Loogle!

Result

Found 132 declarations mentioning MvPolynomial.eval.

MvPolynomial.eval 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} [CommSemiring R] (f : σ → R) : MvPolynomial σ R →+* R
MvPolynomial.eval_zero' 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} [CommSemiring R] : (MvPolynomial.eval fun x => 0) = MvPolynomial.constantCoeff
MvPolynomial.eval_zero 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} [CommSemiring R] : MvPolynomial.eval 0 = MvPolynomial.constantCoeff
MvPolynomial.eval_X 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} [CommSemiring R] {f : σ → R} (n : σ) : (MvPolynomial.eval f) (MvPolynomial.X n) = f n
MvPolynomial.eval₂_id 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} [CommSemiring R] {g : σ → R} (p : MvPolynomial σ R) : MvPolynomial.eval₂ (RingHom.id R) g p = (MvPolynomial.eval g) p
MvPolynomial.eval_ofNat 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} [CommSemiring R] {f : σ → R} (n : ℕ) [n.AtLeastTwo] : (MvPolynomial.eval f) (OfNat.ofNat n) = OfNat.ofNat n
MvPolynomial.eval_C 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} [CommSemiring R] {f : σ → R} (a : R) : (MvPolynomial.eval f) (MvPolynomial.C a) = a
MvPolynomial.eval₂_comp 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {S₁ : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S₁] (f : R →+* S₁) (g : σ → R) (p : MvPolynomial σ R) : f ((MvPolynomial.eval g) p) = MvPolynomial.eval₂ f (⇑f ∘ g) p
MvPolynomial.eval_sum 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} [CommSemiring R] {ι : Type u_2} (s : Finset ι) (f : ι → MvPolynomial σ R) (g : σ → R) : (MvPolynomial.eval g) (∑ i ∈ s, f i) = ∑ i ∈ s, (MvPolynomial.eval g) (f i)
MvPolynomial.eval_prod 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} [CommSemiring R] {ι : Type u_2} (s : Finset ι) (f : ι → MvPolynomial σ R) (g : σ → R) : (MvPolynomial.eval g) (∏ i ∈ s, f i) = ∏ i ∈ s, (MvPolynomial.eval g) (f i)
MvPolynomial.aeval_eq_eval 📋 Mathlib.Algebra.MvPolynomial.Eval
{S₁ : Type v} {σ : Type u_1} [CommSemiring S₁] (f : σ → S₁) : ⇑(MvPolynomial.aeval f) = ⇑(MvPolynomial.eval 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.eval_pow 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} [CommSemiring R] {p : MvPolynomial σ R} {f : σ → R} (n : ℕ) : (MvPolynomial.eval f) (p ^ n) = (MvPolynomial.eval f) p ^ n
MvPolynomial.eval_eq' 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} [CommSemiring R] [Fintype σ] (X : σ → R) (f : MvPolynomial σ R) : (MvPolynomial.eval X) f = ∑ d ∈ f.support, f.coeff d * ∏ i, X i ^ d i
MvPolynomial.eval_mem 📋 Mathlib.Algebra.MvPolynomial.Eval
{σ : Type u_1} {S : Type u_2} {subS : Type u_3} [CommSemiring S] [SetLike subS S] [SubsemiringClass subS S] {p : MvPolynomial σ S} {s : subS} (hs : ∀ i ∈ p.support, p.coeff i ∈ s) {v : σ → S} (hv : ∀ (i : σ), v i ∈ s) : (MvPolynomial.eval v) p ∈ s
MvPolynomial.eval_eq 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} [CommSemiring R] (X : σ → R) (f : MvPolynomial σ R) : (MvPolynomial.eval X) f = ∑ d ∈ f.support, f.coeff d * ∏ i ∈ d.support, X i ^ d i
MvPolynomial.smul_eval 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} [CommSemiring R] (x : σ → R) (p : MvPolynomial σ R) (s : R) : (MvPolynomial.eval x) (s • p) = s * (MvPolynomial.eval x) p
MvPolynomial.eval_eval₂ 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} [CommSemiring R] {S : Type u_2} {τ : Type u_3} {x : τ → S} [CommSemiring S] (f : R →+* MvPolynomial τ S) (g : σ → MvPolynomial τ S) (p : MvPolynomial σ R) : (MvPolynomial.eval x) (MvPolynomial.eval₂ f g p) = MvPolynomial.eval₂ ((MvPolynomial.eval x).comp f) (fun s => (MvPolynomial.eval x) (g s)) p
