|
Hilbert–Kunz Function
In algebra, the Hilbert–Kunz function of a local ring (''R'', ''m'') of prime characteristic ''p'' is the function :f(q) = \operatorname_R(R/m^) where ''q'' is a power of ''p'' and ''m'' 'q''/sup> is the ideal generated by the ''q''-th powers of elements of the maximal ideal ''m''. The notion was introduced by Ernst Kunz, who used it to characterize a regular ring as a Noetherian ring in which the Frobenius morphism is flat. If d is the dimension of the local ring, Monsky showed that f(q)/(q^d) is c+O(1/q) for some real constant c. This constant, the "Hilbert-Kunz multiplicity", is greater than or equal to 1. Watanabe and Yoshida strengthened some of Kunz's results, showing that in the unmixed case, the ring is regular precisely when c=1. Hilbert–Kunz functions and multiplicities have been studied for their own sake. Brenner and Trivedi have treated local rings coming from the homogeneous co-ordinate rings of smooth projective curves, using techniques from algebraic geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
 |
Algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication. Elementary algebra is the main form of algebra taught in schools. It examines mathematical statements using variables for unspecified values and seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called '' systems of linear equations''. It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. Abstract algebra studies algebraic structures, which consist of a set of mathemati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
Paul Monsky
Paul Monsky (born June 17, 1936) is an American mathematician and professor at Brandeis University. After earning a bachelor's degree from Swarthmore College, he received his Ph.D. in 1962 from the University of Chicago under the supervision of Walter Lewis Baily, Jr. He has introduced the Monsky–Washnitzer cohomology and he has worked intensively on Hilbert–Kunz functions and Hilbert–Kunz multiplicity. In 2007, Monsky and Holger Brenner gave an example showing that tight closure does not commute with localization. The first proof of Monsky's theorem In geometry, Monsky's theorem states that it is not possible to dissect a square into an odd number of triangles of equal area. In other words, a square does not have an odd equidissection. The problem was posed by Fred Richman in the ''American ..., published in 1970, which states that a square cannot be divided into an odd number of equal-area triangles, is named after him. In the mid-1970s, Monsky stopped paying U.S. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
|
David Eisenbud
David Eisenbud (born 8 April 1947 in New York City) is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and former director of the then Mathematical Sciences Research Institute (MSRI), now known as Simons Laufer Mathematical Sciences Institute (SLMath). He served as Director of MSRI from 1997 to 2007, and then again from 2013 to 2022. Biography Eisenbud is the son of mathematical physicist Leonard Eisenbud, who was a student and collaborator of the renowned physicist Eugene Wigner. Eisenbud received his Ph.D. in 1970 from the University of Chicago, where he was a student of Saunders Mac Lane and, unofficially, James Christopher Robson. He then taught at Brandeis University from 1970 to 1997, during which time he had visiting positions at Harvard University, Institut des Hautes Études Scientifiques (IHÉS), University of Bonn, and Centre national de la recherche scientifique (CNRS). He joined the staff at MSRI in 1997, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
 |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
Craig Huneke
Craig Lee Huneke (born August 27, 1951) is an American mathematician specializing in commutative algebra. He is a professor at the University of Virginia. Huneke graduated from Oberlin College with a bachelor's degree in 1973 and in 1978 earned a Ph.D. from the Yale University under Nathan Jacobson and David Eisenbud (''Determinantal ideal and questions related to factoriality''). As a post-doctoral fellow, he was at the University of Michigan. In 1979 he became an assistant professor and was at the Massachusetts Institute of Technology and the University of Bonn (1980). In 1981 he became an assistant professor at Purdue University, where in 1984 he became an associate professor and became a professor in 1987. From 1994 to 1995 he was a visiting professor at the University of Michigan and in 1999 was at the Max Planck Institute for Mathematics in Bonn (as a Fulbright Scholar). In 1999, he was Henry J. Bischoff professor at the University of Kansas. In 2002 he was at MSRI. Since 20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
|
Melvin Hochster
Melvin Hochster (born August 2, 1943) is an American mathematician working in commutative algebra. He is currently the Jack E. McLaughlin Distinguished University Professor Emeritus of Mathematics at the University of Michigan. Education Hochster attended Stuyvesant High School, where he was captain of the Math Team, and received a B.A. from Harvard University. While at Harvard, he was a Putnam Fellow in 1960. He earned his Ph.D. in 1967 from Princeton University, where he wrote a dissertation under Goro Shimura characterizing the prime spectra of commutative rings. Career He held positions at the University of Minnesota and Purdue University before joining the faculty at Michigan in 1977. Hochster's work is primarily in commutative algebra, especially the study of modules over local rings. He has established classic theorems concerning Cohen–Macaulay rings, invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
|
Hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally. A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface. For example, the equation :x_1^2+x_2^2+\cdots+x_n^2-1=0 defines an algebraic hypersurface of dimension in the Euclidean space of dimension . This hypersurface is also a smooth manifold, and is called a hypersphere or an -sphere. Smooth hypersurface A hypersurface that is a smooth manifold is called a ''smooth hypersurface''. In , a smooth hypersurface is ori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
|
Diagonal Form
In mathematics, a diagonal form is an algebraic form (homogeneous polynomial) without cross-terms involving different indeterminates. That is, it is of the form :\sum_^n a_i ^m\ for some degree ''m''. Such forms ''F'', and the hypersurfaces ''F'' = 0 they define in projective space, are very special in geometric terms, with many symmetries. They also include famous cases like the Fermat curves, and other examples well known in the theory of Diophantine equations. A great deal has been worked out about their theory: algebraic geometry, local zeta-functions via Jacobi sums, Hardy-Littlewood circle method. Diagonalization Any degree-2 homogeneous polynomial can be transformed to a diagonal form by variable substitution. Higher-degree homogeneous polynomials can be diagonalized if and only if their catalecticant is non-zero. The process is particularly simple for degree-2 forms (quadratic forms), based on the eigenvalues of the symmetric matrix representing the quadratic f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
 |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
Regular Local Ring
In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be any Noetherian local ring with unique maximal ideal \mathfrak, and suppose a_1,\cdots,a_n is a minimal set of generators of \mathfrak. Then Krull's principal ideal theorem implies that n\geq\dim A, and A is regular whenever n=\dim A. The concept is motivated by its geometric meaning. A point x on an algebraic variety X is nonsingular (a smooth point) if and only if the local ring \mathcal_ of germs at x is regular. (See also: regular scheme.) Regular local rings are ''not'' related to von Neumann regular rings. For Noetherian local rings, there is the following chain of inclusions: Characterizations There are a number of useful definitions of a regular local ring, one of which is mentioned above. In particular, if A is a Noetherian local ring with maximal ideal \mathfrak, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
|
Krull Dimension
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules. The Krull dimension was introduced to provide an algebraic definition of the dimension of an algebraic variety: the dimension of the affine variety defined by an ideal ''I'' in a polynomial ring ''R'' is the Krull dimension of ''R''/''I''. A field ''k'' has Krull dimension 0; more generally, ''k'' 'x''1, ..., ''x''''n''has Krull dimension ''n''. A principal ideal domain that is not a field has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximal ideal is nilpotent. There are several other ways that have been used to define the dimension of a ring. Most of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |