|
Bernstein–Kushnirenko Theorem
The Bernstein–Kushnirenko theorem (or Bernstein–Khovanskii–Kushnirenko (BKK) theorem), proven by David Bernstein and in 1975, is a theorem in algebra. It states that the number of non-zero complex solutions of a system of Laurent polynomial equations f_1= \cdots = f_n=0 is equal to the mixed volume of the Newton polytopes of the polynomials f_1, \ldots, f_n, assuming that all non-zero coefficients of f_n are generic. Statement Let A be a finite subset of \Z^n. Consider the subspace L_A of the Laurent polynomial algebra \Complex \left x_1^, \ldots, x_n^ \right /math> consisting of Laurent polynomials whose exponents are in A. That is: :L_A = \left \, \right. where for each \alpha = (a_1, \ldots, a_n) \in \Z^n we have used the shorthand notation x^\alpha to denote the monomial x_1^ \cdots x_n^. Now take n finite subsets A_1, \ldots, A_n of \Z^n , with the corresponding subspaces of Laurent polynomials, L_, \ldots, L_. Consider a generic system of equations from these ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) () is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stammbach (1997, 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Science+Business Media
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, op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Bernstein (mathematician) )
{{hndis, Bernstein, David ...
David Bernstein may refer to: * David P. Bernstein (born 1956), professor of forensic psychotherapy * David Bernstein (law professor) (born 1967), American law professor * David I. Bernstein, dean of Pardes Institute of Jewish Studies, Jerusalem and New York City * David Bernstein (architect) (1937–2018), co-founder of Circle 33, see Levitt Bernstein * David Bernstein (businessman) (born 1943), chairman of the British Red Cross, formerly chairman of French Connection, of Manchester City F.C., and of the English Football Association * David Bernstein (activist), American political activist * David Bernstein, Israeli chess player (see Israeli Chess Championship The Israeli Chess Championship is a chess event held every year in Israel. History From 1951 to 1971, the men's and women's championships were held every two years, eventually becoming an annual event. Winners Notes References Bibliogra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
Laurent Polynomial
In mathematics, a Laurent polynomial (named after Pierre Alphonse Laurent) in one variable over a field \mathbb is a linear combination of positive and negative powers of the variable with coefficients in \mathbb. Laurent polynomials in X form a ring denoted \mathbb , X^/math>. They differ from ordinary polynomials in that they may have terms of negative degree. The construction of Laurent polynomials may be iterated, leading to the ring of Laurent polynomials in several variables. Laurent polynomials are of particular importance in the study of complex variables. Definition A Laurent polynomial with coefficients in a field \mathbb is an expression of the form : p = \sum_k p_k X^k, \quad p_k \in \mathbb where X is a formal variable, the summation index k is an integer (not necessarily positive) and only finitely many coefficients p_ are non-zero. Two Laurent polynomials are equal if their coefficients are equal. Such expressions can be added, multiplied, and brought back t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixed Volume
In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to a tuple of convex bodies in \mathbb^n. This number depends on the size and shape of the bodies, and their relative orientation to each other. Definition Let K_1, K_2, \dots, K_r be convex bodies in \mathbb^n and consider the function : f(\lambda_1, \ldots, \lambda_r) = \mathrm_n (\lambda_1 K_1 + \cdots + \lambda_r K_r), \qquad \lambda_i \geq 0, where \text_n stands for the n-dimensional volume, and its argument is the Minkowski sum of the scaled convex bodies K_i. One can show that f is a homogeneous polynomial of degree n, so can be written as : f(\lambda_1, \ldots, \lambda_r) = \sum_^r V(K_, \ldots, K_) \lambda_ \cdots \lambda_, where the functions V are symmetric. For a particular index function j \in \^n , the coefficient V(K_, \dots, K_) is called the mixed volume of K_, \dots, K_. Properties * The mixed volume is uniquely determined by the fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton Polytope
In mathematics, the Newton polytope is an integral polytope associated with a multivariate polynomial that can be used in the asymptotic analysis of those polynomials. It is a generalization of the KruskalNewton diagram developed for the analysis of bivariant polynomials. Given a vector \mathbf=(x_1,\ldots,x_n) of variables and a finite family (\mathbf_k)_k of pairwise distinct vectors from \mathbb^n each encoding the exponents within a monomial, consider the multivariate polynomial f(\mathbf)=\sum_k c_k\mathbf^ where we use the shorthand notation (x_1,\ldots,x_n)^ for the monomial x_1^x_2^\cdots x_n^. Then the Newton polytope associated to f is the convex hull of the vectors \mathbf_k; that is \operatorname(f)=\left\\!. In order to make this well-defined, we assume that all coefficients c_k are non-zero. The Newton polytope satisfies the following homomorphism-type property: \operatorname(fg)=\operatorname(f)+\operatorname(g) where the addition is in the sense of Minkowski. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Addition
In geometry, the Minkowski sum of two set (mathematics), sets of position vectors ''A'' and ''B'' in Euclidean space is formed by vector addition, adding each vector in ''A'' to each vector in ''B'': A + B = \ The Minkowski difference (also ''Minkowski subtraction'', ''Minkowski decomposition'', or ''geometric difference'') is the corresponding inverse, where (A - B) produces a set that could be summed with ''B'' to recover ''A''. This is defined as the Complement (set theory), complement of the Minkowski sum of the complement of ''A'' with the reflection of ''B'' about the origin. \begin -B &= \\\ A - B &= (A^\complement + (-B))^\complement \end This definition allows a symmetrical relationship between the Minkowski sum and difference. Note that alternately taking the sum and difference with ''B'' is not necessarily equivalent. The sum can fill gaps which the difference may not re-open, and the difference can erase small islands which the sum cannot recreate from nothing. \be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a Bounded set, bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its projective duality, dual problem of intersecting Half-space (geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Lattice Polytope
In geometry and polyhedral combinatorics, an integral polytope is a convex polytope whose vertices all have integer Cartesian coordinates. That is, it is a polytope that equals the convex hull of its integer points. Integral polytopes are also called lattice polytopes or Z-polytopes. The special cases of two- and three-dimensional integral polytopes may be called polygons or polyhedra instead of polytopes, respectively. Examples An n-dimensional regular simplex can be represented as an integer polytope in \mathbb^, the convex hull of the integer points for which one coordinate is one and the rest are zero. Another important type of integral simplex, the orthoscheme, can be formed as the convex hull of integer points whose coordinates begin with some number of consecutive ones followed by zeros in all remaining coordinates. The n-dimensional unit cube in \mathbb^n has as its vertices all integer points whose coordinates are zero or one. A permutahedron has vertices whose coordi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Bernstein
Joseph Bernstein (sometimes spelled I. N. Bernshtein; ; ; born 18 April 1945) is a Soviet-born Israeli mathematician working at Tel Aviv University. He works in algebraic geometry, representation theory, and number theory. Biography Bernstein received his Ph.D. in 1972 under Israel Gelfand at Moscow State University. In 1981, he emigrated to the United States due to growing antisemitism in the Soviet Union. Bernstein was a professor at Harvard during 1983-1993. He was a visiting scholar at the Institute for Advanced Study in 1985-86 and again in 1997-98. In 1993, he moved to Israel to take a professorship at Tel Aviv University (emeritus since 2014). Awards and honors Bernstein received a gold medal at the 1962 International Mathematical Olympiad. He was elected to the Israel Academy of Sciences and Humanities in 2002 and was elected to the United States National Academy of Sciences in 2004. In 2004, Bernstein was awarded the Israel Prize for mathematics. In 1998, he was an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
