input polynomial and its derivative. More precisely, if is the time needed to compute the GCD of two polynomials of degree and the quotient of these polynomial by the GCD, then is an upper bound for the time needed to compute the square free decomposition. There are also known algorithms for the computation of the square-free decomposition of multivariate polynomials, that proceed generally by considering a multivariate polynomial as a univariate polynomial with polynomial coefficients, and applying recursively a univariate algorithm. Yun's algorithm This section describes Yun's algorithm for the square-free decomposition of univariate polynomials over a field of characteristic 0. It proceeds by a succession of GCD computations and exact divisions. The input is thus a non-zero polynomial f, and the first step of the algorithm consists of computing the GCD a0 of f and its formal derivative f'''. If is the desired factorization, we have thus and If we set , and , we get that and Iterating this process until we find all the This is formalized into an algorithm as follows: The degree of and is one less than the degree of As is the product of the the sum of the degrees of the is the degree of As the complexity of GCD computations and divisions increase more than linearly with the degree, it follows that the total running time of the "repeat" loop is less than the running time of the first line of the algorithm, and that the total running time of Yun's algorithm is upper bounded by twice the time needed to compute the GCD of and and the quotient of and by their GCD. Square root In general, a polynomial has no square root. More precisely, most polynomials cannot be written as the square of another polynomial. A polynomial has a square root if and only if all exponents of the square-free decomposition are even. In this case, a square root is obtained by dividing these exponents by 2. Thus the problem of deciding if a polynomial has a square root, and of computing it if it exists, is a special case of square-free factorization. Notes References Polynomials Computer algebra Polynomials factorization algorithms