Loogle!

Result

Found 107 declarations mentioning MvPolynomial.map.

MvPolynomial.map 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) : MvPolynomial σ R →+* MvPolynomial σ S₁
MvPolynomial.algebraMap_def 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u_2} {S : Type u_3} {σ : Type u_4} [CommSemiring R] [CommSemiring S] [Algebra R S] : algebraMap (MvPolynomial σ R) (MvPolynomial σ S) = MvPolynomial.map (algebraMap R S)
MvPolynomial.map_id 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) : (MvPolynomial.map (RingHom.id R)) p = p
MvPolynomial.map_X 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) (n : σ) : (MvPolynomial.map f) (MvPolynomial.X n) = MvPolynomial.X n
MvPolynomial.constantCoeff_comp_map 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) : MvPolynomial.constantCoeff.comp (MvPolynomial.map f) = f.comp MvPolynomial.constantCoeff
MvPolynomial.support_map_subset 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) (p : MvPolynomial σ R) : ((MvPolynomial.map f) p).support ⊆ p.support
MvPolynomial.map_injective 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) (hf : Function.Injective ⇑f) : Function.Injective ⇑(MvPolynomial.map f)
MvPolynomial.map_surjective 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) (hf : Function.Surjective ⇑f) : Function.Surjective ⇑(MvPolynomial.map f)
MvPolynomial.map_injective_iff 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) : Function.Injective ⇑(MvPolynomial.map f) ↔ Function.Injective ⇑f
MvPolynomial.map_surjective_iff 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) : Function.Surjective ⇑(MvPolynomial.map f) ↔ Function.Surjective ⇑f
MvPolynomial.coeff_map 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) (p : MvPolynomial σ R) (m : σ →₀ ℕ) : MvPolynomial.coeff m ((MvPolynomial.map f) p) = f (MvPolynomial.coeff m p)
MvPolynomial.coeffs_map 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) (p : MvPolynomial σ R) [DecidableEq S₁] : ((MvPolynomial.map f) p).coeffs ⊆ Finset.image (⇑f) p.coeffs
MvPolynomial.support_map_of_injective 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (p : MvPolynomial σ R) {f : R →+* S₁} (hf : Function.Injective ⇑f) : ((MvPolynomial.map f) p).support = p.support
MvPolynomial.eval₂_map 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {S₂ : Type w} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] [CommSemiring S₂] (f : R →+* S₁) (g : σ → S₂) (φ : S₁ →+* S₂) (p : MvPolynomial σ R) : MvPolynomial.eval₂ φ g ((MvPolynomial.map f) p) = MvPolynomial.eval₂ (φ.comp f) g p
MvPolynomial.coe_coeffs_map 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) (p : MvPolynomial σ R) : ↑((MvPolynomial.map f) p).coeffs ⊆ ⇑f '' ↑p.coeffs
MvPolynomial.mem_range_map_iff_coeffs_subset 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] {f : R →+* S₁} {x : MvPolynomial σ S₁} : x ∈ Set.range ⇑(MvPolynomial.map f) ↔ ↑x.coeffs ⊆ Set.range ⇑f
MvPolynomial.eval_map 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) (g : σ → S₁) (p : MvPolynomial σ R) : (MvPolynomial.eval g) ((MvPolynomial.map f) p) = MvPolynomial.eval₂ f g p
MvPolynomial.eval₂_eq_eval_map 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) (g : σ → S₁) (p : MvPolynomial σ R) : MvPolynomial.eval₂ f g p = (MvPolynomial.eval g) ((MvPolynomial.map f) p)
MvPolynomial.mapAlgHom.eq_1 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {S₂ : Type w} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] [CommSemiring S₂] [Algebra R S₁] [Algebra R S₂] (f : S₁ →ₐ[R] S₂) : MvPolynomial.mapAlgHom f = { toRingHom := MvPolynomial.map ↑f, commutes' := ⋯ }
MvPolynomial.map_ofNat 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) (n : ℕ) [n.AtLeastTwo] : (MvPolynomial.map f) (OfNat.ofNat n) = OfNat.ofNat n
MvPolynomial.eval₂_map_comp_C 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] {ι : Type u_2} (f : R →+* S₁) (h : ι → MvPolynomial σ S₁) (p : MvPolynomial ι R) : MvPolynomial.eval₂ ((MvPolynomial.map f).comp MvPolynomial.C) h p = MvPolynomial.eval₂ MvPolynomial.C h ((MvPolynomial.map f) p)
