On 3/12/25 19:00, Waldek Hebisch wrote:
No, there is also 'rootOf'.
Ah, OK, that makes sense. And as you noted my code only makes sense for
radical expressions. But that is currently exactly what I need.
minPoly is in terms of lower oder kernels. For 'rootOf' minPoly is
not a binomial.
Yes, clear, that is to consider for the general case, but radicals are
currently sufficient for me.
The code fails in 'rootFactor' (which looks like a bug in
'rootFactor').
In fact, I was not sure whether I should actually use rootFactor, but it
was necessary for one case where I got from radicalRoots a tower with
kernels like sqrt(1+sqrt(7/5)), sqrt(7), sqrt(5). And for computing the
minimal polynomial application of rootFactor should not change the
minimal poly.
There is also question what to do with final 'factor'.
Well, of course, I wouldn't include factor in the code, but for my
demonstrated input radical expression it basically showed that it can be
expressed by unnested roots, since the minpoly factors.
In fact, for me the game is also like rsimp. Some radicals that come out
of my computation cannot be simplified by rsimp, so I look for other
means to produce radicals with lower depth.
If all intermediate root are independent, and generic case we will
get irreducible polynomial. However, in non-generic case, that is
when 'q' generates proper subfield of the field generated by roots
(I think that such cases are extremally rare, but possible) I am not
sure what will come out from Groebner basis computation.
Well... the only thing that would trouble me would be if there would be
no univariate polynomial in x over ZZ in the Groebner basis. But why
would that be? What might happen is that the polynomial is not minimal.
But then factorization would help and checking which factor evaluates to
0. (Oh, oh... plugging in the initial radical into a factor might not
simplify to 0 in our AN, although the expression represents 0. It would
need a function to produce a canonical representation for AN.)
Do you see that your argument can ever apply to radical numbers? If I
apply rootFactor, only primes survive under the radical.
Ralf
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