MvPolynomial.eval_assoc 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} [CommSemiring R] {τ : Type u_2} (f : σ → MvPolynomial τ R) (g : τ → R) (p : MvPolynomial σ R) : (MvPolynomial.eval (⇑(MvPolynomial.eval g) ∘ f)) p = (MvPolynomial.eval g) (MvPolynomial.eval₂ MvPolynomial.C f p)
MvPolynomial.eval_monomial 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} {a : R} {s : σ →₀ ℕ} [CommSemiring R] {f : σ → R} : (MvPolynomial.eval f) ((MvPolynomial.monomial s) a) = a * s.prod fun n e => f n ^ e
MvPolynomial.eval_mul 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} [CommSemiring R] {p q : MvPolynomial σ R} {f : σ → R} : (MvPolynomial.eval f) (p * q) = (MvPolynomial.eval f) p * (MvPolynomial.eval f) q
MvPolynomial.eval_add 📋 Mathlib.Algebra.MvPolynomial.Eval
{R : Type u} {σ : Type u_1} [CommSemiring R] {p q : MvPolynomial σ R} {f : σ → R} : (MvPolynomial.eval f) (p + q) = (MvPolynomial.eval f) p + (MvPolynomial.eval f) q
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.coe_aeval_eq_eval 📋 Mathlib.Algebra.MvPolynomial.Eval
{S₁ : Type v} {σ : Type u_1} [CommSemiring S₁] (f : σ → S₁) : ↑(MvPolynomial.aeval f) = MvPolynomial.eval f
MvPolynomial.eval_rename 📋 Mathlib.Algebra.MvPolynomial.Rename
{σ : Type u_1} {τ : Type u_2} {R : Type u_4} [CommSemiring R] (k : σ → τ) (g : τ → R) (p : MvPolynomial σ R) : (MvPolynomial.eval g) ((MvPolynomial.rename k) p) = (MvPolynomial.eval (g ∘ k)) p
MvPolynomial.eval_rename_prod_mk 📋 Mathlib.Algebra.MvPolynomial.Rename
{σ : Type u_1} {τ : Type u_2} {R : Type u_4} [CommSemiring R] (g : σ × τ → R) (i : σ) (p : MvPolynomial τ R) : (MvPolynomial.eval g) ((MvPolynomial.rename (Prod.mk i)) p) = (MvPolynomial.eval fun j => g (i, j)) p
MvPolynomial.eval_toMvPolynomial 📋 Mathlib.Algebra.MvPolynomial.Equiv
{R : Type u_1} {σ : Type u_3} [CommSemiring R] (f : σ → R) (i : σ) (p : Polynomial R) : (MvPolynomial.eval f) ((Polynomial.toMvPolynomial i) p) = Polynomial.eval (f i) p
MvPolynomial.coeff_eval_eq_eval_coeff 📋 Mathlib.Algebra.MvPolynomial.Equiv
{R : Type u} (S₁ : Type v) [CommSemiring R] (s' : S₁ → R) (f : Polynomial (MvPolynomial S₁ R)) (i : ℕ) : (Polynomial.map (MvPolynomial.eval s') f).coeff i = (MvPolynomial.eval s') (f.coeff i)
MvPolynomial.optionEquivLeft_elim_eval 📋 Mathlib.Algebra.MvPolynomial.Equiv
(R : Type u) (S₁ : Type v) [CommSemiring R] (s : S₁ → R) (y : R) (f : MvPolynomial (Option S₁) R) : (MvPolynomial.eval fun x => x.elim y s) f = Polynomial.eval y (Polynomial.map (MvPolynomial.eval s) ((MvPolynomial.optionEquivLeft R S₁) f))
MvPolynomial.eval_comp_toMvPolynomial 📋 Mathlib.Algebra.MvPolynomial.Equiv
{R : Type u_1} {σ : Type u_3} [CommSemiring R] (f : σ → R) (i : σ) : (MvPolynomial.eval f).comp ↑(Polynomial.toMvPolynomial i) = Polynomial.evalRingHom (f i)
MvPolynomial.eval_eq_eval_mv_eval' 📋 Mathlib.Algebra.MvPolynomial.Equiv
{R : Type u} [CommSemiring R] {n : ℕ} (s : Fin n → R) (y : R) (f : MvPolynomial (Fin (n + 1)) R) : (MvPolynomial.eval (Fin.cons y s)) f = Polynomial.eval y (Polynomial.map (MvPolynomial.eval s) ((MvPolynomial.finSuccEquiv R n) f))
MvPolynomial.eval_neg 📋 Mathlib.Algebra.MvPolynomial.CommRing
{R : Type u} {σ : Type u_1} [CommRing R] (p : MvPolynomial σ R) (f : σ → R) : (MvPolynomial.eval f) (-p) = -(MvPolynomial.eval f) p
MvPolynomial.eval_sub 📋 Mathlib.Algebra.MvPolynomial.CommRing
{R : Type u} {σ : Type u_1} [CommRing R] (p : MvPolynomial σ R) {q : MvPolynomial σ R} (f : σ → R) : (MvPolynomial.eval f) (p - q) = (MvPolynomial.eval f) p - (MvPolynomial.eval f) q
Matrix.mvPolynomialX_mapMatrix_eval 📋 Mathlib.LinearAlgebra.Matrix.MvPolynomial