MvPolynomial.eval₂_comp_right 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] {S₂ : Type u_2} [CommSemiring S₂] (k : S₁ →+* S₂) (f : R →+* S₁) (g : σ → S₁) (p : MvPolynomial σ R) : k (MvPolynomial.eval₂ f g p) = MvPolynomial.eval₂ k (⇑k ∘ g) ((MvPolynomial.map f) p)
MvPolynomial.map_mapRange_eq_iff 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) (g : S₁ → R) (hg : g 0 = 0) (φ : MvPolynomial σ S₁) : (MvPolynomial.map f) (Finsupp.mapRange g hg φ) = φ ↔ ∀ (d : σ →₀ ℕ), f (g (MvPolynomial.coeff d φ)) = MvPolynomial.coeff d φ
MvPolynomial.map_C 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) (a : R) : (MvPolynomial.map f) (MvPolynomial.C a) = MvPolynomial.C (f a)
MvPolynomial.constantCoeff_map 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) (φ : MvPolynomial σ R) : MvPolynomial.constantCoeff ((MvPolynomial.map f) φ) = f (MvPolynomial.constantCoeff φ)
MvPolynomial.map_leftInverse 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] {f : R →+* S₁} {g : S₁ →+* R} (hf : Function.LeftInverse ⇑f ⇑g) : Function.LeftInverse ⇑(MvPolynomial.map f) ⇑(MvPolynomial.map g)
MvPolynomial.map_rightInverse 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] {f : R →+* S₁} {g : S₁ →+* R} (hf : Function.RightInverse ⇑f ⇑g) : Function.RightInverse ⇑(MvPolynomial.map f) ⇑(MvPolynomial.map g)
MvPolynomial.eval₂Hom_map_hom 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {S₂ : Type w} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] [CommSemiring S₂] (f : R →+* S₁) (g : σ → S₂) (φ : S₁ →+* S₂) (p : MvPolynomial σ R) : (MvPolynomial.eval₂Hom φ g) ((MvPolynomial.map f) p) = (MvPolynomial.eval₂Hom (φ.comp f) g) p
MvPolynomial.map_eval 📋 Mathlib.Algebra.MvPolynomial.Eval
{S₁ : Type v} {σ : Type u_1} [CommSemiring S₁] {S₂ : Type u_2} [CommSemiring S₂] (q : S₁ →+* S₂) (g : σ → S₁) (p : MvPolynomial σ S₁) : q ((MvPolynomial.eval g) p) = (MvPolynomial.eval (⇑q ∘ g)) ((MvPolynomial.map q) p)
MvPolynomial.C_dvd_iff_map_hom_eq_zero 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (q : R →+* S₁) (r : R) (hr : ∀ (r' : R), q r' = 0 ↔ r ∣ r') (φ : MvPolynomial σ R) : MvPolynomial.C r ∣ φ ↔ (MvPolynomial.map q) φ = 0
MvPolynomial.map_map 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {S₂ : Type w} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) [CommSemiring S₂] (g : S₁ →+* S₂) (p : MvPolynomial σ R) : (MvPolynomial.map g) ((MvPolynomial.map f) p) = (MvPolynomial.map (g.comp f)) p
MvPolynomial.map_eval₂ 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {S₂ : Type w} {S₃ : Type x} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) (g : S₂ → MvPolynomial S₃ R) (p : MvPolynomial S₂ R) : (MvPolynomial.map f) (MvPolynomial.eval₂ MvPolynomial.C g p) = MvPolynomial.eval₂ MvPolynomial.C (⇑(MvPolynomial.map f) ∘ g) ((MvPolynomial.map f) p)
MvPolynomial.map_monomial 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) (s : σ →₀ ℕ) (a : R) : (MvPolynomial.map f) ((MvPolynomial.monomial s) a) = (MvPolynomial.monomial s) (f a)
MvPolynomial.mapAlgHom_coe_ringHom 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {S₂ : Type w} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] [CommSemiring S₂] [Algebra R S₁] [Algebra R S₂] (f : S₁ →ₐ[R] S₂) : ↑(MvPolynomial.mapAlgHom f) = MvPolynomial.map ↑f
MvPolynomial.map_comp_rename 📋 Mathlib.Algebra.MvPolynomial.Rename
{σ : Type u_1} {τ : Type u_2} {R : Type u_4} {S : Type u_5} [CommSemiring R] [CommSemiring S] (f : R →+* S) (g : σ → τ) : (MvPolynomial.map f).comp (MvPolynomial.rename g).toRingHom = (MvPolynomial.rename g).comp (MvPolynomial.map f)
MvPolynomial.map_rename 📋 Mathlib.Algebra.MvPolynomial.Rename
{σ : Type u_1} {τ : Type u_2} {R : Type u_4} {S : Type u_5} [CommSemiring R] [CommSemiring S] (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) : (MvPolynomial.map f) ((MvPolynomial.rename g) p) = (MvPolynomial.rename g) ((MvPolynomial.map f) p)
MvPolynomial.degrees_map_le 📋 Mathlib.Algebra.MvPolynomial.Degrees
{R : Type u} {S : Type v} {σ : Type u_1} [CommSemiring R] {p : MvPolynomial σ R} [CommSemiring S] {f : R →+* S} : ((MvPolynomial.map f) p).degrees ≤ p.degrees
MvPolynomial.degrees_map_of_injective 📋 Mathlib.Algebra.MvPolynomial.Degrees
{R : Type u} {S : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S] (p : MvPolynomial σ R) {f : R →+* S} (hf : Function.Injective ⇑f) : ((MvPolynomial.map f) p).degrees = p.degrees
