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Triangular Decomposition
In computer algebra, a triangular decomposition of a polynomial system is a set of simpler polynomial systems such that a point is a solution of if and only if it is a solution of one of the systems . When the purpose is to describe the solution set of in the algebraic closure of its coefficient field, those simpler systems are regular chains. If the coefficients of the polynomial systems are real numbers, then the real solutions of can be obtained by a triangular decomposition into regular semi-algebraic systems. In both cases, each of these simpler systems has a triangular shape and remarkable properties, which justifies the terminology. History The Characteristic Set Method is the first factorization-free algorithm, which was proposed for decomposing an algebraic variety into equidimensional components. Moreover, the Author, Wen-Tsun Wu, realized an implementation of this method and reported experimental data in his 1987 pioneer article titled "A zero structure the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial System
A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations where the are polynomials in several variables, say , over some field . A ''solution'' of a polynomial system is a set of values for the s which belong to some algebraically closed field extension of , and make all equations true. When is the field of rational numbers, is generally assumed to be the field of complex numbers, because each solution belongs to a field extension of , which is isomorphic to a subfield of the complex numbers. This article is about the methods for solving, that is, finding all solutions or describing them. As these methods are designed for being implemented in a computer, emphasis is given on fields in which computation (including equality testing) is easy and efficient, that is the field of rational numbers and finite fields. Searching for solutions that belong to a specific set is a problem which is generally much more difficult, and is ou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Algebra
In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Although computer algebra could be considered a subfield of scientific computing, they are generally considered as distinct fields because scientific computing is usually based on numerical computation with approximate floating point numbers, while symbolic computation emphasizes ''exact'' computation with expressions containing variables that have no given value and are manipulated as symbols. Software applications that perform symbolic calculations are called ''computer algebra systems'', with the term ''system'' alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user programming language (usually different from the lang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomials
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' joi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. The word ''alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equations
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in French an ''équation'' is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. ''Solving'' an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation. There are two kinds of equations: identities and conditional equations. An identity is true for all values of the variables. A conditional equation is only true for particular values of the variables. An equation is written as two expressions, connected by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Semi-algebraic System
In computer algebra, a regular semi-algebraic system is a particular kind of triangular system of multivariate polynomials over a real closed field. Introduction Regular chains and triangular decompositions are fundamental and well-developed tools for describing the complex solutions of polynomial systems. The notion of a regular semi-algebraic system is an adaptation of the concept of a regular chain focusing on solutions of the real analogue: semi-algebraic systems. Any semi-algebraic system S can be decomposed into finitely many regular semi-algebraic systems S_1, \ldots, S_e such that a point (with real coordinates) is a solution of S if and only if it is a solution of one of the systems S_1, \ldots, S_e.Changbo Chen, James H. Davenport, John P. May, Marc Moreno-Maza, Bican Xia, Rong XiaoTriangular decomposition of semi-algebraic systems Proceedings of 2010 International Symposium on Symbolic and Algebraic Computation (ISSAC 2010), ACM Press, pp. 187–194, 2010. Formal d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Closed Field
In mathematics, a real closed field is a field ''F'' that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. Definitions A real closed field is a field ''F'' in which any of the following equivalent conditions is true: #''F'' is elementarily equivalent to the real numbers. In other words, it has the same first-order properties as the reals: any sentence in the first-order language of fields is true in ''F'' if and only if it is true in the reals. #There is a total order on ''F'' making it an ordered field such that, in this ordering, every positive element of ''F'' has a square root in ''F'' and any polynomial of odd degree with coefficients in ''F'' has at least one root in ''F''. #''F'' is a formally real field such that every polynomial of odd degree with coefficients in ''F'' has at least one root in ''F'', and for every element ''a'' of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generic Point
In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point. In classical algebraic geometry, a generic point of an affine or projective algebraic variety of dimension ''d'' is a point such that the field generated by its coordinates has transcendence degree ''d'' over the field generated by the coefficients of the equations of the variety. In scheme theory, the spectrum of an integral domain has a unique generic point, which is the zero ideal. As the closure of this point for the Zariski topology is the whole spectrum, the definition has been extended to general topology, where a generic point of a topological space ''X'' is a point whose closure is ''X''. Definition and motivation A generic point of the topological space ''X'' is a point ''P'' whose closure is all of ''X'', that is, a point that is dense in ''X'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers. Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have been introduced for generalizing some properties of polynomial rings. A closely related notion is tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maple (software)
Maple is a symbolic and numeric computing environment as well as a multi-paradigm programming language. It covers several areas of technical computing, such as symbolic mathematics, numerical analysis, data processing, visualization, and others. A toolbox, MapleSim, adds functionality for multidomain physical modeling and code generation. Maple's capacity for symbolic computing include those of a general-purpose computer algebra system. For instance, it can manipulate mathematical expressions and find symbolic solutions to certain problems, such as those arising from ordinary and partial differential equations. Maple is developed commercially by the Canadian software company Maplesoft. The name 'Maple' is a reference to the software's Canadian heritage. Overview Core functionality Users can enter mathematics in traditional mathematical notation. Custom user interfaces can also be created. There is support for numeric computations, to arbitrary precision, as well as symbol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom (computer Algebra System)
Axiom is a free, general-purpose computer algebra system. It consists of an interpreter environment, a compiler and a library, which defines a strongly typed hierarchy. History Two computer algebra systems named Scratchpad were developed by IBM. The first one was started in 1965 by James Griesmer at the request of Ralph Gomory, and written in Fortran. The development of this software was stopped before any public release. The second Scratchpad, originally named Scratchpad II, was developed from 1977 on, at Thomas J. Watson Research Center, under the direction of Richard Dimick Jenks. The design is principally due to Richard D. Jenks (IBM Research), James H. Davenport (University of Bath), Barry M. Trager (IBM Research), David Y.Y. Yun (Southern Methodist University) and Victor S. Miller (IBM Research). Early consultants on the project were David Barton (University of California, Berkeley) and James W. Thatcher (IBM Research). Implementation included Robert Sutor (IBM Rese ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
