oldk1331 wrote:
>
> > IIUC Sturm sequence is essentially
> > consists of two parts: purely algebraic computation and
> > determination of signs. So it make sense to expose
> > computational part also for unordered rings.
>
> Yes, that's what I'm trying to do.
>
> > However,
> > machinations with sign of leading coefficient look strange.
>
> Why is that? Doesn't the leading coefficient determine sign
> at infinity?
Yes. But my impression (maybe wrong) was that algebraic
calculation can be done without looking at signs.
> > ToolsForSign realy is rather weak and I would
> > avoid it if possible.
>
> > Also, in analogy with other domains we could pass sign
> > determining function as a parameter.
>
> Why not put sign determining function into ToolsForSign to
> make it not weak?
If you mean passing sign determining function to ToolsForSign
that makes no sense: if you have sign determining function
you just call it without need of extra intermediate stage.
If you mean implementing better ToolsForSign, then take
into account that ToolsForSign wants to be generic, while
strong methods tend to be specialized. Also, strong methods
tend to be slower, sometimes user may prefer weake but
faster method.
However, after some thought I think that sign determining
function as parameter helps only a little. Namely, main
reason for lack of linear order is because we want to
solve parametric problems. In such case results of
comparisons depend on parameters and say root counting
must produce result depending on parameters -- no
sign determining function producing boolean results
will work.
So now I think that if calculations are really independent
of sign that we should return result of calculations
without any machinations with signs. If calculations
depend on sign, meaning that no simple postprocessing
can produce current result from computation independent
of signs, then we can expose dependence: add signs
as extra parameters.
--
Waldek Hebisch
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