Re: More on radcan()

Le 21/08/2026 à 18:41, Richard Fateman a écrit :
> Ultimately, the problem is that there are insufficient mechanisms in 
> Maxima (or competitive systems) for
> dealing with 'multi-valued' objects.  sqrt(x^2) is not x, abs(x),  or 
> -x.  It might be   z = root_of(x=y^2, y).
> But then one must extend the whole system to work on, say,   if z>0 
> then .... ,  or exp(z^3), ....
> Even simple 'solutions'   like enumerated sets  {x,-x} would require 
> much more complication.
> Ambitious graduate students are invited to investigate.


Unfortunately this is an unfixable problem. Quoting J. Dieudonné's 
Elements of analysis 1, page 192

"As we have announced in Chapter 1 the reader will find no mention in 
this chapter of the so-called "multiple-valued" or "multiform" 
functions. It is of course a great nuisance that one cannot define in 
the field C a genuine continuous function √z which would satisfy the 
relation (√z)^2=z; but the solution to this difficulty is certainly not 
to be sought in a deliberate perversion of the general concept of 
mapping, by which one suddenly decrees that there is after all such a 
"function", which, however the uncommon feature that for each z≠0 it has 
two distinct "values". The penalty for this indecent and silly behavior 
is immediate: it is impossible to perform even the simplest algebraic 
operations with any reasonable confidence; for example the relation 
2√z=√z+√z is certainly not true, for if we follow the "definition" of √z 
we are compelled to attribute for z≠0 two distinct values to the 
left-hand side , and three distinct values to the right-and side! 
Fortunately there is a solution to the difficulty, which has nothing to 
do with such nonsense; it was discovered more than 100 years ago by 
Riemann, and consists in restoring the uniqueness of value of √z by 
"doubling" so to speak the domain of the variable z...."

Which means that √z is not an element of C but a point in the "Riemann 
surface" of √z which consists of two copies of C, except at 0 and 
infinity (branch points), the choice of sheet corresponding to the 
choice of sign. A simple way to visualize this is to consider the curve 
x-y^2=0 in the complex plane One would be tempted to take x as a 
coordinate on the curve, but then you have two points above x, (x,√x) 
and (x,-√x), and moreover the situation is unmanageable at x=0. In fact 
y is the good coordinate on the curve, describing it continuously and 
analytically even at (0,0). All this extends to any curves P(x,y)=0 (P 
polynomial) as explained in Springer G. "Introduction to Riemann surfaces".


It is highly dubious that such considerations may be implemented in a CAS.


-- 
Michel Talon

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