English
Under an aeval map, polyCharpolyAux transforms compatibly with the aeval operator on coefficients.
Русский
При отображении через aeval polyCharpolyAux сводится к operadores aeval на коэффициентах.
LaTeX
$$ (polyCharpolyAux φ b b_m).map (MvPolynomial.aeval x).toRingHom = LinearMap.charpoly ( (tensorProduct R A M M).comp (baseChange A φ) (...))$$
Lean4
theorem polyCharpolyAux_map_aeval (A : Type*) [CommRing A] [Algebra R A] [Module.Finite A (A ⊗[R] M)]
[Module.Free A (A ⊗[R] M)] (x : ι → A) :
(polyCharpolyAux φ b bₘ).map (MvPolynomial.aeval x).toRingHom =
LinearMap.charpoly
((tensorProduct R A M M).comp (baseChange A φ) ((basis A b).repr.symm (Finsupp.equivFunOnFinite.symm x))) :=
by
rw [← polyCharpolyAux_map_eval (tensorProduct R A M M ∘ₗ baseChange A φ) _ (basis A bₘ), polyCharpolyAux_baseChange,
Polynomial.map_map]
congr
exact DFunLike.ext _ _ fun f ↦ (MvPolynomial.eval_map (algebraMap R A) x f).symm