MvPolynomial.vars_map 📋 Mathlib.Algebra.MvPolynomial.Variables
{R : Type u} {S : Type v} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) [CommSemiring S] (f : R →+* S) : ((MvPolynomial.map f) p).vars ⊆ p.vars
MvPolynomial.vars_map_of_injective 📋 Mathlib.Algebra.MvPolynomial.Variables
{R : Type u} {S : Type v} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) [CommSemiring S] {f : R →+* S} (hf : Function.Injective ⇑f) : ((MvPolynomial.map f) p).vars = p.vars
MvPolynomial.mapAlgEquiv_apply 📋 Mathlib.Algebra.MvPolynomial.Equiv
{R : Type u} (σ : Type u_1) [CommSemiring R] {A₁ : Type u_2} {A₂ : Type u_3} [CommSemiring A₁] [CommSemiring A₂] [Algebra R A₁] [Algebra R A₂] (e : A₁ ≃ₐ[R] A₂) (a : MvPolynomial σ A₁) : (MvPolynomial.mapAlgEquiv σ e) a = (MvPolynomial.map ↑e) a
MvPolynomial.commAlgEquiv_C 📋 Mathlib.Algebra.MvPolynomial.Equiv
{R : Type u_2} {S₁ : Type u_3} {S₂ : Type u_4} [CommSemiring R] (p : MvPolynomial S₂ R) : (MvPolynomial.commAlgEquiv R S₁ S₂) (MvPolynomial.C p) = (MvPolynomial.map MvPolynomial.C) p
MvPolynomial.mapEquiv_apply 📋 Mathlib.Algebra.MvPolynomial.Equiv
{S₁ : Type v} {S₂ : Type w} (σ : Type u_1) [CommSemiring S₁] [CommSemiring S₂] (e : S₁ ≃+* S₂) (a : MvPolynomial σ S₁) : (MvPolynomial.mapEquiv σ e) a = (MvPolynomial.map ↑e) a
MvPolynomial.ker_map 📋 Mathlib.RingTheory.Polynomial.Basic
{R : Type u} {S : Type u_1} {σ : Type v} [CommRing R] [CommRing S] (f : R →+* S) : RingHom.ker (MvPolynomial.map f) = Ideal.map MvPolynomial.C (RingHom.ker f)
MvPolynomial.mapRange_eq_map 📋 Mathlib.RingTheory.MvPolynomial.Basic
(σ : Type u) {R : Type u_1} {S : Type u_2} [CommSemiring R] [CommSemiring S] (p : MvPolynomial σ R) (f : R →+* S) : Finsupp.mapRange ⇑f ⋯ p = (MvPolynomial.map f) p
MvPolynomial.aeval_map_algebraMap 📋 Mathlib.RingTheory.MvPolynomial.Tower
{R : Type u_1} (A : Type u_2) {B : Type u_3} {σ : Type u_4} [CommSemiring R] [CommSemiring A] [CommSemiring B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] (x : σ → B) (p : MvPolynomial σ R) : (MvPolynomial.aeval x) ((MvPolynomial.map (algebraMap R A)) p) = (MvPolynomial.aeval x) p
MvPolynomial.algebraTensorAlgEquiv_tmul 📋 Mathlib.RingTheory.TensorProduct.MvPolynomial
(R : Type u) [CommSemiring R] {σ : Type u_1} (A : Type u_4) [CommSemiring A] [Algebra R A] (a : A) (p : MvPolynomial σ R) : (MvPolynomial.algebraTensorAlgEquiv R A) (a ⊗ₜ[R] p) = a • (MvPolynomial.map (algebraMap R A)) p
MvPolynomial.algebraTensorAlgEquiv_symm_map 📋 Mathlib.RingTheory.TensorProduct.MvPolynomial
(R : Type u) [CommSemiring R] {σ : Type u_1} (A : Type u_4) [CommSemiring A] [Algebra R A] (x : MvPolynomial σ R) : (MvPolynomial.algebraTensorAlgEquiv R A).symm ((MvPolynomial.map (algebraMap R A)) x) = 1 ⊗ₜ[R] x
MvPolynomial.exists_dvd_map_of_isAlgebraic 📋 Mathlib.RingTheory.Algebraic.Integral
{R : Type u_1} {S : Type u_2} [CommRing R] [CommRing S] [Algebra R S] [alg : Algebra.IsAlgebraic R S] {σ : Type u_4} [NoZeroDivisors S] {f : MvPolynomial σ S} (hf : f ≠ 0) : ∃ g, g ≠ 0 ∧ f ∣ (MvPolynomial.map (algebraMap R S)) g
MvPolynomial.map_comp_C 📋 Mathlib.Algebra.MvPolynomial.Monad
{σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* S) : (MvPolynomial.map f).comp MvPolynomial.C = MvPolynomial.C.comp f
MvPolynomial.join₂_comp_map 📋 Mathlib.Algebra.MvPolynomial.Monad
{σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* MvPolynomial σ S) : MvPolynomial.join₂.comp (MvPolynomial.map f) = MvPolynomial.bind₂ f
MvPolynomial.bind₂_map 📋 Mathlib.Algebra.MvPolynomial.Monad
{σ : Type u_1} {R : Type u_3} {S : Type u_4} {T : Type u_5} [CommSemiring R] [CommSemiring S] [CommSemiring T] (f : S →+* MvPolynomial σ T) (g : R →+* S) (φ : MvPolynomial σ R) : (MvPolynomial.bind₂ f) ((MvPolynomial.map g) φ) = (MvPolynomial.bind₂ (f.comp g)) φ
MvPolynomial.map_bind₂ 📋 Mathlib.Algebra.MvPolynomial.Monad
{σ : Type u_1} {R : Type u_3} {S : Type u_4} {T : Type u_5} [CommSemiring R] [CommSemiring S] [CommSemiring T] (f : R →+* MvPolynomial σ S) (g : S →+* T) (φ : MvPolynomial σ R) : (MvPolynomial.map g) ((MvPolynomial.bind₂ f) φ) = (MvPolynomial.bind₂ ((MvPolynomial.map g).comp f)) φ
