1) Why does not sqrt(a^2) return abs(a)?
IIUC:
When operating on a symbolic expression x in Expression R,
the operator sqrt "represents an arbitrary (but always the same through the code)
solution y of the algebraic equation y^2=x".
Even when R = Integer, there are still 2 real roots y that satisfy the equation.
Similar semantics for the n-th root nthRoot:
"an arbitrary (but always the same through the code) solution of the algebraic
equation y^n=x".
Note that sqrt(x) may not be imagined to be a set with two elements
because the (pointwise) product of two sets would be a set including also
non-diagonal products of the elements.
(IE sqrt(4)={+2,-2} ==> sqrt(4)*sqrt(4) = {(+2)*(+2),...} = {+4, -4}).
So, FriCAS deliberately tries not to make a choice of roots
and consequently sqrt has a different semantics (on expressions)
as compared to all other CAS in my knowledge.
As for the (sometimes) weird behavior of abs in FriCAS
this should be related to the fact that using a smart abs one might construct
zero divisors,
(nonzero x1,x2 such that x1 * x2 = 0). This would violate the fact that
Expression Integer
is meant to be a differential field, which is essential for the indefinite
integration algorithms.
To recover more information I advice you to track the above keywords in the
past messages:
many of your (present and future) questions have been already asked many times
in the last 10 years.
riccardo
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