{m : Type u_1} {R : Type u_3} [Fintype m] [DecidableEq m] [CommSemiring R] (A : Matrix m m R) : (MvPolynomial.eval fun p => A p.1 p.2).mapMatrix (Matrix.mvPolynomialX m m R) = A
Matrix.eval_det_mvPolynomialX 📋 Mathlib.LinearAlgebra.Matrix.MvPolynomial
(m : Type u_1) (R : Type u_3) [DecidableEq m] [Fintype m] [CommRing R] (s : m × m → R) : (MvPolynomial.eval s) (Matrix.mvPolynomialX m m R).det = (Matrix.of fun i j => s (i, j)).det
Ideal.mem_span_iff_exists_isHomogeneous 📋 Mathlib.RingTheory.MvPolynomial.Homogeneous
{ι : Type u_1} {R : Type u_2} [CommSemiring R] (x : ι → R) (y : R) : y ∈ Ideal.span (Set.range x) ↔ ∃ p, p.IsHomogeneous 1 ∧ (MvPolynomial.eval x) p = y
Ideal.mem_span_pow_iff_exists_isHomogeneous 📋 Mathlib.RingTheory.MvPolynomial.Homogeneous
{ι : Type u_1} {R : Type u_2} [CommSemiring R] {n : ℕ} (x : ι → R) (y : R) : y ∈ Ideal.span (Set.range x) ^ n ↔ ∃ p, p.IsHomogeneous n ∧ (MvPolynomial.eval x) p = y
MvPolynomial.IsHomogeneous.eq_zero_of_forall_eval_eq_zero 📋 Mathlib.RingTheory.MvPolynomial.Homogeneous
{R : Type u_5} {σ : Type u_6} [CommRing R] [IsDomain R] [Infinite R] {F : MvPolynomial σ R} {n : ℕ} (hF : F.IsHomogeneous n) (h : ∀ (r : σ → R), (MvPolynomial.eval r) F = 0) : F = 0
MvPolynomial.IsHomogeneous.eq_zero_of_forall_eval_eq_zero_of_le_card 📋 Mathlib.RingTheory.MvPolynomial.Homogeneous
{R : Type u_5} {σ : Type u_6} [CommRing R] [IsDomain R] {F : MvPolynomial σ R} {n : ℕ} (hF : F.IsHomogeneous n) (h : ∀ (r : σ → R), (MvPolynomial.eval r) F = 0) (hnR : ↑n ≤ Cardinal.mk R) : F = 0
MvPolynomial.IsHomogeneous.funext 📋 Mathlib.RingTheory.MvPolynomial.Homogeneous
{R : Type u_5} {σ : Type u_6} [CommRing R] [IsDomain R] [Infinite R] {F G : MvPolynomial σ R} {n : ℕ} (hF : F.IsHomogeneous n) (hG : G.IsHomogeneous n) (h : ∀ (r : σ → R), (MvPolynomial.eval r) F = (MvPolynomial.eval r) G) : F = G
MvPolynomial.IsHomogeneous.funext_of_le_card 📋 Mathlib.RingTheory.MvPolynomial.Homogeneous
{R : Type u_5} {σ : Type u_6} [CommRing R] [IsDomain R] {F G : MvPolynomial σ R} {n : ℕ} (hF : F.IsHomogeneous n) (hG : G.IsHomogeneous n) (h : ∀ (r : σ → R), (MvPolynomial.eval r) F = (MvPolynomial.eval r) G) (hnR : ↑n ≤ Cardinal.mk R) : F = G
Matrix.toMvPolynomial_eval_eq_apply 📋 Mathlib.Algebra.Module.LinearMap.Polynomial
{m : Type u_1} {n : Type u_2} {R : Type u_4} [Fintype n] [CommSemiring R] (M : Matrix m n R) (i : m) (c : n → R) : (MvPolynomial.eval c) (M.toMvPolynomial i) = M.mulVec c i
LinearMap.isNilRegular_iff_coeff_polyCharpoly_nilRank_ne_zero 📋 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) [Module.Finite R L] [Module.Free R L] (x : L) : φ.IsNilRegular x ↔ (MvPolynomial.eval ⇑(b.repr x)) ((φ.polyCharpoly b).coeff φ.nilRank) ≠ 0
LinearMap.polyCharpoly_map_eq_charpoly 📋 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) (x : L) : Polynomial.map (MvPolynomial.eval ⇑(b.repr x)) (φ.polyCharpoly b) = LinearMap.charpoly (φ x)
LinearMap.polyCharpolyAux_map_eq_charpoly 📋 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) [Module.Finite R M] [Module.Free R M] (x : L) : Polynomial.map (MvPolynomial.eval ⇑(b.repr x)) (φ.polyCharpolyAux b bₘ) = LinearMap.charpoly (φ x)
LinearMap.toMvPolynomial_eval_eq_apply 📋 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) (c : ι₁ →₀ R) : (MvPolynomial.eval ⇑c) (LinearMap.toMvPolynomial b₁ b₂ f i) = (b₂.repr (f (b₁.repr.symm c))) i
LinearMap.polyCharpoly_coeff_eval 📋 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) (x : L) (i : ℕ) : (MvPolynomial.eval ⇑(b.repr x)) ((φ.polyCharpoly b).coeff i) = (LinearMap.charpoly (φ x)).coeff i
