' is pseudo-canonical. It is expected that α(''D'') is always a rational number. Connection with height zeta function The Nevanlinna invariant has similar formal properties to the abscissa of convergence of the height zeta function and it is conjectured that they are essentially the same. More precisely, Batyrev–Manin conjectured the following. Let ''X'' be a projective variety over a number field ''K'' with ample divisor ''D'' giving rise to an embedding and he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stamm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business international ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Geometry
In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study these equations. Four theorems in Diophantine geometry which are of fundamental importance include: * Mordell–Weil Theorem * Roth's Theorem * Siegel's Theorem * Faltings's Theorem Background Serge Lang published a book ''Diophantine Geometry'' in the area in 1962, and by this book he coined the term "Diophantine Geometry". The traditional arrangement of material on Diophantine equations was by degree and number of variables, as in Mordell's ''Diophantine Equations'' (1969). Mordell's book starts with a remark on homogeneous equations ''f'' = 0 over the rational field, attributed to C. F. Gauss, that non-zero solutions in integers (even primitive lattice points) exist if non-zero rational solutions do, and notes a caveat of L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
