> Yes.  But my impression (maybe wrong) was that algebraic
> calculation can be done without looking at signs.

In this case, to determine the number of roots of UP(x, POLY INT),
we have to compute SturmHabichtSequence first, during this
process, we need the sign of leading coefficient sc1, it is used
afterwards like "sc1^(p-r-1)".

The example from the reference is this:
Q : UP(x, POLY INT) := x^8 - 2*x^6 + t*x^5 - x^2 - x + 1
SturmHabichtCoefficients(Q,1)  -- gives same result in the reference

Top five coefficients are:
1, 8, 32, - 72t^2  + 192, - 135t^4  - 272t^2  + 1152, ......

To computer the number of roots, we need to know the sign of these
coefficients, we have to evaluate t at some number to do so.


This usage of sign is pretty simple: typically the input will have a
integer leadingcoefficient instead of (t^2+1), and I use ToolsForSign
simply to avoid writing
if R has RetractableTo Fraction Integer then
retractIfCan(r)@Union(Fraction Integer, "failed")
if R has RetractableTo Integer then retractIfCan(r)@Union(Integer, "failed")
again.

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