LinearMap.polyCharpolyAux_map_eval 📋 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) [Module.Finite R M] [Module.Free R M] (x : ι → R) : Polynomial.map (MvPolynomial.eval x) (φ.polyCharpolyAux b bₘ) = LinearMap.charpoly (φ (b.repr.symm (Finsupp.equivFunOnFinite.symm x)))
LinearMap.polyCharpolyAux_coeff_eval 📋 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) [Module.Finite R M] [Module.Free R M] (x : L) (i : ℕ) : (MvPolynomial.eval ⇑(b.repr x)) ((φ.polyCharpolyAux b bₘ).coeff i) = (LinearMap.charpoly (φ x)).coeff i
LinearMap.polyCharpolyAux_map_eq_toMatrix_charpoly 📋 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) (x : L) : Polynomial.map (MvPolynomial.eval ⇑(b.repr x)) (φ.polyCharpolyAux b bₘ) = ((LinearMap.toMatrix bₘ bₘ) (φ x)).charpoly
LinearMap.polyCharpolyAux_eval_eq_toMatrix_charpoly_coeff 📋 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) (x : L) (i : ℕ) : (MvPolynomial.eval ⇑(b.repr x)) ((φ.polyCharpolyAux b bₘ).coeff i) = ((LinearMap.toMatrix bₘ bₘ) (φ x)).charpoly.coeff i
LieAlgebra.isRegular_iff_coeff_polyCharpoly_rank_ne_zero 📋 Mathlib.Algebra.Lie.Rank
(R : Type u_1) {L : Type u_2} {ι : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] [Fintype ι] (b : Module.Basis ι R L) (x : L) [DecidableEq ι] : LieAlgebra.IsRegular R x ↔ (MvPolynomial.eval ⇑(b.repr x)) (((↑(LieAlgebra.ad R L)).polyCharpoly b).coeff (LieAlgebra.rank R L)) ≠ 0
LieModule.isRegular_iff_coeff_polyCharpoly_rank_ne_zero 📋 Mathlib.Algebra.Lie.Rank
(R : Type u_1) {L : Type u_2} (M : Type u_3) {ι : Type u_4} [CommRing R] [LieRing L] [LieAlgebra R L] [Module.Finite R L] [Module.Free R L] [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M] [Module.Finite R M] [Module.Free R M] [Fintype ι] (b : Module.Basis ι R L) (x : L) [DecidableEq ι] : LieModule.IsRegular R M x ↔ (MvPolynomial.eval ⇑(b.repr x)) (((↑(LieModule.toEnd R L M)).polyCharpoly b).coeff (LieModule.rank R L M)) ≠ 0
MvPolynomial.expand_zero_apply 📋 Mathlib.Algebra.MvPolynomial.Expand
{σ : Type u_1} {R : Type u_3} [CommSemiring R] (f : MvPolynomial σ R) : (MvPolynomial.expand 0) f = MvPolynomial.C ((MvPolynomial.eval 1) f)
MvPolynomial.eval_expand 📋 Mathlib.Algebra.MvPolynomial.Expand
{σ : Type u_1} {R : Type u_3} [CommSemiring R] (p : ℕ) (f : σ → R) (φ : MvPolynomial σ R) : (MvPolynomial.eval f) ((MvPolynomial.expand p) φ) = (MvPolynomial.eval (f ^ p)) φ
MvPolynomial.eval_polynomial_eval_finSuccEquiv 📋 Mathlib.Algebra.MvPolynomial.Polynomial
{R : Type u_1} {n : ℕ} {x : Fin n → R} [CommSemiring R] (f : MvPolynomial (Fin (n + 1)) R) (q : MvPolynomial (Fin n) R) : (MvPolynomial.eval x) (Polynomial.eval q ((MvPolynomial.finSuccEquiv R n) f)) = (MvPolynomial.eval fun i => Fin.cases ((MvPolynomial.eval x) q) x i) f
MvPolynomial.funext 📋 Mathlib.Algebra.MvPolynomial.Funext
{R : Type u_1} [CommRing R] [IsDomain R] [Infinite R] {σ : Type u_2} {p q : MvPolynomial σ R} (h : ∀ (x : σ → R), (MvPolynomial.eval x) p = (MvPolynomial.eval x) q) : p = q
MvPolynomial.funext_iff 📋 Mathlib.Algebra.MvPolynomial.Funext
{R : Type u_1} [CommRing R] [IsDomain R] [Infinite R] {σ : Type u_2} {p q : MvPolynomial σ R} : p = q ↔ ∀ (x : σ → R), (MvPolynomial.eval x) p = (MvPolynomial.eval x) q
MvPolynomial.funext_set 📋 Mathlib.Algebra.MvPolynomial.Funext
{R : Type u_1} [CommRing R] [IsDomain R] {σ : Type u_2} {p q : MvPolynomial σ R} (s : σ → Set R) (hs : ∀ (i : σ), (s i).Infinite) (h : ∀ x ∈ Set.univ.pi s, (MvPolynomial.eval x) p = (MvPolynomial.eval x) q) : p = q
MvPolynomial.funext_set_iff 📋 Mathlib.Algebra.MvPolynomial.Funext
{R : Type u_1} [CommRing R] [IsDomain R] {σ : Type u_2} {p q : MvPolynomial σ R} (s : σ → Set R) (hs : ∀ (i : σ), (s i).Infinite) : p = q ↔ ∀ x ∈ Set.univ.pi s, (MvPolynomial.eval x) p = (MvPolynomial.eval x) q
