Edwin Woollett wrote:
> On May 19, 2009, on thread "ratsimp bad, rectform good",
> Edwin Woollett wrote:
> " Maxima's solve function can return a solution in terms of z =
> (-1)^(1/n), which is any (in general complex) number z
> such that z^n = -1. "
>
> On May 20, 2009, on the same thread, Richard Fateman wrote:
> " Actually, I think the expression (-1)^(1/6) in Maxima is not "any"
> number z such that z^6=-1. It is a particular one.
> Just as 3^(1/2) is only the positive value. This is a notational
> issue, and Maxima is not, generally, able to deal with
> multiple values."
>
> Let's look at the 4th roots of ( - 1 ) as an example.
>
> solve ( a^4 + 1) returns a list of four answers, each
> with a factor ( - 1 )^(1/4). Since the latter factor
> can be four possible complex numbers, one might
> think that maxima is returning more than four answers.
>
You might think this, but of course you would be wrong, since a
polynomial of degree 4 has exactly 4 roots.
There is a fundamental challenge to the correctness of the result of
solve if you separate the 4 answers and treat them as though each one
can be further bifurcated into 4 separate results regardless of the
other 3 answers that come from solve.
> For some reason, solve offers the user the choice of
> replacing ( - 1 )^(1/4) (using ratsubst, say) throughout
> the list with a single choice and that will result in a list of
> four complex numbers.
>
That seems like an OK thing to do, to me.
> For any user choice, you get the same list of four
> complex numbers, just in a different (cyclic) order.
>
Exactly why it is OK to do it.
I am not sure I understand what you are doing in the rest of the message.
It is not adequate to determine that you have 4 distinct roots of a
quartic by taking each of the roots, inserting it into the quartic, and
simplifying to zero.
All that tells you is that you have 4 expressions, each of which might
be equivalent to some root, but maybe not the 4 distinct ones.
How could this be? It is well known that 1/sqrt(2) and 2/sqrt(2) are
equal, but not identical. If you posed them as the two roots of the
equation x^2-2=0, what would your exploration say?
Let me try to be clear: If you use a single expression that includes
(-1)^(1/4), you are asking for trouble. If solve produces an
expression that includes (-1)^(1/4), it probably also includes 3 more
expressions with that form. That's sort of OK, if you keep the 4
expressions together as a set of solutions.
Or you can pick a particular solution (in the complex plane) and then
instead of having the 4 solutions that circulate around, you have fixed
them all in place. But then you should remove (-1)^(1/4) in all the
roots in the solution set.