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Vojta's Conjecture
In mathematics, Vojta's conjecture is a conjecture introduced by about heights of points on algebraic varieties over number fields. The conjecture was motivated by an analogy between diophantine approximation and Nevanlinna theory (value distribution theory) in complex analysis. It implies many other conjectures in Diophantine approximation, Diophantine equations, arithmetic geometry, and mathematical logic. Statement of the conjecture Let F be a number field, let X/F be a non-singular algebraic variety, let D be an effective divisor on X with at worst normal crossings, let H be an ample divisor on X, and let K_X be a canonical divisor on X. Choose Weil height functions h_H and h_ and, for each absolute value v on F, a local height function \lambda_. Fix a finite set of absolute values S of F, and let \epsilon>0. Then there is a constant C and a non-empty Zariski open set U\subseteq X, depending on all of the above choices, such that :: \sum_ \lambda_(P) + h_(P) \le \epsilon h_H(P) ... [...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] |
Absolute Value (algebra)
In algebra, an absolute value (also called a valuation, magnitude, or norm, although "norm (mathematics), norm" usually refers to a specific kind of absolute value on a Field (mathematics), field) is a Function (mathematics), function which measures the "size" of elements in a field or integral domain. More precisely, if ''D'' is an integral domain, then an absolute value is any mapping , x, from ''D'' to the real numbers R satisfying: It follows from these axioms that , 1, = 1 and , -1, = 1. Furthermore, for every positive integer ''n'', :, ''n'', = , 1 + 1 + ... + 1 (''n'' times), = , −1 − 1 − ... − 1 (''n'' times), ≤ ''n''. The classical "absolute value" is one in which, for example, , 2, =2, but many other functions fulfill the requirements stated above, for instance the square root of the classical absolute value (but not the square thereof). An ab ... [...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 internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ABC Conjecture
The ''abc'' conjecture (also known as the Oesterlé–Masser conjecture) is a conjecture in number theory that arose out of a discussion of Joseph Oesterlé and David Masser in 1985. It is stated in terms of three positive integers ''a'', ''b'' and ''c'' (hence the name) that are relatively prime and satisfy ''a'' + ''b'' = ''c''. The conjecture essentially states that the product of the distinct prime factors of ''abc'' is usually not much smaller than ''c''. A number of famous conjectures and theorems in number theory would follow immediately from the ''abc'' conjecture or its versions. Mathematician Dorian Goldfeld described the ''abc'' conjecture as "The most important unsolved problem in Diophantine analysis". The ''abc'' conjecture originated as the outcome of attempts by Oesterlé and Masser to understand the Szpiro conjecture about elliptic curves, which involves more geometric structures in its statement than the ''abc'' conjecture. The ''abc'' conjecture was sho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bombieri–Lang Conjecture
In arithmetic geometry, the Bombieri–Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variety of general type. Statement The weak Bombieri–Lang conjecture for surfaces states that if X is a smooth surface of general type defined over a number field k, then the points of X do not form a dense set in the Zariski topology on X. The general form of the Bombieri–Lang conjecture states that if X is a positive-dimensional algebraic variety of general type defined over a number field k, then the points of X do not form a dense set in the Zariski topology. The refined form of the Bombieri–Lang conjecture states that if X is an algebraic variety of general type defined over a number field k, then there is a dense open subset U of X such that for all number field extensions k' over k, the set of points in U is finite. History The Bombieri–Lang conjecture was independent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Type
In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''. Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the notation ''κ''. Shigeru Iitaka extended it and defined the Kodaira dimension for higher dimensional varieties (under the name of canonical dimension), and later named it after Kunihiko Kodaira. The plurigenera The canonical bundle of a smooth scheme, smooth algebraic variety ''X'' of dimension ''n'' over a field is the line bundle of ''n''-forms, :\,\!K_X = \bigwedge^n\Omega^1_X, which is the ''n''th exterior power of the cotangent bundle of ''X''. For an integer ''d'', the ''d''th tensor power of ''K''''X'' is again a line bundle. For ''d'' ≥ 0, the vector space of global sections ''H''0(''X'',''K''''X''''d'') has the remarkable property that it is a birational invariant of smooth projective varieties ''X''. That is, this vector spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Serge Lang
Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the influential ''Algebra''. He received the Frank Nelson Cole Prize in 1960 and was a member of the Bourbaki group. As an activist, Lang campaigned against the Vietnam War, and also successfully fought against the nomination of the political scientist Samuel P. Huntington to the National Academies of Science. Later in his life, Lang was an HIV/AIDS denialist. He claimed that HIV had not been proven to cause AIDS and protested Yale's research into HIV/AIDS. Biography and mathematical work Lang was born in Saint-Germain-en-Laye, close to Paris, in 1927. He had a twin brother who became a basketball coach and a sister who became an actress. Lang moved with his family to California as a teenager, where he graduated in 1943 from Beverly Hills H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calabi-Yau Variety
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry. Their name was coined by , after who first conjectured that such surfaces might exist, and who proved the Calabi conjecture. Calabi–Yau manifolds are complex manifolds that are generalizations of K3 surfaces in any number of complex dimensions (i.e. any even number of real dimensions). They were originally defined as compact Kähler manifolds with a vanishing first Chern class and a Ricci-flat metric, though many other similar but inequivalent definitions are sometimes used. Definitions The motivational definition given by Shing-Tung Yau is of a compact Kähle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K3 Surface
In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface :x^4+y^4+z^4+w^4=0 in complex projective 3-space. Together with two-dimensional compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two. As such, they are at the center of the classification of algebraic surfaces, between the positively curved del Pezzo surfaces (which are easy to classify) and the negatively curved surfaces of general type (which are essentially unclassifiable). K3 surfaces can be considered the simplest algebraic varieti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined ''over'' that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localization techniques lead naturally fr ... [...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's t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Varieties
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 determined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