MvPolynomial.schwartz_zippel_totalDegree 📋 Mathlib.Algebra.MvPolynomial.SchwartzZippel
{R : Type u_1} [CommRing R] [IsDomain R] [DecidableEq R] {n : ℕ} {p : MvPolynomial (Fin n) R} (hp : p ≠ 0) (S : Finset R) : ↑{f ∈ Fintype.piFinset fun x => S | (MvPolynomial.eval f) p = 0}.card / ↑S.card ^ n ≤ ↑p.totalDegree / ↑S.card
MvPolynomial.schwartz_zippel_sum_degreeOf 📋 Mathlib.Algebra.MvPolynomial.SchwartzZippel
{R : Type u_1} [CommRing R] [IsDomain R] [DecidableEq R] {n : ℕ} {p : MvPolynomial (Fin n) R} (hp : p ≠ 0) (S : Fin n → Finset R) : ↑{x ∈ Fintype.piFinset fun i => S i | (MvPolynomial.eval x) p = 0}.card / ∏ i, ↑(S i).card ≤ ∑ i, ↑(MvPolynomial.degreeOf i p) / ↑(S i).card
MvPolynomial.schwartz_zippel_sup_sum 📋 Mathlib.Algebra.MvPolynomial.SchwartzZippel
{R : Type u_1} [CommRing R] [IsDomain R] [DecidableEq R] {n : ℕ} {p : MvPolynomial (Fin n) R} (hp : p ≠ 0) (S : Fin n → Finset R) : ↑{x ∈ Fintype.piFinset fun i => S i | (MvPolynomial.eval x) p = 0}.card / ∏ i, ↑(S i).card ≤ p.support.sup fun s => ∑ i, ↑(s i) / ↑(S i).card
Polynomial.eval_homogenize 📋 Mathlib.Algebra.Polynomial.Homogenize
{K : Type u_1} [Semifield K] {p : Polynomial K} {n : ℕ} (hn : p.natDegree ≤ n) (x : Fin 2 → K) (hx : x 1 ≠ 0) : (MvPolynomial.eval x) (p.homogenize n) = Polynomial.eval (x 0 / x 1) p * x 1 ^ n
Polynomial.eval_eq_div_eval_toTupleMvPolynomial 📋 Mathlib.Algebra.Polynomial.Homogenize
{R : Type u_2} [Field R] (p : Polynomial R) (x : R) : Polynomial.eval x p = (MvPolynomial.eval ![x, 1]) (p.toTupleMvPolynomial 0) / (MvPolynomial.eval ![x, 1]) (p.toTupleMvPolynomial 1)
WeierstrassCurve.addSubMap_ne_zero 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Affine.AddSubMap
{R : Type u_1} [CommRing R] (W : WeierstrassCurve R) [W.IsElliptic] [IsReduced R] {x : Fin 3 → R} (hx : x ≠ 0) : (fun i => (MvPolynomial.eval x) (W.addSubMap i)) ≠ 0
WeierstrassCurve.addSubMapCoeff_condition 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Affine.AddSubMap
{R : Type u_1} [CommRing R] (W : WeierstrassCurve R) [W.IsElliptic] (x : Fin 3 → R) (i : Fin 3) : ∑ j, (MvPolynomial.eval x) (MvPolynomial.C ↑W.Δ'⁻¹ * W.addSubMapCoeff (i, j)) * (MvPolynomial.eval x) (W.addSubMap j) = x i ^ 4
WeierstrassCurve.Jacobian.eval_polynomialY 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic
{R : Type r} [CommRing R] {W' : WeierstrassCurve.Jacobian R} (P : Fin 3 → R) : (MvPolynomial.eval P) W'.polynomialY = 2 * P 1 + W'.a₁ * P 0 * P 2 + W'.a₃ * P 2 ^ 3
WeierstrassCurve.Jacobian.nonsingular_iff_of_Z_ne_zero 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic
{F : Type u} [Field F] {W : WeierstrassCurve.Jacobian F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) : W.Nonsingular P ↔ W.Equation P ∧ ((MvPolynomial.eval P) W.polynomialX ≠ 0 ∨ (MvPolynomial.eval P) W.polynomialY ≠ 0)
WeierstrassCurve.Jacobian.eval_polynomialX_of_Z_ne_zero 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic
{F : Type u} [Field F] {W : WeierstrassCurve.Jacobian F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) : (MvPolynomial.eval P) W.polynomialX / P 2 ^ 4 = Polynomial.evalEval (P 0 / P 2 ^ 2) (P 1 / P 2 ^ 3) W.toAffine.polynomialX
WeierstrassCurve.Jacobian.eval_polynomialY_of_Z_ne_zero 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic
{F : Type u} [Field F] {W : WeierstrassCurve.Jacobian F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) : (MvPolynomial.eval P) W.polynomialY / P 2 ^ 3 = Polynomial.evalEval (P 0 / P 2 ^ 2) (P 1 / P 2 ^ 3) W.toAffine.polynomialY
WeierstrassCurve.Jacobian.eval_polynomial_of_Z_ne_zero 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic
