Hi Roberto,
On Wed, 16 Jan 2002, Roberto Bagnara wrote:
the definition of square-free polynomial we have in mind is semantic, not syntactic. In other words, we believe it is unnecessary to expand() the lhs and rhs of
p(X) = q(X)^2*r(X)
since what is meant is that the lhs and the rhs are the same function.
Sure, but I would still recommend considering inclusion of expansion and the use of `==' instead of `=' since this then puts it into the syntatic domain of of C++ and GiNaC. Just to avoid potential confusion...
We have also thought about a definition of square-free decomposition that could safely accommodate both the univariate and the multivariate case. Here is a summary of what we would like to add to GiNaC's documentation (both the tutorial and the developer's reference) just before the introduction of the sqrfree() function.
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Definition 1
A polynomial p(X) in C[X] is said <EM>square-free</EM> if, whenever any two polynomials q(X) and r(X) in C[X] are such that p(X) = q(X)^2*r(X), q(X) is constant.
Note: we mean that p(X) has no repeated factors, apart eventually from constants.
Definition 2
Given a polynomial p(X) in C[X], we say that the decomposition
p(X) = b * p_1(X)^a_1 * p_2(X)^a_2 * ... * p_r(X)^a_r
is a <EM>square-free decomposition</EM> of p(X) if the following conditions hold:
b is a non-zero constant;
a_j is a positive integer for j=1, ..., r;
the degree of the polynomial p_j is strictly positive for j=1, ..., r;
Err, otherwise they wouldn't be polynomials.
- the polynomial p_1(X) * p_2(X) * ... * p_r(X) is square-free.
Note: this need not be unique. For example, if a_j is even, we could change the polynomial p_j(X) into (-p_j(X)). We do not ask that the factors p_j(X) are irreducible polynomials.
Specification of sqrfree()
Given a polynomial p(X) in C[X], the function sqrfree() returns a square-free decomposition of p(X).
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If you agree, we would adapt this for the two different contexts, translate it into Doxygen and LaTeX and provide a patch against GiNaC 1.0.3.
Please do go ahead! We are only seemingly disagreeing because I have the implementor's viewpoint and you the mathematician's. I see it as a mere finding of all GCDs of the original polynomial p(X) and all its derivatives. Hence my slight confusion.
Best wishes -richy.