MvPolynomial.join₂_map 📋 Mathlib.Algebra.MvPolynomial.Monad
{σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* MvPolynomial σ S) (φ : MvPolynomial σ R) : MvPolynomial.join₂ ((MvPolynomial.map f) φ) = (MvPolynomial.bind₂ f) φ
MvPolynomial.map_bind₁ 📋 Mathlib.Algebra.MvPolynomial.Monad
{σ : Type u_1} {τ : Type u_2} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* S) (g : σ → MvPolynomial τ R) (φ : MvPolynomial σ R) : (MvPolynomial.map f) ((MvPolynomial.bind₁ g) φ) = (MvPolynomial.bind₁ fun i => (MvPolynomial.map f) (g i)) ((MvPolynomial.map f) φ)
MvPolynomial.IsHomogeneous.map 📋 Mathlib.RingTheory.MvPolynomial.Homogeneous
{σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] {φ : MvPolynomial σ R} {n : ℕ} (hφ : φ.IsHomogeneous n) (f : R →+* S) : ((MvPolynomial.map f) φ).IsHomogeneous n
MvPolynomial.IsHomogeneous.of_map 📋 Mathlib.RingTheory.MvPolynomial.Homogeneous
{σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] {φ : MvPolynomial σ R} {n : ℕ} {f : R →+* S} (hf : Function.Injective ⇑f) (h : ((MvPolynomial.map f) φ).IsHomogeneous n) : φ.IsHomogeneous n
MvPolynomial.map.eq_1 📋 Mathlib.LinearAlgebra.Matrix.Charpoly.Univ
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) : MvPolynomial.map f = MvPolynomial.eval₂Hom (MvPolynomial.C.comp f) MvPolynomial.X
Matrix.charpoly.univ_map_map 📋 Mathlib.LinearAlgebra.Matrix.Charpoly.Univ
{R : Type u_1} {S : Type u_2} (n : Type u_3) [CommRing R] [CommRing S] [Fintype n] [DecidableEq n] (f : R →+* S) : Polynomial.map (MvPolynomial.map f) (Matrix.charpoly.univ R n) = Matrix.charpoly.univ S n
Matrix.toMvPolynomial_map 📋 Mathlib.Algebra.Module.LinearMap.Polynomial
{m : Type u_1} {n : Type u_2} {R : Type u_4} {S : Type u_5} [Fintype n] [CommSemiring R] [CommSemiring S] (f : R →+* S) (M : Matrix m n R) (i : m) : (M.map ⇑f).toMvPolynomial i = (MvPolynomial.map f) (M.toMvPolynomial i)
LinearMap.toMvPolynomial_baseChange 📋 Mathlib.Algebra.Module.LinearMap.Polynomial
{R : Type u_1} {M₁ : Type u_2} {M₂ : Type u_3} {ι₁ : Type u_4} {ι₂ : Type u_5} [CommRing R] [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] [Fintype ι₁] [Finite ι₂] [DecidableEq ι₁] (b₁ : Module.Basis ι₁ R M₁) (b₂ : Module.Basis ι₂ R M₂) (f : M₁ →ₗ[R] M₂) (i : ι₂) (A : Type u_6) [CommRing A] [Algebra R A] : LinearMap.toMvPolynomial (Algebra.TensorProduct.basis A b₁) (Algebra.TensorProduct.basis A b₂) (LinearMap.baseChange A f) i = (MvPolynomial.map (algebraMap R A)) (LinearMap.toMvPolynomial b₁ b₂ f i)
LinearMap.polyCharpolyAux_baseChange 📋 Mathlib.Algebra.Module.LinearMap.Polynomial
{R : Type u_1} {L : Type u_2} {M : Type u_3} {ι : Type u_5} {ιM : Type u_7} [CommRing R] [AddCommGroup L] [Module R L] [AddCommGroup M] [Module R M] (φ : L →ₗ[R] Module.End R M) [Fintype ι] [Fintype ιM] [DecidableEq ι] [DecidableEq ιM] (b : Module.Basis ι R L) (bₘ : Module.Basis ιM R M) (A : Type u_8) [CommRing A] [Algebra R A] : (LinearMap.tensorProduct R A M M ∘ₗ LinearMap.baseChange A φ).polyCharpolyAux (Algebra.TensorProduct.basis A b) (Algebra.TensorProduct.basis A bₘ) = Polynomial.map (MvPolynomial.map (algebraMap R A)) (φ.polyCharpolyAux b bₘ)
LinearMap.polyCharpoly_baseChange 📋 Mathlib.Algebra.Module.LinearMap.Polynomial
{R : Type u_1} {L : Type u_2} {M : Type u_3} {ι : Type u_5} [CommRing R] [AddCommGroup L] [Module R L] [AddCommGroup M] [Module R M] (φ : L →ₗ[R] Module.End R M) [Fintype ι] [DecidableEq ι] [Module.Free R M] [Module.Finite R M] (b : Module.Basis ι R L) (A : Type u_8) [CommRing A] [Algebra R A] : (LinearMap.tensorProduct R A M M ∘ₗ LinearMap.baseChange A φ).polyCharpoly (Algebra.TensorProduct.basis A b) = Polynomial.map (MvPolynomial.map (algebraMap R A)) (φ.polyCharpoly b)
MvPolynomial.pderiv_map 📋 Mathlib.Algebra.MvPolynomial.PDeriv
{R : Type u} {σ : Type v} [CommSemiring R] {S : Type u_1} [CommSemiring S] {φ : R →+* S} {f : MvPolynomial σ R} {i : σ} : (MvPolynomial.pderiv i) ((MvPolynomial.map φ) f) = (MvPolynomial.map φ) ((MvPolynomial.pderiv i) f)
Algebra.Presentation.baseChange_relation 📋 Mathlib.RingTheory.Extension.Presentation.Basic