{F : Type u} [Field F] {W : WeierstrassCurve.Jacobian F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) : (MvPolynomial.eval P) W.polynomial / P 2 ^ 6 = Polynomial.evalEval (P 0 / P 2 ^ 2) (P 1 / P 2 ^ 3) W.toAffine.polynomial
WeierstrassCurve.Jacobian.eval_polynomialX 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic
{R : Type r} [CommRing R] {W' : WeierstrassCurve.Jacobian R} (P : Fin 3 → R) : (MvPolynomial.eval P) W'.polynomialX = W'.a₁ * P 1 * P 2 - (3 * P 0 ^ 2 + 2 * W'.a₂ * P 0 * P 2 ^ 2 + W'.a₄ * P 2 ^ 4)
WeierstrassCurve.Jacobian.eval_polynomialZ 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic
{R : Type r} [CommRing R] {W' : WeierstrassCurve.Jacobian R} (P : Fin 3 → R) : (MvPolynomial.eval P) W'.polynomialZ = W'.a₁ * P 0 * P 1 + 3 * W'.a₃ * P 1 * P 2 ^ 2 - (2 * W'.a₂ * P 0 ^ 2 * P 2 + 4 * W'.a₄ * P 0 * P 2 ^ 3 + 6 * W'.a₆ * P 2 ^ 5)
WeierstrassCurve.Jacobian.polynomial_relation 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic
{R : Type r} [CommRing R] {W' : WeierstrassCurve.Jacobian R} (P : Fin 3 → R) : 6 * (MvPolynomial.eval P) W'.polynomial = 2 * P 0 * (MvPolynomial.eval P) W'.polynomialX + 3 * P 1 * (MvPolynomial.eval P) W'.polynomialY + P 2 * (MvPolynomial.eval P) W'.polynomialZ
WeierstrassCurve.Jacobian.eval_polynomial 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic
{R : Type r} [CommRing R] {W' : WeierstrassCurve.Jacobian R} (P : Fin 3 → R) : (MvPolynomial.eval P) W'.polynomial = P 1 ^ 2 + W'.a₁ * P 0 * P 1 * P 2 + W'.a₃ * P 1 * P 2 ^ 3 - (P 0 ^ 3 + W'.a₂ * P 0 ^ 2 * P 2 ^ 2 + W'.a₄ * P 0 * P 2 ^ 4 + W'.a₆ * P 2 ^ 6)
WeierstrassCurve.Jacobian.nonsingular_iff_of_Y_eq_negY 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula
{F : Type u} [Field F] {W : WeierstrassCurve.Jacobian F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) (hy : P 1 = W.negY P) : W.Nonsingular P ↔ W.Equation P ∧ (MvPolynomial.eval P) W.polynomialX ≠ 0
WeierstrassCurve.Projective.eval_polynomialX_of_Z_ne_zero 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic
{F : Type u} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) : (MvPolynomial.eval P) W.polynomialX / P 2 ^ 2 = Polynomial.evalEval (P 0 / P 2) (P 1 / P 2) W.toAffine.polynomialX
WeierstrassCurve.Projective.eval_polynomialY_of_Z_ne_zero 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic
{F : Type u} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) : (MvPolynomial.eval P) W.polynomialY / P 2 ^ 2 = Polynomial.evalEval (P 0 / P 2) (P 1 / P 2) W.toAffine.polynomialY
WeierstrassCurve.Projective.eval_polynomial_of_Z_ne_zero 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic
{F : Type u} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) : (MvPolynomial.eval P) W.polynomial / P 2 ^ 3 = Polynomial.evalEval (P 0 / P 2) (P 1 / P 2) W.toAffine.polynomial
WeierstrassCurve.Projective.nonsingular_iff_of_Z_ne_zero 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic
{F : Type u} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) : W.Nonsingular P ↔ W.Equation P ∧ ((MvPolynomial.eval P) W.polynomialX ≠ 0 ∨ (MvPolynomial.eval P) W.polynomialY ≠ 0)
WeierstrassCurve.Projective.eval_polynomialY 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic
{R : Type r} [CommRing R] {W' : WeierstrassCurve.Projective R} (P : Fin 3 → R) : (MvPolynomial.eval P) W'.polynomialY = 2 * P 1 * P 2 + W'.a₁ * P 0 * P 2 + W'.a₃ * P 2 ^ 2
WeierstrassCurve.Projective.eval_polynomialX 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic
{R : Type r} [CommRing R] {W' : WeierstrassCurve.Projective R} (P : Fin 3 → R) : (MvPolynomial.eval P) W'.polynomialX = W'.a₁ * P 1 * P 2 - (3 * P 0 ^ 2 + 2 * W'.a₂ * P 0 * P 2 + W'.a₄ * P 2 ^ 2)
