sfactors

Sfactors is a term used in computational algebra to denote the squarefree factors of a polynomial, or more generally the distinct components that appear when factoring with multiplicities stripped away. In this context, a squarefree factor is a factor that does not contain any repeated irreducible factor.

In polynomial factorization over a field, any nonzero polynomial f can be written as a product of irreducible polynomials with multiplicities: f = ∏ p_i^{e_i}, where the p_i are distinct irreducible polynomials and the e_i are positive integers. The sfactors are the irreducible factors p_i themselves, while the exponents e_i capture how many times each sfactor appears in f. A related notion is the squarefree part of f, which is ∏ p_i (the product of the distinct irreducible factors), obtained by removing the multiplicities.

Sfactors are central to the squarefree factorization process, an essential preprocessing step in many polynomial factoring algorithms, especially over finite fields. The squarefree factorization algorithm repeatedly computes gcds with derivatives and divides out repeated factors to separate f into its squarefree components. This yields a decomposition of f into a product of squarefree factors, often written as f = ∏ s_i^{e_i} with each s_i squarefree and the s_i pairwise coprime.

In practice, software systems such as computer algebra systems provide a function or operation called sfactor or squarefree_factorization. Users obtain either the list of squarefree factors and their multiplicities or a structured factorization that highlights the distinct sfactors and their exponents.

Applications of sfactors include simplifying polynomial division, speeding up complete factorization, and aiding symbolic integration and differential equation solving where repeated factors complicate algebraic manipulation.