{R : Type u} {S : Type v} {ι : Type w} {σ : Type t} [CommRing R] [CommRing S] [Algebra R S] (T : Type u_1) [CommRing T] [Algebra R T] (P : Algebra.Presentation R S ι σ) (i : σ) : (Algebra.Presentation.baseChange T P).relation i = (MvPolynomial.map (algebraMap R T)) (P.relation i)
MvPolynomial.map_expand 📋 Mathlib.Algebra.MvPolynomial.Expand
{σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] (f : R →+* S) (p : ℕ) (φ : MvPolynomial σ R) : (MvPolynomial.map f) ((MvPolynomial.expand p) φ) = (MvPolynomial.expand p) ((MvPolynomial.map f) φ)
Polynomial.homogenize_map 📋 Mathlib.Algebra.Polynomial.Homogenize
{R : Type u_1} [CommSemiring R] {S : Type u_2} [CommSemiring S] (f : R →+* S) (p : Polynomial R) (n : ℕ) : (Polynomial.map f p).homogenize n = (MvPolynomial.map f) (p.homogenize n)
AlgebraicGeometry.AffineSpace.mapSpecMap 📋 Mathlib.AlgebraicGeometry.AffineSpace
{n : Type v} {R S : CommRingCat} (φ : R ⟶ S) : CategoryTheory.Arrow.mk (AlgebraicGeometry.AffineSpace.map n (AlgebraicGeometry.Spec.map φ)) ≅ CategoryTheory.Arrow.mk (AlgebraicGeometry.Spec.map (CommRingCat.ofHom (MvPolynomial.map (CommRingCat.Hom.hom φ))))
AlgebraicGeometry.AffineSpace.map_SpecMap 📋 Mathlib.AlgebraicGeometry.AffineSpace
{n : Type v} {R S : CommRingCat} (φ : R ⟶ S) : AlgebraicGeometry.AffineSpace.map n (AlgebraicGeometry.Spec.map φ) = CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.AffineSpace.SpecIso n S).hom (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Spec.map (CommRingCat.ofHom (MvPolynomial.map (CommRingCat.Hom.hom φ)))) (AlgebraicGeometry.AffineSpace.SpecIso n R).inv)
AlgebraicGeometry.AffineSpace.map_Spec_map 📋 Mathlib.AlgebraicGeometry.AffineSpace
{n : Type v} {R S : CommRingCat} (φ : R ⟶ S) : AlgebraicGeometry.AffineSpace.map n (AlgebraicGeometry.Spec.map φ) = CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.AffineSpace.SpecIso n S).hom (CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Spec.map (CommRingCat.ofHom (MvPolynomial.map (CommRingCat.Hom.hom φ)))) (AlgebraicGeometry.AffineSpace.SpecIso n R).inv)
WeierstrassCurve.Jacobian.map_polynomial 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic
{R : Type r} {S : Type s} [CommRing R] [CommRing S] {W' : WeierstrassCurve.Jacobian R} (f : R →+* S) : (WeierstrassCurve.map W' f).toJacobian.polynomial = (MvPolynomial.map f) W'.polynomial
WeierstrassCurve.Jacobian.map_polynomialX 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic
{R : Type r} {S : Type s} [CommRing R] [CommRing S] {W' : WeierstrassCurve.Jacobian R} (f : R →+* S) : (WeierstrassCurve.map W' f).toJacobian.polynomialX = (MvPolynomial.map f) W'.polynomialX
WeierstrassCurve.Jacobian.map_polynomialY 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic
{R : Type r} {S : Type s} [CommRing R] [CommRing S] {W' : WeierstrassCurve.Jacobian R} (f : R →+* S) : (WeierstrassCurve.map W' f).toJacobian.polynomialY = (MvPolynomial.map f) W'.polynomialY
WeierstrassCurve.Jacobian.map_polynomialZ 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic
{R : Type r} {S : Type s} [CommRing R] [CommRing S] {W' : WeierstrassCurve.Jacobian R} (f : R →+* S) : (WeierstrassCurve.map W' f).toJacobian.polynomialZ = (MvPolynomial.map f) W'.polynomialZ
WeierstrassCurve.Jacobian.baseChange_polynomial 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic
{R : Type r} {S : Type s} {A : Type u} {B : Type v} [CommRing R] [CommRing S] [CommRing A] [CommRing B] {W' : WeierstrassCurve.Jacobian R} [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Algebra R B] [Algebra S B] [IsScalarTower R S B] (f : A →ₐ[S] B) : (WeierstrassCurve.baseChange W' B).toJacobian.polynomial = (MvPolynomial.map ↑f) (WeierstrassCurve.baseChange W' A).toJacobian.polynomial
WeierstrassCurve.Jacobian.baseChange_polynomialX 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic
{R : Type r} {S : Type s} {A : Type u} {B : Type v} [CommRing R] [CommRing S] [CommRing A] [CommRing B] {W' : WeierstrassCurve.Jacobian R} [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Algebra R B] [Algebra S B] [IsScalarTower R S B] (f : A →ₐ[S] B) : (WeierstrassCurve.baseChange W' B).toJacobian.polynomialX = (MvPolynomial.map ↑f) (WeierstrassCurve.baseChange W' A).toJacobian.polynomialX
WeierstrassCurve.Jacobian.baseChange_polynomialY 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic
