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Weierstrass Point
In mathematics, a Weierstrass point P on a nonsingular algebraic curve C defined over the complex numbers is a point such that there are more functions on C, with their poles restricted to P only, than would be predicted by the Riemann–Roch theorem. The concept is named after Karl Weierstrass. Consider the vector spaces :L(0), L(P), L(2P), L(3P), \dots where L(kP) is the space of meromorphic functions on C whose order at P is at least -k and with no other poles. We know three things: the dimension is at least 1, because of the constant functions on C; it is non-decreasing; and from the Riemann–Roch theorem the dimension eventually increments by exactly 1 as we move to the right. In fact if g is the genus of C, the dimension from the k-th term is known to be :l(kP) = k - g + 1, for k \geq 2g - 1. Our knowledge of the sequence is therefore :1, ?, ?, \dots, ?, g, g + 1, g + 2, \dots. What we know about the ? entries is that they can increment by at most 1 each time (this is ... [...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] |
Ramified Covering
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as when a covering map degenerates at a point of a space, with some collapsing of the fibers of the mapping. In complex analysis In complex analysis, the basic model can be taken as the ''z'' → ''z''''n'' mapping in the complex plane, near ''z'' = 0. This is the standard local picture in Riemann surface theory, of ramification of order ''n''. It occurs for example in the Riemann–Hurwitz formula for the effect of mappings on the genus. See also branch point. In algebraic topology In a covering map the Euler–Poincaré characteristic should multiply by the number of sheets; ramification can therefore be detected by some dropping from that. The ''z'' → ''z''''n'' mapping shows this as a local ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic (field)
In mathematics, the characteristic of a ring (mathematics), ring , often denoted , is defined to be the smallest number of times one must use the ring's identity element, multiplicative identity (1) in a sum to get the additive identity (0). If this sum never reaches the additive identity the ring is said to have characteristic zero. That is, is the smallest positive number such that: :\underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent (group theory), exponent of the ring's additive group, that is, the smallest positive integer such that: :\underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). Some authors do not include the multiplicative identity element in their r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subsemigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup. As in the case of groups or magmas, the semigroup operation need not be commutative, so ''x''·''y'' is not necessarily equal to ''y''·''x''; a well-known example of an operation that is ass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adolf Hurwitz
Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory. Early life He was born in Hildesheim, then part of the Kingdom of Hanover, to a Jewish family and died in Zürich, in Switzerland. His father Salomon Hurwitz, a merchant, was not wealthy. Hurwitz's mother, Elise Wertheimer, died when he was three years old. Family records indicate that he had siblings and cousins, but their names have yet to be confirmed except for an older brother, Julius, with whom he developed an arithmetical theory for complex continued fractions circa 1890. Hurwitz entered the in Hildesheim in 1868. He was taught mathematics there by Hermann Schubert. Schubert persuaded Hurwitz's father to allow him to attend university, and arranged for Hurwitz to study with Felix Klein at Munich. Salomon Hurwitz could not afford to send his son to university, but his friend, Mr. Edwards, assisted financially. Educational career Hur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Semigroup
In mathematics, a numerical semigroup is a special kind of a semigroup. Its underlying set is the set of all nonnegative integers except a finite number and the binary operation is the operation of addition of integers. Also, the integer 0 must be an element of the semigroup. For example, while the set is a numerical semigroup, the set is not because 1 is in the set and 1 + 1 = 2 is not in the set. Numerical semigroups are commutative monoids and are also known as numerical monoids. The definition of numerical semigroup is intimately related to the problem of determining nonnegative integers that can be expressed in the form ''x''1''n''1 + ''x''2 ''n''2 + ... + ''x''''r'' ''n''''r'' for a given set of positive integers and for arbitrary nonnegative integers ''x''1, ''x''2, ..., ''x''''r''. This problem had been considered by several mathematicians like Frobenius (1849–1917) and Sylvester (1814–1897) at the end of the 19th century. During the second half of the twentieth c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clifford's Theorem On Special Divisors
In mathematics, Clifford's theorem on special divisors is a result of on algebraic curves, showing the constraints on special linear systems on a curve ''C''. Statement A divisor on a Riemann surface ''C'' is a formal sum \textstyle D = \sum_P m_P P of points ''P'' on ''C'' with integer coefficients. One considers a divisor as a set of constraints on meromorphic functions in the function field of ''C,'' defining L(D) as the vector space of functions having poles only at points of ''D'' with positive coefficient, ''at most as bad'' as the coefficient indicates, and having zeros at points of ''D'' with negative coefficient, with ''at least'' that multiplicity. The dimension of L(D) is finite, and denoted \ell(D). The linear system of divisors attached to ''D'' is the corresponding projective space of dimension \ell(D)-1. The other significant invariant of ''D'' is its degree ''d'', which is the sum of all its coefficients. A divisor is called ''special'' if ''ℓ''(''K'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; for example, two distinct projective lines in a projective plane meet in exactly one point (there is no "parallel" case). There are many equivalent ways to formally define a projective line; one of the most common is to define a projective line over a field ''K'', commonly denoted P1(''K''), as the set of one-dimensional subspaces of a two-dimensional ''K''-vector space. This definition is a special instance of the general definition of a projective space. The projective line over the reals is a manifold; see real projective line for details. Homogeneous coordinates An arbitrary point in the projective line P1(''K'') may be represented by an equivalence class of ''homogeneous coordinates'', which take the form of a pair : _1 : x_2/mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. The main interest in Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm. Every Riemann surface is a two-dimensional real analytic manifold (i.e., a surface), but it contains more structure (specifically a complex structure) which is needed for the unambiguous definitio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is an algebraic variety that is birationally equivalent to an algebraic plane curve. If the curve is contained in an affine space or a projective space, one can take a projection for such a ... [...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] |
