gh-40113: Support implicitly defined species inside compositions

[mwhansen]

Description

This makes define_implicitly work when an implicitly defined species occurs inside a composition (plethysm), resolving #40113. The motivating example is the implicit compositional inverse of E_{\geq 1}:

sage: L.<X> = LazyCombinatorialSpecies(QQ)
sage: E1 = L.Sets().restrict(1)
sage: Omega = L.undefined(1)
sage: L.define_implicitly([Omega], [E1(Omega) - X])
sage: Omega[:6] == E1.revert()[:6]
True

The obstruction and its resolution

The difficulty raised in #40113 was computing, for instance, E(f_0 X)_3 with f_0 an undetermined coefficient, since the result depends polynomially rather than linearly on f_0.

The resolution is to recognise which kind of \lambda-ring element an undetermined coefficient is: the f_{n,M} in \Omega_n = \sum_M f_{n,M} M are molecular multiplicities — eventually non-negative integers — hence constants, so the power-sum (Adams) operations fix them, p_k[f_{n,M}] = f_{n,M}. Only the genuine weight variables (your q) are monomials, with p_k[q] = q^k. With this convention,

E(f_0 X)_3 = f_0*E_3 + (f_0^2 - f_0)*X*E_2 + (1/6*f_0^3 - 1/2*f_0^2 + 1/3*f_0)*X^3

which specialises at f_0 = 3 to 3*E_3 + 6*X*E_2 + X^3, as it must.

The polynomial dependence does not break the solver: for an equation F(\Omega) = 0 of positive valuation, \Omega_n enters the degree-n comparison only through the degree-one (identity) part of the outer factors, hence linearly; any product or genuine plethysm of \Omega_n needs total degree > n. So the higher powers f_0^2, f_0^3 only ever multiply already determined lower-degree coefficients, and the existing "keep the linear equations" strategy still determines everything.

Implementation

• p_k (the stretch in PolynomialSpecies._exponential) now fixes undetermined coefficients and stretches only base-ring monomials, via the new CoefficientRing._power_sum_plethysm.
• The plethysm in _compose_with_weighted_singletons is computed over the coefficient ring extended by the solver's variables.
• The solver is taught to see the new variables: the composite coefficient is built over its true parent (so back-substitution recognises it); Stream_function.input_streams now also detects streams captured inside a list/tuple in the closure, so the composition's argument streams (in particular derived ones such as q*\Omega) are substituted into; and the per-call argument snapshots are re-read rather than memoised while still undetermined.

Tested cases

Correctness is checked by cross-validating independent code paths — against revert(), against an explicit define() (which feeds determined coefficients into the composition), self-consistency, and counts:

Equation	Cross-check
E_{>=1}(\Omega) = X	matches revert() (implicit Lagrange inversion)
A = X*E(A)	rooted trees; explicit define == implicit; count n^{n-1}
A = X + E_2(A)	valuation-2 outer
A = X + E_2(B), B = X + E_2(A)	coupled system; A = B
A = Y + q*E_2(A)	weight on the composition term (over QQ[q])
B = Y + E_2(q*B)	weight inside the argument
D = Y*E(q*D)	weighted, full E, valuation 1
C = Y + (E(q*C) - 1)	over QQ(q); needs the fraction field

One boundary

When the linear part scales the unknown by a weight (e.g. E_1(q\Omega) = q\Omega occurs in the equation), then (1 - q) f_{n,M} = (\text{lower}), so the solution lies in Frac(k) and the problem has to be posed over, say, QQ(q) rather than QQ[q]. This is the same reason revert() currently requires a field base ring.

Dependencies

• gh-40438: Fast inclusion of univariate species into multivariate species #42441 (check performance of the inclusion of univariate species into multivariate species. #40438), on which the composition code here is based.

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