{R : Type r} {S : Type s} {A : Type u} {B : Type v} [CommRing R] [CommRing S] [CommRing A] [CommRing B] {W' : WeierstrassCurve.Jacobian R} [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Algebra R B] [Algebra S B] [IsScalarTower R S B] (f : A →ₐ[S] B) : (WeierstrassCurve.baseChange W' B).toJacobian.polynomialY = (MvPolynomial.map ↑f) (WeierstrassCurve.baseChange W' A).toJacobian.polynomialY
WeierstrassCurve.Jacobian.baseChange_polynomialZ 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic
{R : Type r} {S : Type s} {A : Type u} {B : Type v} [CommRing R] [CommRing S] [CommRing A] [CommRing B] {W' : WeierstrassCurve.Jacobian R} [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Algebra R B] [Algebra S B] [IsScalarTower R S B] (f : A →ₐ[S] B) : (WeierstrassCurve.baseChange W' B).toJacobian.polynomialZ = (MvPolynomial.map ↑f) (WeierstrassCurve.baseChange W' A).toJacobian.polynomialZ
WeierstrassCurve.Projective.map_polynomial 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic
{R : Type r} {S : Type s} [CommRing R] [CommRing S] {W' : WeierstrassCurve.Projective R} (f : R →+* S) : (WeierstrassCurve.map W' f).toProjective.polynomial = (MvPolynomial.map f) W'.polynomial
WeierstrassCurve.Projective.map_polynomialX 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic
{R : Type r} {S : Type s} [CommRing R] [CommRing S] {W' : WeierstrassCurve.Projective R} (f : R →+* S) : (WeierstrassCurve.map W' f).toProjective.polynomialX = (MvPolynomial.map f) W'.polynomialX
WeierstrassCurve.Projective.map_polynomialY 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic
{R : Type r} {S : Type s} [CommRing R] [CommRing S] {W' : WeierstrassCurve.Projective R} (f : R →+* S) : (WeierstrassCurve.map W' f).toProjective.polynomialY = (MvPolynomial.map f) W'.polynomialY
WeierstrassCurve.Projective.map_polynomialZ 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic
{R : Type r} {S : Type s} [CommRing R] [CommRing S] {W' : WeierstrassCurve.Projective R} (f : R →+* S) : (WeierstrassCurve.map W' f).toProjective.polynomialZ = (MvPolynomial.map f) W'.polynomialZ
WeierstrassCurve.Projective.baseChange_polynomial 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic
{R : Type r} {S : Type s} {A : Type u} {B : Type v} [CommRing R] [CommRing S] [CommRing A] [CommRing B] {W' : WeierstrassCurve.Projective R} [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Algebra R B] [Algebra S B] [IsScalarTower R S B] (f : A →ₐ[S] B) : (WeierstrassCurve.baseChange W' B).toProjective.polynomial = (MvPolynomial.map ↑f) (WeierstrassCurve.baseChange W' A).toProjective.polynomial
WeierstrassCurve.Projective.baseChange_polynomialX 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic
{R : Type r} {S : Type s} {A : Type u} {B : Type v} [CommRing R] [CommRing S] [CommRing A] [CommRing B] {W' : WeierstrassCurve.Projective R} [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Algebra R B] [Algebra S B] [IsScalarTower R S B] (f : A →ₐ[S] B) : (WeierstrassCurve.baseChange W' B).toProjective.polynomialX = (MvPolynomial.map ↑f) (WeierstrassCurve.baseChange W' A).toProjective.polynomialX
WeierstrassCurve.Projective.baseChange_polynomialY 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic
{R : Type r} {S : Type s} {A : Type u} {B : Type v} [CommRing R] [CommRing S] [CommRing A] [CommRing B] {W' : WeierstrassCurve.Projective R} [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Algebra R B] [Algebra S B] [IsScalarTower R S B] (f : A →ₐ[S] B) : (WeierstrassCurve.baseChange W' B).toProjective.polynomialY = (MvPolynomial.map ↑f) (WeierstrassCurve.baseChange W' A).toProjective.polynomialY
WeierstrassCurve.Projective.baseChange_polynomialZ 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic
{R : Type r} {S : Type s} {A : Type u} {B : Type v} [CommRing R] [CommRing S] [CommRing A] [CommRing B] {W' : WeierstrassCurve.Projective R} [Algebra R S] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Algebra R B] [Algebra S B] [IsScalarTower R S B] (f : A →ₐ[S] B) : (WeierstrassCurve.baseChange W' B).toProjective.polynomialZ = (MvPolynomial.map ↑f) (WeierstrassCurve.baseChange W' A).toProjective.polynomialZ
