Hi, On Thu, Aug 05, 2010 at 05:09:26PM -0600, Aaron S. Meurer wrote: > I didn't want to open an issue before asking here, because for all I know > there could be a very good reason for this. > > In [1]: Poly(-5*x**3, x).div(Poly(3*x, x)) > Out[1]: (Poly(0, x, domain='ZZ'), Poly(-5*x**3, x, domain='ZZ')) > > In [2]: Poly(-5*x**3, x, field=True).div(Poly(3*x, x, field=True)) > Out[2]: (Poly(-5/3*x**2, x, domain='QQ'), Poly(0, x, domain='QQ')) > > I understand that it doesn't want to create field coefficients, but I thought > that the definition of polynomial division and remainder was that a.div(b) > returns q and r such that a == b*q + r and **either r == 0 or deg(r) < > deg(b)**. [1] obviously does not satisfy this second condition. So what > definition does it use? > > This is causing problems in the Risch algorithm. It seems like I need to be > using field=True every time I create a Poly, or else I will get wrong > results. But then that gives things like gcd() problems. > > By the way, I know about pdiv. But I think regular div() should be returning > results with field coefficients if it must to satisfy the definition, unless > I am seriously missing something. >
The conditions are met only over a field or when b is monic. Over integers with non-monic b you arrive with the problem you showed. If you rely on those conditions (e.g. you are creating a sequence of polynomials with decreasing degrees) then you have to perform computations over a field (unless your polynomials are monic --- not a very interesting case in general). I guess you run into this problem somewhere around EEA (Extended Euclidean algorithm). In this case (and a few other, e.g. Strum sequences) user-level polynomial manipulation functions switch ground domain to a field automatically (on lower levels you get DomainError exception to avoid infinite loops). So, you will have to use field=True whenever the conditions must be met. We may also think about adding this automatic behaviour to div() (as it is done in gcdex(), half_gcdex(), sturm() ...). You said that there are problems with gcd() computations. What is specifically wrong here? Maybe it's an issue that GCD over a field is always monic in SymPy? > Aaron Meurer > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To post to this group, send email to sy...@googlegroups.com. > To unsubscribe from this group, send email to > sympy+unsubscr...@googlegroups.com. > For more options, visit this group at > http://groups.google.com/group/sympy?hl=en. > -- Mateusz
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