Concepts inQuantifier elimination for real algebra—the cubic case
Quantifier elimination
Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. One way of classifying formulas is by the amount of quantification. Formulae with less depth of quantifier alternation are thought of as being simpler, with the quantifier-free formulae as the simplest. A theory has quantifier elimination if for every formula, there exists another formula without quantifiers which is equivalent to it.
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Algebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it is an algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as an algebra, this multiplication must satisfy certain compatibility axioms with the given vector space structure, such as distributivity.
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Inequality (mathematics)
In mathematics, an inequality is a relation that holds between two values when they are different. The notation a ¿ b means that a is not equal to b. It does not say that one is greater than the other, or even that they can be compared in size. If the values in question are elements of an ordered set, such as the integers or the real numbers, they can be compared in size. The notation a < b means that a is less than b. The notation a > b means that a is greater than b.
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Degree of a polynomial
The degree of a polynomial is the highest degree of its terms, when the polynomial is expressed in canonical form (i.e. as a linear combination of monomials). The degree of a term is the sum of the exponents of the variables that appear in it. The word degree is now standard, but in some older books, the word order may be used instead. For example, the polynomial has three terms. (Notice, this polynomial can also be expressed as .
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Quadratic function
A quadratic function, in mathematics, is a polynomial function of the form The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis. The expression in the definition of a quadratic function is a polynomial of degree 2 or second order, or a 2nd degree polynomial, because the highest exponent of x is 2. If the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the equation are called the roots of the equation.
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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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Parametric equation
the butterfly curve. ]] In mathematics, parametric equation is a method of defining a relation using parameters. A simple kinematic example is when one uses a time parameter to determine the position, velocity, and other information about a body in motion. Abstractly, a parametric equation defines a relation as a set of equations. Therefore, it is somewhat more accurately defined as a parametric representation. It is part of regular parametric representation.
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Bracket
Brackets are tall punctuation marks used in matched pairs within text, to set apart or interject other text. Used unqualified, brackets refer to different types of brackets in different parts of the world and in different contexts.
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