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The root separation is a fundamental parameter of the computational complexity of root-finding algorithms for polynomials. In fact, the root separation determines the precision of number representation that is needed for being certain of distinguishing distinct roots.
Separable polynomials are used to define separable extensions: A field extension K ⊂ L is a separable extension if and only if for every α in L which is algebraic over K, the minimal polynomial of α over K is a separable polynomial. Inseparable extensions (that is, extensions which are not separable) may occur only in positive characteristic.
Sturm's theorem. In mathematics, the Sturm sequence of a univariate polynomial p is a sequence of polynomials associated with p and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of p located in an interval in terms of the number of changes of signs of the values of ...
For polynomials with integer coefficients, the minimum distance sep(p) may be lower bounded in terms of the degree of the polynomial and the maximal absolute value of its coefficients; see Properties of polynomial roots § Root separation. This allows the analysis of worst-case complexity of algorithms based on Vincent's theorems. However ...
Finding the root of a linear polynomial (degree one) is easy and needs only one division: the general equation has solution For quadratic polynomials (degree two), the quadratic formula produces a solution, but its numerical evaluation may require some care for ensuring numerical stability. For degrees three and four, there are closed-form ...
Laguerre polynomials. In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer. Sometimes the name Laguerre polynomials is ...
Polynomial long division is an algorithm that implements the Euclidean division of polynomials, which starting from two polynomials A (the dividend) and B (the divisor) produces, if B is not zero, a quotient Q and a remainderR such that. and either R = 0 or the degree of R is lower than the degree of B. These conditions uniquely define Q and R ...
Finding roots of polynomials has been an important problem since the time of the ancient Greeks. Some polynomials, however, such as x 2 + 1 over R, the real numbers, have no roots. By constructing the splitting field for such a polynomial one can find the roots of the polynomial in the new field.