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Tarski's theorem

Tarski's theorem says that the first-order theory of reals with , , , and allows quantifier elimination. Algorithmic quantifier elimination implies decidability assuming that the truth values of sentences involving only constants can be computed. However, the converse is not true. For example, the first-order theory of reals with , , and is decidable, but does not allow quantifier elimination.Tarski's theorem means that the solution set of a quantified system of real algebraic equations and inequations is a semialgebraic set (Tarski 1951, Strzebonski 2000).Although Tarski proved that quantifier elimination was possible, his method was totally impractical (Davenport and Heintz 1988). A much more efficient procedure for implementing quantifier elimination is called cylindrical algebraic decomposition. It was developed by Collins (1975) and is implemented as CylindricalDecomposition[ineqs, vars]...

Primitive part

The primitive part of a polynomial is , where is the content.For a general univariate polynomial , the Wolfram Language function FactorTermsList[poly, x] returns a list of three elements, the first being the integer content , the second being the polynomial content, i.e., a primitive (with respect to all variables) polynomial that does not depend on and which divides all coefficients of , and the third element being the primitive part of . The original polynomial is then the product of these three parts. For example, FactorTermsList[9E x^3 + 3E, x] returns 3, E, 1 + 3x^2.

Conjunctive normal form

A statement is in conjunctive normal form if it is a conjunction (sequence of ANDs) consisting of one or more conjuncts, each of which is a disjunction (OR) of one or more literals (i.e., statement letters and negations of statement letters; Mendelson 1997, p. 30). Examples of conjunctive normal forms include (1)(2)(3)(4)where denotes OR, denotes AND, and denotes NOT (Mendelson 1997, p. 30).Every statement in logic consisting of a combination of multiple , , and s can be written in conjunctive normal form.An expression can be put in conjunctive normal form using the WolframLanguage using the following code: ConjunctiveNormalForm[f_] := Not[LogicalExpand[Not[f]]] //. { Not[a_Or] :> And @@ (Not /@ List @@ a), Not[a_And] :> Or @@ (Not /@ List @@ a) }

Cylindrical algebraic decomposition

Define a cell in as an open interval or a point. A cell in then has one of two forms,(1)or(2)where , is a cell in , and are either (1) continuous functions on such that for some polynomials and , and , or (2) , and for all .A cylindrical algebraic decomposition of is a representation of as a finite union of disjoint cells. Let be finite set of polynomials in variables. A cylindrical algebraic decomposition of is said to be -invariant if each of the polynomials from has a constant sign on each cell of the decomposition.The cylindrical algebraic decomposition (CAD) algorithm, given a finite set of polynomials in variables, computes an -invariant cylindrical algebraic decomposition of . Given a logical combination of polynomial equations and inequalities in real unknowns, one can use the CAD algorithm to find a cylindrical algebraic decomposition of its solution set. For example, the decomposition of(3)is given by(4)The command CylindricalDecomposition[ineqs,..

Polynomial discriminant

A polynomial discriminant is the product of the squares of the differences of the polynomial roots . The discriminant of a polynomial is defined only up to constant factor, and several slightly different normalizations can be used. For a polynomial(1)of degree , the most common definition of the discriminant is(2)which gives a homogenous polynomial of degree in the coefficients of .The discriminant of a polynomial is given in terms of a resultant as(3)where is the derivative of and is the degree of . For fields of infinite characteristic, so the formula reduces to(4)The discriminant of a univariate polynomial is implemented in the Wolfram Language as Discriminant[p, x].The discriminant of the quadratic equation(5)is given by(6)The discriminant of the cubic equation(7)is given by(8)The discriminant of a quartic equation(9)is(10)(Schroeppel 1972)...

Content

There are several meanings of the word content in mathematics.The content of a polytope or other -dimensional object is its generalized volume (i.e., its "hypervolume"). Just as a three-dimensional object has volume, surface area, and generalized diameter, an -dimensional object has "measures" of order 1, 2, ..., . The content of a region can be computed in the Wolfram Language using RegionMeasure[reg].The content of an integer polynomial , denoted , is the largest integer such that also has integer coefficients. Gauss's lemma for contents states that if and are two polynomials with integer coefficients, then (Séroul 2000, p. 287).For a general univariate polynomial , the Wolfram Language command FactorTermsList[poly, x] returns a list of three elements, the first being the integer content , the second being the polynomial content, i.e., a primitive (with respect to all variables) polynomial that..

Resultant

Given a polynomial(1)of degree with roots , , ..., and a polynomial(2)of degree with roots , , ..., , the resultant , also denoted and also called the eliminant, is defined by(3)(Trott 2006, p. 26).Amazingly, the resultant is also given by the determinantof the corresponding Sylvester matrix.Kronecker gave a series of lectures on resultants during the summer of 1885 (O'Connor and Robertson 2005).An important application of the resultant is the elimination of one variable from a system of two polynomial equations (Trott 2006, p. 26).The resultant of two polynomials can be computed using the Wolfram Language function Resultant[poly1, poly2, var]. This command accepts the following methods: Automatic, SylvesterMatrix, BezoutMatrix, Subresultants, and Modular, where the optimal choice depends dramatically on the concrete polynomial pair under consideration and typically requires some experimentation. For high-order..

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