On 11/21/23 10:02, Waldek Hebisch wrote:
On Tue, Nov 21, 2023 at 08:24:39AM +0800, Qian Yun wrote:
On 11/20/23 20:48, Waldek Hebisch wrote:
Extra remark: it should be possible to get much of effect
of RealClosure by considering pairs of AlgebraicNumber
and floating point approximation. Namely, we can compute
minimal polynomial of algebraic number and moderately
accurate floating point approximation uniquely determines
corresponding exact root. In case when we get reducible
polynomials we could use floating point approximations to
find right factor.
Does this mean the right thing to do is to have two kinds
of algebraic numbers? For first kind, its position is not
fixed (like current status of 'rootOf), for the other kind,
the position is fixed (you described above).
Because some computation doesn't care which root is which
(for example sum of all roots or a polynomial), while other
computation cares.
We may have more kinds. Current AlgebraicNumber may run into
trouble if you create depenednt roots. Ralf posted code where
roots have consistent but unkown "order". That is it avoid
troubles due to dependent roots. There is correspondence
with roots in a finite field, and relation of roots in finite
field and roots in complex numbers is hard to compute.
Roots in finite field have rather simple relation to
roots in completition of p-adic numbers, to we can get
several competing variants of algebraic numbers.
If so, I want to confirm that is 'nthRoot already a kind of
algebraic number?
'nthRoot constructs radicals, a subset of algebraic numbers.
And clearly code in different places treats 'nthRoot by
using its real valued branch, different than 'rootOf.
- Qian
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