On 11/21/23 09:52, Waldek Hebisch wrote:
On Tue, Nov 21, 2023 at 08:33:50AM +0800, Qian Yun wrote:
On 11/20/23 10:08, Waldek Hebisch wrote:
There is question in which cases your apprach works. Examples
in mapleok.input which work have minimal polynomial for residues
which is variation of x^4 + a^4 where a is a rational constant
or x^4 + 2*x^2 + 2 or x^4 + x^2/64 + 1/8192. x^4 + a^4 factors
in extention by square root of 2:
In theory this approach should work with higher degree of polynomial.
After all as you said, it is about finding a algebraic extension to
factor the polynomial into simpler ones.
Yes. However, explict formulas for root are possible only in
special cases. And even when possible are likely to use
complex expressions for real roots. AFAICS biquadratic case,
that is degree 4 polynomial containing only even powers
should work. I am not sure if out solver is smart enough,
but symmetric degree 4 polynomial should work. Basically
polynomials which one can solve repeatedly solving quadratic
equations. Once solution involves general formula for degree 3
or degree 4 polynomial we will get complex expressions.
How do you suggest to proceed? Write a special case for biquadratic?
Let's see if I can find out such an example.
Groebner produces sensible thing:
(50) -> groebner([PP::POLY(INT), QQ::POLY(INT), 8*u^3 - 6*u + 1])
2 2 3
(50) [v + u - 1, 8 u - 6 u + 1]
and solution of this could be passed to Rioboo. But TransSolvePackage
can not usefully handle it.
How can this go further? Return rootSum in Rioboo?
Possibly. Or do what we do now, that is use symbolic roots.
For degree 3 after you add symbolic root polynomial splits
into linear term and quadratic (or into linear terms).
rootSum is the future, but as long as limit can not handle
rootSum we have to use symbolic roots.
If current limit can handle result from integrate, then the limit
can expand the rootSum just like integrate does. Not an improvement,
but also not a step back, right?
- Qian
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