sfactor

Sfactor is a term used in polynomial algebra to denote the square-free factorization of a polynomial. Square-free factorization decomposes a polynomial into factors that are square-free (no factor is repeated inside any factor) and are pairwise coprime, with the multiplicity of each irreducible factor recorded separately. The primary purpose is to expose the multiplicity structure of the polynomial’s roots and to simplify subsequent factoring over a given field.

In practice, computing sfactor typically involves the polynomial’s derivative and greatest common divisor operations. By comparing a polynomial with its derivative, factors that correspond to repeated roots are identified, and successive gcd-based steps separate different multiplicities into a canonical square-free form. The resulting representation allows subsequent full factorization to be performed more efficiently, especially when factoring over fields such as the rationals or finite fields.

Sfactor is widely implemented in computer algebra systems as a callable routine or command. Examples include dedicated functions in Maple, Magma, and Sage, and the SymPy library in Python provides a square-free factorization routine as well. The output expresses the original polynomial as a product of square-free polynomials, with multiplicities indicated so that the original factorization can be recovered.

See also: Factorization, Square-free polynomial, Polynomial derivative, Greatest common divisor (gcd). This concept is foundational in factorization algorithms and has applications in algebra, number theory, and algebraic geometry.