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Re: equations
*To*: mathgroup at smc.vnet.net
*Subject*: [mg32011] Re: equations
*From*: Andrzej Kozlowski <andrzej at tuins.ac.jp>
*Date*: Tue, 18 Dec 2001 02:34:20 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
The point I was trying to make was somewhat different, I think. Of
course any polynomial equation with symbolic coefficients can be
regarded as an equation over some function field in characteristic zero,
and since the function field has an algebraic closure we know that there
will be n-roots, where n is the degree of the equation. We can also
express the roots in terms of parametrized Root objects, in the manner
indicated. But these Root objects are of course "multivalued functions"
with branch points depending on the parameters. I think what people mean
when they want a "formula" is an expression that will always give ,in
some algorithmic way, a root of the original equation for any value of
the parameters. However, for each choice of the parameters it will be
necessary to decide which branches to take, and I this can't be
specified until the parameters are assigned values. So while we can say
that roots exist, and we can refer to them as "the roots" I don't think
we can write a computer program that will automatically choose the
correct branches to give us a solution for very choice of parameters.
(Or can we?) The point of my comments is that a formula in this context
must mean "an algorithm which can be implemented on a computer", and
since the issue of choosing the correct branch is basically topological
and not algebraic I do not think such an algorithm can exist at present.
However I may be wrong here, so I would like to be corrected on this
point.
(That would then raise another issue, why the solutions returned by
eliminate do not seem to work for certain values of the parameters).
Andrzej Kozlowski
Toyama International University
JAPAN
http://platon.c.u-tokyo.ac.jp/andrzej/
On Tuesday, December 18, 2001, at 12:15 PM, Daniel Lichtblau wrote:
> 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
>>
>> [...]
>
> 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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