WeierstrassCurve.Projective.polynomial_relation 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic
{R : Type r} [CommRing R] {W' : WeierstrassCurve.Projective R} (P : Fin 3 → R) : 3 * (MvPolynomial.eval P) W'.polynomial = P 0 * (MvPolynomial.eval P) W'.polynomialX + P 1 * (MvPolynomial.eval P) W'.polynomialY + P 2 * (MvPolynomial.eval P) W'.polynomialZ
WeierstrassCurve.Projective.eval_polynomialZ 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic
{R : Type r} [CommRing R] {W' : WeierstrassCurve.Projective R} (P : Fin 3 → R) : (MvPolynomial.eval P) W'.polynomialZ = P 1 ^ 2 + W'.a₁ * P 0 * P 1 + 2 * W'.a₃ * P 1 * P 2 - (W'.a₂ * P 0 ^ 2 + 2 * W'.a₄ * P 0 * P 2 + 3 * W'.a₆ * P 2 ^ 2)
WeierstrassCurve.Projective.eval_polynomial 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic
{R : Type r} [CommRing R] {W' : WeierstrassCurve.Projective R} (P : Fin 3 → R) : (MvPolynomial.eval P) W'.polynomial = P 1 ^ 2 * P 2 + W'.a₁ * P 0 * P 1 * P 2 + W'.a₃ * P 1 * P 2 ^ 2 - (P 0 ^ 3 + W'.a₂ * P 0 ^ 2 * P 2 + W'.a₄ * P 0 * P 2 ^ 2 + W'.a₆ * P 2 ^ 3)
WeierstrassCurve.Projective.nonsingular_iff_of_Y_eq_negY 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula
{F : Type u} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hPz : P 2 ≠ 0) (hy : P 1 = W.negY P) : W.Nonsingular P ↔ W.Equation P ∧ (MvPolynomial.eval P) W.polynomialX ≠ 0
WeierstrassCurve.Projective.negDblY_of_Y_eq' 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula
{R : Type r} [CommRing R] {W' : WeierstrassCurve.Projective R} [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 = Q 1 * P 2) (hy' : P 1 * Q 2 = W'.negY Q * P 2) : W'.negDblY P * P 2 ^ 2 = -(MvPolynomial.eval P) W'.polynomialX ^ 3
WeierstrassCurve.Projective.dblY_of_Y_eq' 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula
{R : Type r} [CommRing R] {W' : WeierstrassCurve.Projective R} [NoZeroDivisors R] {P Q : Fin 3 → R} (hP : W'.Equation P) (hPz : P 2 ≠ 0) (hQz : Q 2 ≠ 0) (hx : P 0 * Q 2 = Q 0 * P 2) (hy : P 1 * Q 2 = Q 1 * P 2) (hy' : P 1 * Q 2 = W'.negY Q * P 2) : W'.dblY P * P 2 ^ 2 = (MvPolynomial.eval P) W'.polynomialX ^ 3
WeierstrassCurve.Projective.dblX_eq' 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula
{R : Type r} [CommRing R] {W' : WeierstrassCurve.Projective R} {P : Fin 3 → R} (hP : W'.Equation P) : W'.dblX P * P 2 = ((MvPolynomial.eval P) W'.polynomialX ^ 2 - W'.a₁ * (MvPolynomial.eval P) W'.polynomialX * P 2 * (P 1 - W'.negY P) - W'.a₂ * P 2 ^ 2 * (P 1 - W'.negY P) ^ 2 - 2 * P 0 * P 2 * (P 1 - W'.negY P) ^ 2) * (P 1 - W'.negY P)
WeierstrassCurve.Projective.dblX_eq 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula
{F : Type u} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hP : W.Equation P) (hPz : P 2 ≠ 0) : W.dblX P = ((MvPolynomial.eval P) W.polynomialX ^ 2 - W.a₁ * (MvPolynomial.eval P) W.polynomialX * P 2 * (P 1 - W.negY P) - W.a₂ * P 2 ^ 2 * (P 1 - W.negY P) ^ 2 - 2 * P 0 * P 2 * (P 1 - W.negY P) ^ 2) * (P 1 - W.negY P) / P 2
WeierstrassCurve.Projective.negDblY_eq' 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula
{R : Type r} [CommRing R] {W' : WeierstrassCurve.Projective R} {P : Fin 3 → R} (hP : W'.Equation P) : W'.negDblY P * P 2 ^ 2 = -(MvPolynomial.eval P) W'.polynomialX * ((MvPolynomial.eval P) W'.polynomialX ^ 2 - W'.a₁ * (MvPolynomial.eval P) W'.polynomialX * P 2 * (P 1 - W'.negY P) - W'.a₂ * P 2 ^ 2 * (P 1 - W'.negY P) ^ 2 - 2 * P 0 * P 2 * (P 1 - W'.negY P) ^ 2 - P 0 * P 2 * (P 1 - W'.negY P) ^ 2) + P 1 * P 2 ^ 2 * (P 1 - W'.negY P) ^ 3
WeierstrassCurve.Projective.negDblY_eq 📋 Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula
