Square-free Polynomial

In mathematics, a square-free polynomial is a polynomial defined over a field or, more generally, a unique factorization domain that is not a multiple of the square of a non unit factor. In the important case of univariate polynomials over a field k, this means that, is square-free if and only if for every polynomial of positive degree. In applications in physics and engineering, a square-free polynomial is commonly called a polynomial with no repeated roots.

In the context of field theory, square-free polynomials are also called separable polynomials.

A square-free decomposition or square-free factorization of a polynomial is a factorization into powers of square-free factors

where the a_k that are not equal to 1 are pairwise coprime square-free polynomials. Every non-zero polynomial with coefficients in a field admits a square-free factorization, which is unique up to the multiplication of the factors by non zero constants. The square-free factorization is much easier to compute than the complete factorization into irreducible factors, and is thus often preferred when the complete factorization is not really needed, like for the partial fraction decomposition and the symbolic integration of rational fractions. Square-free factorization is the first step of the polynomial factorization algorithms which are implemented in computer algebra systems. Therefore, the algorithm of square-free factorization is basic in computer algebra.

In the case of univariate polynomials over a field, any multiple factor of a polynomial introduces a nontrivial common factor of f and its formal derivative f ′, so a sufficient condition for f to be square-free is that the greatest common divisor of f and f ′ is 1. Over a perfect field, all irreducible polynomials are separable, so that condition is also necessary. If the polynomial is not square free, the product of the in the above square free decomposition may be obtained as the quotient of by its GCD with its derivative. Further GCD computations and exact divisions allow to compute the square-free factorization (see square-free factorization over a finite field). In characteristic zero, a better algorithm is known, Yun's algorithm, which is described below. Its computational complexity is, at most, twice that of the GCD computation of the 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.