[PURRS-devel] Re: The specification of sqrfree()

[Richard B. Kreckel]

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.
> 
> ======================================================================
> 
> 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: 
> 
> 1) b is a non-zero constant; 
> 
> 2) a_j is a positive integer for j=1, ..., r;
> 
> 3) the degree of the polynomial p_j is strictly positive 
>    for j=1, ..., r; 

Err, otherwise they wouldn't be polynomials.

> 4) 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).
> 
> ======================================================================
> 
> 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.
-- 
Richard B. Kreckel
<Richard.Kreckel at Uni-Mainz.DE>
<http://wwwthep.physik.uni-mainz.de/~kreckel/>