|
Siegel Identity
In mathematics, Siegel's identity refers to one of two formulae that are used in the resolution of Diophantine equations. Statement The first formula is : \frac + \frac = 1 . The second is : \frac \cdot\frac + \frac \cdot \frac = 1 . Application The identities are used in translating Diophantine problems connected with integral points on hyperelliptic curves into S-unit equations. See also * Siegel formula References * * * * * {{cite book , title=The Algorithmic Resolution of Diophantine Equations , volume=41 , series=London Mathematical Society Student Texts , first=N. P. , last=Smart , authorlink=Nigel Smart (cryptographer) , publisher=Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ... , year=1998 , isbn=0-521-64633-2 , page36–37, ... [...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] |
Diophantine Equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents. Diophantine problems have fewer equations than unknowns and involve finding integers that solve simultaneously all equations. As such systems of equations define algebraic curves, algebraic surfaces, or, more generally, algebraic sets, their study is a part of algebraic geometry that is called ''Diophantine geometry''. The word ''Diophantine'' refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine problems that Di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperelliptic Curve
In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form y^2 + h(x)y = f(x) where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' distinct roots, and ''h''(''x'') is a polynomial of degree 3. Therefore, in giving such an equation to specify a non-singular curve, it is almost always assumed that a non-singular model (also called a smooth completion), equivalent in the sense of birational geometry, is meant. To be more precise, the equation defines a quadratic extension of C(''x''), and it is that function field that is meant. The singular point at infinity can be removed (since this is a curve) by the normalization ( integral closure) process. It turns out that after doing this, there is an open cover of the curve by two affine charts: the one already given by y^2 = f(x) and another one given by w^2 = v^f(1/v) . The glueing maps between the two charts are given by (x, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
S-unit Equation
In mathematics, in the field of algebraic number theory, an ''S''-unit generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for ''S''-units. Definition Let ''K'' be a number field with ring of integers ''R''. Let ''S'' be a finite set of prime ideals of ''R''. An element ''x'' of ''K'' is an ''S''-unit if the principal fractional ideal (''x'') is a product of primes in ''S'' (to positive or negative powers). For the ring of rational integers Z one may take ''S'' to be a finite set of prime numbers and define an ''S''-unit to be a rational number whose numerator and denominator are divisible only by the primes in ''S''. Properties The ''S''-units form a multiplicative group containing the units of ''R''. Dirichlet's unit theorem holds for ''S''-units: the group of ''S''-units is finitely generated, with rank (maximal number of multiplicatively independent elements) equal to ''r'' + ''s'', where ''r'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siegel Formula
Siegel (also Segal or Segel), is a German and Ashkenazi Jewish surname. it can be traced to 11th century Bavaria and was used by people who made wax seals for or sealed official documents (each such male being described as a ''Siegelbeamter''). Alternate spellings include Sigel, Sigl, Siegl, and others. "Siegel" is also the modern German word for seal. The name ultimately derives from the Latin ''sigillum,'' meaning "seal" as in the Seal of the City of New York: ''Sigillum Civitatis Novi Eboraci''. The Germanicized derivative of the name was given to professional seal makers and engravers. Some researchers have attributed the surname to Sigel, referring to Sól (Sun), the goddess of the sun in Germanic mythology (Siȝel or sigel in Old English / Anglo-Saxon), but that is highly speculative. Variants, and false cognates Other variants may routinely include Siegelman, Siegle, Sigl, and Sigel. Presumably, some bearers of these names are lineal descendants of ethnic Jews who chang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...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] |
Algebraic Identities
In mathematics, an identity is an equality relating one mathematical expression ''A'' to another mathematical expression ''B'', such that ''A'' and ''B'' (which might contain some variables) produce the same value for all values of the variables within a certain range of validity. In other words, ''A'' = ''B'' is an identity if ''A'' and ''B'' define the same functions, and an identity is an equality between functions that are differently defined. For example, (a+b)^2 = a^2 + 2ab + b^2 and \cos^2\theta + \sin^2\theta =1 are identities. Identities are sometimes indicated by the triple bar symbol instead of , the equals sign. Common identities Algebraic identities Certain identities, such as a+0=a and a+(-a)=0, form the basis of algebra, while other identities, such as (a+b)^2 = a^2 + 2ab +b^2 and a^2 - b^2 = (a+b)(a-b), can be useful in simplifying algebraic expressions and expanding them. Trigonometric identities Geometrically, trigonometric ide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
