Cleanup and speedup in QQbar

[vdelecroix]

This is a task ticket for speed up and cleaning in sage.rings.qqbar

Prerequisites on number fields/polynomials/interval arithmetic

• #18332: implement is_integer and is_rational on number field elements
• #18334: implement unique_sign and unique_trunc on interval field elements
• #18337: implement real and imag on real intervals
• #17830: consider the natural order induced from RR for number fields with real embedding
• #19362, #28191: refine root
• #18356: implement composed_op from [BFSS]
• #19824: faster comparison for (real) embedded number fields
• #24503: enhanced LazyAlgebraic

Tasks in AA/QQbar:

• #18303: better comparisons
• #17886, #18242: faster exactification operations using power series and resultants (see also #18356)
• #15600: exactification is slow in do_polred
• #16222, #18122: enhanced minpoly
• #19954 and #19955: remove all descriptors except ANRoot, ANExtension and ANRational
• #20074: simplify method dispatcher for binary operators
• #12745: conversion issue QQbar -> AA
• #18106: introduce a polynomial descriptor, make sum and product be n-ary instead of binary and get rid of recursive calls
• #12715: number field embeddings should always have target in AA or QQbar
• #19356: QQbar.polynomial_root(): allow approximate root
• #21095: slowness in exactification
• #21838: do not make appear the imaginary (or real) part in QQbar when it is known to be zero
• #5574: Implement QQbar^QQ as action
• use firstly interval arithmetic for sign/floor/ceil/trunc/round instead of the _floor_ceil helper
• remove the following methods from ANDescr:
  • rational_value
  • is_field_element
  • is_exact
• (?) enhanced sage_input for ANExtension
• better powering (__pow__) using ANExtension and fix convention for powering in AA vs QQbar
• reimplement _real_refine_interval without these ugly dictionaries (see also #17895)
• use directly embedded number fields in AlgebraicGenerator as comparison there is faster (see #17830 and #19824). We might want that these embedded number field carries an interval of fixed precision with them (let say 64) to avoid reevaluation in most cases
• #26944: improve hashing to avoid exact computations
• more generally, track down code in qqbar.py that may lead to larger extensions than necessary if an exact computation is triggered (for example, see if the formulas in ANUnaryExpr.exactify() can be optimized).

Bugs

• #20209: QQbar -> RIF conversion fails
• #12745: Coercion problem with Algebraic Real Field
• #20288: representation and mutability of the description

[zimmerma]

about slowness of QQbar, see #19910.

[nbruin]

There's an issue on QQBar element powering reported in ​#18836 comment 9 that may have interaction with what you're doing here (perhaps you're solving the issue?). Perhaps it's worthwhile including that problem in this QQbar cleanup operation somewhere.