Please let me know if my file NahayCubicResolvent.PDF makes any sense to you. Thanks.
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Yes, I tried it. That works! Thank you, Mariner!

Yes, I tried it. That works! Thank you, Mariner!

I've found a use, if one can call it that, for these so-called "useless"
closed-form solutions. Specifically, I am examining the structure of
differential resolvents of polynomials. In April 1999
I found a way to factor certain terms of my powersum formula for a
differential resolvent. I have wrestled with how to factor the remaining
terms.
By factor, in my case, I mean to give a matrix determinantal formula,
e.g. my formula for the k-th term of a differential resolvent is given
as det(A(k)) of a certain matrix A(k). Since 1999 I have tried to
factor the A(k)'s, i.e determine the entries of matrices B(k)
such that
A(k) = M*B(k) where M is the matrix which is known to factor out of
A(k) for SOME of the k (but I can't figure out how it
factors out for the others).
I started a paper this year, and am still writing it, for joint
differential resolvents of polynomials.
In all cases, I run up to computational difficulty. Namely,
enormous intermediate blowup problems. For example, just last night,
I calculated that my powersum formula for a 6-th order
joint differential resolvent of two polynomials would consist
of computing the determinant of roughly a 684 x 684 matrix with
polynomial entries.
These closed-form solutions of polynomials may give me insights,
even if I never use them for direct computations.
Thank you for informing me of Lauricella functions. I have never
heard of them.
I am just furious and frustrated at always hitting a mathematical
complexity wall no matter which approach I take in any math paper
I start to write.

Thank you, Robert Israel! Yes, the Mumford book "Tata Lectures on Theta II".
In the appendix is Umemura's formula - as much as you wish to
call it a formula. I've seen it nowhere else.
Many of the polynomials with which I deal ARE highly structured.
Or, if the polynomials themselves are not highly structured,
then the specific things I do WITH them have a lot of structure -
for example, computing differential resolvents of such polynomials.
Ideally, at the end of the day (i.e. at the end of my lifetime)
I would like some sort of "advanced category-theory type
description" of all this computational work I've done with polynomials
and differential equations.

You are half correct. Thank you for catching my error. I didn't care
about complex vs real solutions or initial conditions of my artificial
differential equation. I just want to test Maple's powers.
I'll worry about all that later after I get the main solution-generating
algorithm going.
(-5)^(2/5) = ((-5)^(1/5))^2 = (- (5^(1/5)))^2 = (5^(1/5))^2 = 5^(2/5)
One of the fifth roots of a negative number is a real negative number.
Though it certainly IS very sloppy math to ignore this fact in a
published math paper (or in a math class), often in practical applications,
one might come across this situation for which one DOES seek the
real negative solution.
(-1)^5 = -1 implies (in a universe without complex numbers!) -1 = (-1)^(1/5)
x^5+1=0 has 1 real solution x=-1 and 4 complex solutions {-1*h, -1*h^2,
-1*h^3, -1*h^4} where h = exp(2*pi*sqrt(-1)/5)

You are half correct. Thank you for catching my error. I didn't care
about complex vs real solutions or initial conditions of my artificial
differential equation. I just want to test Maple's powers.
I'll worry about all that later after I get the main solution-generating
algorithm going.
(-5)^(2/5) = ((-5)^(1/5))^2 = (- (5^(1/5)))^2 = (5^(1/5))^2 = 5^(2/5)
One of the fifth roots of a negative number is a real negative number.
Though it certainly IS very sloppy math to ignore this fact in a
published math paper (or in a math class), often in practical applications,
one might come across this situation for which one DOES seek the
real negative solution.
(-1)^5 = -1 implies (in a universe without complex numbers!) -1 = (-1)^(1/5)
x^5+1=0 has 1 real solution x=-1 and 4 complex solutions {-1*h, -1*h^2,
-1*h^3, -1*h^4} where h = exp(2*pi*sqrt(-1)/5)

Thank you, Robert Israel.
I meant to ask - where in the online help glossary is the syntax x:='x' explained?
I must make a habit of asking this, since this is what I am often really after
when I post a question on this board.
Obviously, I now know to look up 'eval' and 'unassign'.

How could "nops" be considered anywhere NEAR "perfectly suited"?
In C++ and Java, probably the two most common computer programming
languages on earth, "length" or "listlength" or something very similar to these
and obvious is used to return the length of a list or an array.