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Manin Conjecture
In mathematics, the Manin conjecture describes the conjectural distribution of rational points on an algebraic variety relative to a suitable height function. It was proposed by Yuri I. Manin and his collaborators in 1989 when they initiated a program with the aim of describing the distribution of rational points on suitable algebraic varieties. Conjecture Their main conjecture is as follows. Let V be a Fano variety defined over a number field K, let H be a height function which is relative to the anticanonical divisor and assume that V(K) is Zariski dense in V. Then there exists a non-empty Zariski open subset U \subset V such that the counting function of K-rational points of bounded height, defined by :N_(B)=\#\ for B \geq 1, satisfies :N_(B) \sim c B (\log B)^, as B \to \infty. Here \rho is the rank of the Picard group In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Rational Points Of Bounded Height Outside The 27 Lines On Clebsch's Diagonal Cubic Surface
Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ability, as in a rational animal, to a psychological process, like reasoning, to mental states, such as beliefs and intentions, or to persons who possess these other forms of rationality. A thing that lacks rationality is either ''arational'', if it is outside the domain of rational evaluation, or ''irrational'', if it belongs to this domain but does not fulfill its standards. There are many discussions about the essential features shared by all forms of rationality. According to reason-responsiveness accounts, to be rational is to be responsive to reasons. For example, dark clouds are a reason for taking an umbrella, which is why it is rational for an agent to do so in response. An important rival to this approach are coherence-based accou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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 solution set, set of solutions of a system of polynomial equations over the real number, 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 component, irreducible, which means that it is not the Union (set theory), union of two smaller Set (mathematics), sets that are Closed set, 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 mon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Yuri I
Yuri I Vladimirovich (; ; c. 1099 – 15 May 1157), commonly known as Yuri Dolgorukiy (, ) or the Long Arm, was a Monomakhovichi prince of Rostov and Suzdal, acquiring the name ''Suzdalia'' during his reign. Noted for successfully curbing the privileges of the landowning ''boyar'' class in Rostov-Suzdal and his ambitious building programme, Yuri transformed this principality into the independent power that would evolve into early modern Muscovy. Yuri Dolgorukiy was the progenitor of the Yurievichi ( ), a branch of the Monomakhovichi. Yuri spent much of his life in internecine strife with the other Rus' princes for suzerainty over the Kievan Rus, which had been held by his father ( Vladimir Monomakh) and his elder brother before him. Although he twice managed to briefly hold Kiev (in September 1149 – April 1151, again in March 1155 – May 1157) and rule as Grand Prince of Kiev, his autocratic rule and perceived foreigner status made him unpopular with the powerful Kieva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current (2023) managing editors are Jean-Benoît Bost (University of Paris-Sud) and Wilhelm Schlag (Yale University Yale University is a Private university, private Ivy League research university in New Haven, Connecticut, United States. Founded in 1701, Yale is the List of Colonial Colleges, third-oldest institution of higher education in the United Stat ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Academic journals established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Fano Variety
In algebraic geometry, a Fano variety, introduced by Gino Fano , is an algebraic variety that generalizes certain aspects of complete intersections of algebraic hypersurfaces whose sum of degrees is at most the total dimension of the ambient projective space. Such complete intersections have important applications in geometry and number theory, because they typically admit rational points, an elementary case of which is the Chevalley–Warning theorem. Fano varieties provide an abstract generalization of these basic examples for which rationality questions are often still tractable. Formally, a Fano variety is a complete variety ''X'' whose anticanonical bundle ''K''X* is ample. In this definition, one could assume that ''X'' is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities. Recently techniques in differential geometry have been applied to the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Number Field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a field that contains \mathbb and has finite dimension when considered as a vector space over The study of algebraic number fields, that is, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind the rational numbers, by using algebraic methods. Definition Prerequisites The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. These operations make the field into an abelian group under addition, and they make the nonzero elements of the field into another abelian group under multiplication ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Canonical Bundle
In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the nth exterior power of the cotangent bundle \Omega on V. Over the complex numbers, it is the determinant bundle of the holomorphic cotangent bundle T^*V. Equivalently, it is the line bundle of holomorphic n-forms on V. This is the dualising object for Serre duality on V. It may equally well be considered as an invertible sheaf. The canonical class is the divisor class of a Cartier divisor K on V giving rise to the canonical bundle — it is an equivalence class for linear equivalence on V, and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor −K with K canonical. The anticanonical bundle is the corresponding inverse bundle \omega^. When the anticanonical bundle of V is ample, V is called a Fano variety. The adjunction formula Suppose that X is a smooth variety and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Zariski Topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology defined on geometric objects called varieties. It is very different from topologies that are commonly used in real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological space. The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces. The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Picard Group
In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds. Alternatively, the Picard group can be defined as the sheaf cohomology group :H^1 (X, \mathcal_X^).\, For integral Scheme (mathematics), schemes the Picard group is isomorphic to the class group of Cartier divisors. For complex manifolds the exponential sheaf sequence gives basic information on the Picard group. The name is in honour of Émile Picard's theories, in particular of divisors on algebraic surfaces. Examples * The Picard group of the Spectrum of a ring, spectrum of a Dedekind domain is its ''ideal class group''. * The invertible sheaves on projective space P''n''(''k'') for '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Duke Mathematical Journal
''Duke Mathematical Journal'' is a peer-reviewed mathematics journal published by Duke University Press. It was established in 1935. The founding editors-in-chief were David Widder, Arthur Coble, and Joseph Miller Thomas. The first issue included a paper by Solomon Lefschetz. Leonard Carlitz served on the editorial board for 35 years, from 1938 to 1973. The current managing editor is Richard Hain (Duke University). Impact According to the journal homepage, the journal has a 2018 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... of 2.194, ranking it in the top ten mathematics journals in the world. References External links * Mathematics journals Mathematical Journal Academic journals established in 1935 Multilingual journals English-language journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |