From f7d46a20e8946e9a7a81208339596434dbcb745c Mon Sep 17 00:00:00 2001 From: Richard Kreckel Date: Sat, 19 Jan 2002 19:12:24 +0000 Subject: [PATCH] * Docu: elaborate definition of sqrfree() (by Roberto Bagnara) --- ginac/normal.cpp | 34 ++++++++++++++++++++++++++++++++-- 1 file changed, 32 insertions(+), 2 deletions(-) diff --git a/ginac/normal.cpp b/ginac/normal.cpp index 4b490ed0..d88f3dde 100644 --- a/ginac/normal.cpp +++ b/ginac/normal.cpp @@ -1741,11 +1741,41 @@ static exvector sqrfree_yun(const ex &a, const symbol &x) return res; } -/** Compute square-free factorization of multivariate polynomial in Q[X]. +/** Compute a square-free factorization of a multivariate polynomial in Q[X]. * * @param a multivariate polynomial over Q[X] * @param x lst of variables to factor in, may be left empty for autodetection - * @return polynomial a in square-free factored form. */ + * @return a square-free factorization of \p a. + * + * \note + * A polynomial \f$p(X) \in C[X]\f$ is said square-free + * if, whenever any two polynomials \f$q(X)\f$ and \f$r(X)\f$ + * are such that + * \f[ + * p(X) = q(X)^2 r(X), + * \f] + * we have \f$q(X) \in C\f$. + * This means that \f$p(X)\f$ has no repeated factors, apart + * eventually from constants. + * Given a polynomial \f$p(X) \in C[X]\f$, we say that the + * decomposition + * \f[ + * p(X) = b \cdot p_1(X)^{a_1} \cdot p_2(X)^{a_2} \cdots p_r(X)^{a_r} + * \f] + * is a square-free factorization of \f$p(X)\f$ if the + * following conditions hold: + * -# \f$b \in C\f$ and \f$b \neq 0\f$; + * -# \f$a_i\f$ is a positive integer for \f$i = 1, \ldots, r\f$; + * -# the degree of the polynomial \f$p_i\f$ is strictly positive + * for \f$i = 1, \ldots, r\f$; + * -# the polynomial \f$\Pi_{i=1}^r p_i(X)\f$ is square-free. + * + * Square-free factorizations need not be unique. For example, if + * \f$a_i\f$ is even, we could change the polynomial \f$p_i(X)\f$ + * into \f$-p_i(X)\f$. + * Observe also that the factors \f$p_i(X)\f$ need not be irreducible + * polynomials. + */ ex sqrfree(const ex &a, const lst &l) { if (is_a(a) || // algorithm does not trap a==0 -- 2.34.3