MvPolynomial.IsSymmetric.map 📋 Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
{σ : Type u_1} {R : Type u_3} {S : Type u_4} [CommSemiring R] [CommSemiring S] {φ : MvPolynomial σ R} (hφ : φ.IsSymmetric) (f : R →+* S) : ((MvPolynomial.map f) φ).IsSymmetric
MvPolynomial.map_esymm 📋 Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
{S : Type u_4} (σ : Type u_5) (R : Type u_6) [CommSemiring R] [CommSemiring S] [Fintype σ] (n : ℕ) (f : R →+* S) : (MvPolynomial.map f) (MvPolynomial.esymm σ R n) = MvPolynomial.esymm σ S n
MvPolynomial.map_hsymm 📋 Mathlib.RingTheory.MvPolynomial.Symmetric.Defs
{S : Type u_4} (σ : Type u_5) (R : Type u_6) [CommSemiring R] [CommSemiring S] [Fintype σ] [DecidableEq σ] (n : ℕ) (f : R →+* S) : (MvPolynomial.map f) (MvPolynomial.hsymm σ R n) = MvPolynomial.hsymm σ S n
MvPowerSeries.trunc'_map 📋 Mathlib.RingTheory.MvPowerSeries.Trunc
{σ : Type u_1} {R : Type u_2} {S : Type u_3} [DecidableEq σ] [CommSemiring R] [CommSemiring S] (n : σ →₀ ℕ) (f : R →+* S) (p : MvPowerSeries σ R) : (MvPowerSeries.trunc' S n) ((MvPowerSeries.map f) p) = (MvPolynomial.map f) ((MvPowerSeries.trunc' R n) p)
MvPowerSeries.trunc_map 📋 Mathlib.RingTheory.MvPowerSeries.Trunc
{σ : Type u_1} {R : Type u_2} {S : Type u_3} [DecidableEq σ] [CommSemiring R] [CommSemiring S] (n : σ →₀ ℕ) (f : R →+* S) (p : MvPowerSeries σ R) : (MvPowerSeries.trunc S n) ((MvPowerSeries.map f) p) = (MvPolynomial.map f) ((MvPowerSeries.trunc R n) p)
MvPolynomial.C_dvd_iff_zmod 📋 Mathlib.FieldTheory.Finite.Polynomial
{σ : Type u_1} (n : ℕ) (φ : MvPolynomial σ ℤ) : MvPolynomial.C ↑n ∣ φ ↔ (MvPolynomial.map (Int.castRingHom (ZMod n))) φ = 0
Algebra.Presentation.map_relationOfHasCoeffs 📋 Mathlib.RingTheory.Extension.Presentation.Core
{R : Type u_1} {S : Type u_2} {ι : Type u_3} {σ : Type u_4} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Presentation R S ι σ} (R₀ : Type u_5) [CommRing R₀] [Algebra R₀ R] [Algebra R₀ S] [IsScalarTower R₀ R S] [P.HasCoeffs R₀] (r : σ) : (MvPolynomial.map (algebraMap R₀ R)) (Algebra.Presentation.relationOfHasCoeffs R₀ r) = P.relation r
Algebra.Presentation.HasCoeffs.relation_mem_range_map 📋 Mathlib.RingTheory.Extension.Presentation.Core
{R : Type u_1} {S : Type u_2} {ι : Type u_3} {σ : Type u_4} [CommRing R] [CommRing S] [Algebra R S] {P : Algebra.Presentation R S ι σ} (R₀ : Type u_5) [CommRing R₀] [Algebra R₀ R] [Algebra R₀ S] [IsScalarTower R₀ R S] [P.HasCoeffs R₀] (x : σ) : P.relation x ∈ Set.range ⇑(MvPolynomial.map (algebraMap R₀ R))
map_wittPolynomial 📋 Mathlib.RingTheory.WittVector.WittPolynomial
(p : ℕ) {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] (f : R →+* S) (n : ℕ) : (MvPolynomial.map f) (wittPolynomial p R n) = wittPolynomial p S n
wittStructureInt.eq_1 📋 Mathlib.RingTheory.WittVector.StructurePolynomial
(p : ℕ) {idx : Type u_2} [hp : Fact (Nat.Prime p)] (Φ : MvPolynomial idx ℤ) (n : ℕ) : wittStructureInt p Φ n = Finsupp.mapRange Rat.num wittStructureInt._proof_1 (wittStructureRat p ((MvPolynomial.map (Int.castRingHom ℚ)) Φ) n)
map_wittStructureInt 📋 Mathlib.RingTheory.WittVector.StructurePolynomial
(p : ℕ) {idx : Type u_2} [hp : Fact (Nat.Prime p)] (Φ : MvPolynomial idx ℤ) (n : ℕ) : (MvPolynomial.map (Int.castRingHom ℚ)) (wittStructureInt p Φ n) = wittStructureRat p ((MvPolynomial.map (Int.castRingHom ℚ)) Φ) n
witt_structure_prop 📋 Mathlib.RingTheory.WittVector.StructurePolynomial
(p : ℕ) {R : Type u_1} {idx : Type u_2} [CommRing R] [hp : Fact (Nat.Prime p)] (Φ : MvPolynomial idx ℤ) (n : ℕ) : (MvPolynomial.aeval fun i => (MvPolynomial.map (Int.castRingHom R)) (wittStructureInt p Φ i)) (wittPolynomial p ℤ n) = (MvPolynomial.aeval fun i => (MvPolynomial.rename (Prod.mk i)) (wittPolynomial p R n)) Φ
bind₁_rename_expand_wittPolynomial 📋 Mathlib.RingTheory.WittVector.StructurePolynomial
{p : ℕ} {idx : Type u_2} [hp : Fact (Nat.Prime p)] (Φ : MvPolynomial idx ℤ) (n : ℕ) (IH : ∀ m < n + 1, (MvPolynomial.map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((MvPolynomial.map (Int.castRingHom ℚ)) Φ) m) : (MvPolynomial.bind₁ fun b => (MvPolynomial.rename fun i => (b, i)) ((MvPolynomial.expand p) (wittPolynomial p ℤ n))) Φ = (MvPolynomial.bind₁ fun i => (MvPolynomial.expand p) (wittStructureInt p Φ i)) (wittPolynomial p ℤ n)
