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Re: equations

Andrzej Kozlowski wrote:
> Dear Fred and Daniel
> I clearly should read the messages I am responding to more carefully.
> Fred's point is in fact one that I have written about before on this
> list. In general, if you give a set of algebraic equations with symbolic
> coefficients, I do not think it even makes sense to speak of a "general
> solution". There will be a lot of "branching" going on and it is not at
> all clear to me that one can give a "formula" for the roots, in any
> conventional sense anyway, that will Work for all values of the
> parameters.  This is true even for a single equation: if the
> coefficients are numeric one can isolate the roots, but when they are
> not it seems to me that the concept of a general formula does not even
> make sense (for degrees greater than 5). In the case of systems of
> equations it gets even more complicated. Daniel knows a lot more about
> this than me, but I am not at all sure if the concept of "a general
> solutions" (that works for all values of the parameters) makes any
> mathematical sense. It also clearly does not make any practical sense,
> although questions about such formulas are probably the most common
> amongst the postings to this list.
> Andrzej
> [...]
> Andrzej Kozlowski
> Toyama International University

I will agree that often enough huge symbolic parametrized solution sets
are of little practical value. As for the issue of representing general
solution sets in this manner, yes, it can be done. Take the univariate
case. One can get a solution set in terms of parametrized Root[]
objects. These may be regarded as algebraic functions (just as solutions
in terms of parametrized radicals are algebraic functions). Of course
one no longer has the ability to isolate roots in the complex plane,
because our base field is now a closure of Q(parameters) rather than of
Q (where Q denotes the rationals).

Weirdly enough, you can almost do something along these lines by
assigning algebraically independent transcendentals as "values" for your
"indeterminate" parameters. I claim no expertise in this direction, and
am by no means certain that this would have any use. A more common way
to regard parametrized algebraic solutions is as functions that
specialize to values in the closure of Q when we give the parameters
algebraic values.

As for multivariate systems, something similar may be done that works
for generic parameter values. One way to see this is to apply the
methodology in Cox, Little, O'Shea, "Using Algebraic Geometry" chapter 2
section 4. They phrase it in terms of polynomials in C[variables] but in
fact the relevant parts work in Q(parameters)[variables] (all one
requires is a way to construct an eigensystem).

One thing that can be mildly unnerving though still not obviously
"wrong" is as follows. Suppose you have the single equation x^2==a^2.
The obvious way to solve would give solutions x->a and x->-a. But more
blunt approaches might give x->Sqrt[a^2] and x->-Sqrt[a^2] which give
the same pair of solutions for any given value of a but in some sense
are not as nicely behaved. When one works with parametrized Root[]
functions this sort of issue invariably arises. All the same, that does
not make an ensemble of solutions expressed in terms of such objects
incorrect, just difficult to use in practical applications.

I hope this clarifies matters, or at least makes the murk more colorful.

Daniel Lichtblau
Wolfram Research

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