parents with multiple implementations (or backend)

[videlec]

In sage several parents propose several implementations of the same parent such as FiniteField, MatrixSpace, PolynomialRing, Polyhedra

sage: FiniteField(79, impl='modn')
Finite Field of size 79
sage: FiniteField(79, impl='givaro')
Finite Field of size 79
sage: MatrixSpace(QQ, 3, 3, implementation='generic')
Full MatrixSpace of 3 by 3 dense matrices over Rational Field (using Matrix_generic_dense)
sage: MatrixSpace(QQ, 3, 3, implementation='gap')
Full MatrixSpace of 3 by 3 dense matrices over Rational Field (using Matrix_gap)
sage: PolynomialRing(ZZ, 'x', implementation='FLINT')
Univariate Polynomial Ring in x over Integer Ring
sage: PolynomialRing(ZZ, 'x', implementation='NTL')
Univariate Polynomial Ring in x over Integer Ring (using NTL)
sage: Polyhedra(QQ, 3, backend='ppl')
Polyhedra in QQ^3
sage: Polyhedra(QQ, 3, backend='cdd')
Polyhedra in QQ^3

This ticket intends to extend, standardize, optimize and test parents with multiple implementations. Especially regarding

• argument names (implementation vs impl vs backend)
• string representation
• reconstruction (should be part of TestSuite)
• conversion of elements between implementations
• handling of default constructor

Related tickets: #31668

CC: @kliem

Component: misc

Issue created by migration from https://trac.sagemath.org/ticket/31852