C_p_pow_dvd_bind₁_rename_wittPolynomial_sub_sum 📋 Mathlib.RingTheory.WittVector.StructurePolynomial
{p : ℕ} {idx : Type u_2} [hp : Fact (Nat.Prime p)] (Φ : MvPolynomial idx ℤ) (n : ℕ) (IH : ∀ m < n, (MvPolynomial.map (Int.castRingHom ℚ)) (wittStructureInt p Φ m) = wittStructureRat p ((MvPolynomial.map (Int.castRingHom ℚ)) Φ) m) : MvPolynomial.C ↑(p ^ n) ∣ (MvPolynomial.bind₁ fun b => (MvPolynomial.rename fun i => (b, i)) (wittPolynomial p ℤ n)) Φ - ∑ i ∈ Finset.range n, MvPolynomial.C (↑p ^ i) * wittStructureInt p Φ i ^ p ^ (n - i)
WittVector.map_frobeniusPoly 📋 Mathlib.RingTheory.WittVector.Frobenius
(p : ℕ) [hp : Fact (Nat.Prime p)] (n : ℕ) : (MvPolynomial.map (Int.castRingHom ℚ)) (WittVector.frobeniusPoly p n) = WittVector.frobeniusPolyRat p n
WittVector.frobeniusPoly_zmod 📋 Mathlib.RingTheory.WittVector.Frobenius
(p : ℕ) [hp : Fact (Nat.Prime p)] (n : ℕ) : (MvPolynomial.map (Int.castRingHom (ZMod p))) (WittVector.frobeniusPoly p n) = MvPolynomial.X n ^ p
Polynomial.map_map_freeMonic 📋 Mathlib.RingTheory.Polynomial.UniversalFactorizationRing
(R : Type u_1) (S : Type u_2) [CommRing R] [CommRing S] (n : ℕ) (f : R →+* S) : Polynomial.map (MvPolynomial.map f) (Polynomial.freeMonic R n) = Polynomial.freeMonic S n
MvPolynomial.universalFactorizationMapPresentation_σ' 📋 Mathlib.RingTheory.Polynomial.UniversalFactorizationRing
(R : Type u_1) [CommRing R] (n m k : ℕ) (hn : n = m + k) (f : TensorProduct R (MvPolynomial (Fin m) R) (MvPolynomial (Fin k) R)) : (MvPolynomial.universalFactorizationMapPresentation R n m k hn).σ' f = (MvPolynomial.map MvPolynomial.C) ((MvPolynomial.tensorEquivSum R (Fin m) (Fin k) R) f)
MvPolynomial.ker_eval₂Hom_universalFactorizationMap 📋 Mathlib.RingTheory.Polynomial.UniversalFactorizationRing
(R : Type u_1) [CommRing R] (n m k : ℕ) (hn : n = m + k) : RingHom.ker (MvPolynomial.eval₂Hom (↑(MvPolynomial.universalFactorizationMap R n m k hn)) (Sum.elim (fun x => MvPolynomial.X x ⊗ₜ[R] 1) fun x => 1 ⊗ₜ[R] MvPolynomial.X x)) = Ideal.span (Set.range fun i => MvPolynomial.C (MvPolynomial.X i) - (MvPolynomial.map MvPolynomial.C) ((MvPolynomial.tensorEquivSum R (Fin m) (Fin k) R) ((MvPolynomial.universalFactorizationMap R n m k hn) (MvPolynomial.X i))))
MvPolynomial.universalFactorizationMapPresentation_relation 📋 Mathlib.RingTheory.Polynomial.UniversalFactorizationRing
(R : Type u_1) [CommRing R] (n m k : ℕ) (hn : n = m + k) (i : Fin n) : (MvPolynomial.universalFactorizationMapPresentation R n m k hn).relation i = MvPolynomial.C (MvPolynomial.X i) - (MvPolynomial.map MvPolynomial.C) ((MvPolynomial.tensorEquivSum R (Fin m) (Fin k) R) ((MvPolynomial.universalFactorizationMap R n m k hn) (MvPolynomial.X i)))
MvPolynomial.universalFactorizationMap_comp_map 📋 Mathlib.RingTheory.Polynomial.UniversalFactorizationRing
(R : Type u_1) (S : Type u_2) [CommRing R] [CommRing S] [Algebra R S] (n m k : ℕ) (hn : n = m + k) : (MvPolynomial.universalFactorizationMap S n m k hn).comp (MvPolynomial.map (algebraMap R S)) = (Algebra.TensorProduct.lift (Algebra.TensorProduct.includeLeft.comp (MvPolynomial.mapAlgHom (Algebra.ofId R S))) ((AlgHom.restrictScalars R Algebra.TensorProduct.includeRight).comp (MvPolynomial.mapAlgHom (Algebra.ofId R S))) ⋯).comp (MvPolynomial.universalFactorizationMap R n m k hn).toRingHom

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

By constant:
🔍 Real.sin
finds all lemmas whose statement somehow mentions the sine function.
By lemma name substring:
🔍 "differ"
finds all lemmas that have "differ" somewhere in their lemma name.
By subexpression:
🔍 _ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

The pattern can also be non-linear, as in
🔍 Real.sqrt ?a * Real.sqrt ?a

If the pattern has parameters, they are matched in any order. Both of these will find List.map:
🔍 (?a -> ?b) -> List ?a -> List ?b
🔍 List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍 |- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍 |- _ < _ → tsum _ < tsum _
will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.

This is Loogle revision 6ff4759 serving mathlib revision edaf32c