Concepts inTriangular decomposition of semi-algebraic systems
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 coefficient of are real numbers, then the real solutions of can be obtained by a triangular decomposition into regular semi-algebraic systems.
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Semialgebraic set
In mathematics, a semialgebraic set is a subset S of R for some real closed field R (for example R could be the field of real numbers) defined by a finite sequence of polynomial equations (of the form) and inequalities (of the form), or any finite union of such sets. A semialgebraic function is a function with semialgebraic graph. Such sets and functions are mainly studied in real algebraic geometry which is the appropriate framework for algebraic geometry over the real numbers.
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Regular chain
In computer algebra, a regular chain is a particular kind of triangular set in a multivariate polynomial ring over a field. It enhances the notion of characteristic set.
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Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, x − x/4 + 7 is a polynomial, but x − 4/x + 7x is not, because its second term involves division by the variable x (4/x), and also because its third term contains an exponent that is not an integer (3/2).
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Complex number
A complex number is a number which can be put in the form a + bi, where a and b are real numbers and i is called the imaginary unit, where . In this expression, a is called the real part and b the imaginary part of the complex number. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number can be identified with the point (a, b).
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Exponential growth
Exponential growth (including exponential decay when the growth rate is negative) occurs when the growth rate of the value of a mathematical function is proportional to the function's current value. In the case of a discrete domain of definition with equal intervals it is also called geometric growth or geometric decay (the function values form a geometric progression).
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Real number
In mathematics, a real number is a value that represents a quantity along a continuous line. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers such as √2 and π. Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced.
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Algorithm
In mathematics and computer science, an algorithm Listen/ˈælɡərɪðəm/ (originating from al-Khwārizmī, the famous mathematician Muḥammad ibn Mūsā al-Khwārizmī) is a step-by-step procedure for calculations. Algorithms are used for calculation, data processing, and automated reasoning. More precisely, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function.
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