{F : Type u} [Field F] {W : WeierstrassCurve.Projective F} {P : Fin 3 → F} (hP : W.Equation P) (hPz : P 2 ≠ 0) : W.negDblY P = (-(MvPolynomial.eval P) W.polynomialX * ((MvPolynomial.eval P) W.polynomialX ^ 2 - W.a₁ * (MvPolynomial.eval P) W.polynomialX * P 2 * (P 1 - W.negY P) - W.a₂ * P 2 ^ 2 * (P 1 - W.negY P) ^ 2 - 2 * P 0 * P 2 * (P 1 - W.negY P) ^ 2 - P 0 * P 2 * (P 1 - W.negY P) ^ 2) + P 1 * P 2 ^ 2 * (P 1 - W.negY P) ^ 3) / P 2 ^ 2
_private.Mathlib.RingTheory.Smooth.NoetherianDescent.0.Algebra.Smooth.DescentAux.hq 📋 Mathlib.RingTheory.Smooth.NoetherianDescent
{A : Type u} {B : Type u_2} [CommRing A] [CommRing B] [Algebra A B] (self : Algebra.Smooth.DescentAux✝ A B) (i : Algebra.Smooth.DescentAux.vars✝ self) : (MvPolynomial.eval (Algebra.Smooth.DescentAux.P✝ self).relation) (Algebra.Smooth.DescentAux.q✝ self i) = Algebra.Smooth.DescentAux.h✝ self i - MvPolynomial.X i
_private.Mathlib.RingTheory.Smooth.NoetherianDescent.0.Algebra.Smooth.DescentAux.hp 📋 Mathlib.RingTheory.Smooth.NoetherianDescent
{A : Type u} {B : Type u_2} [CommRing A] [CommRing B] [Algebra A B] (self : Algebra.Smooth.DescentAux✝ A B) (j : Algebra.Smooth.DescentAux.rels✝ self) : (MvPolynomial.eval (Algebra.Smooth.DescentAux.P✝ self).relation) (Algebra.Smooth.DescentAux.p✝ self j) = (MvPolynomial.aeval (Algebra.Smooth.DescentAux.h✝ self)) ((Algebra.Smooth.DescentAux.P✝ self).relation j)
AnalyticOnNhd.eval_mvPolynomial 📋 Mathlib.Analysis.Analytic.Polynomial
{𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {σ : Type u_5} [CompleteSpace 𝕜] [Fintype σ] (p : MvPolynomial σ 𝕜) : AnalyticOnNhd 𝕜 (fun x => (MvPolynomial.eval x) p) Set.univ
AnalyticOnNhd.eval_linearMap' 📋 Mathlib.Analysis.Analytic.Polynomial
{𝕜 : Type u_1} {E : Type u_2} {B : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedCommRing B] [NormedAlgebra 𝕜 B] {σ : Type u_5} [CompleteSpace 𝕜] [T2Space E] [FiniteDimensional 𝕜 E] (f : σ → E →ₗ[𝕜] B) (p : MvPolynomial σ B) : AnalyticOnNhd 𝕜 (fun x => (MvPolynomial.eval fun x_1 => (f x_1) x) p) Set.univ
AnalyticOnNhd.eval_continuousLinearMap' 📋 Mathlib.Analysis.Analytic.Polynomial
{𝕜 : Type u_1} {E : Type u_2} {B : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedCommRing B] [NormedAlgebra 𝕜 B] {σ : Type u_5} (f : σ → E →L[𝕜] B) (p : MvPolynomial σ B) : AnalyticOnNhd 𝕜 (fun x => (MvPolynomial.eval fun x_1 => (f x_1) x) p) Set.univ
AnalyticOnNhd.eval_linearMap 📋 Mathlib.Analysis.Analytic.Polynomial
{𝕜 : Type u_1} {E : Type u_2} {B : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedCommRing B] [NormedAlgebra 𝕜 B] {σ : Type u_5} [CompleteSpace 𝕜] [T2Space E] [FiniteDimensional 𝕜 E] (f : E →ₗ[𝕜] σ → B) (p : MvPolynomial σ B) : AnalyticOnNhd 𝕜 (fun x => (MvPolynomial.eval (f x)) p) Set.univ
AnalyticOnNhd.eval_continuousLinearMap 📋 Mathlib.Analysis.Analytic.Polynomial
{𝕜 : Type u_1} {E : Type u_2} {B : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedCommRing B] [NormedAlgebra 𝕜 B] {σ : Type u_5} (f : E →L[𝕜] σ → B) (p : MvPolynomial σ B) : AnalyticOnNhd 𝕜 (fun x => (MvPolynomial.eval (f x)) p) Set.univ
MvPolynomial.sum_eval_eq_zero 📋 Mathlib.FieldTheory.ChevalleyWarning
{K : Type u_1} {σ : Type u_2} [Fintype K] [Field K] [Fintype σ] [DecidableEq σ] (f : MvPolynomial σ K) (h : f.totalDegree < (Fintype.card K - 1) * Fintype.card σ) : ∑ x, (MvPolynomial.eval x) f = 0
char_dvd_card_solutions 📋 Mathlib.FieldTheory.ChevalleyWarning
{K : Type u_1} {σ : Type u_2} [Fintype K] [Field K] [Fintype σ] [DecidableEq σ] [DecidableEq K] (p : ℕ) [CharP K p] {f : MvPolynomial σ K} (h : f.totalDegree < Fintype.card σ) : p ∣ Fintype.card { x // (MvPolynomial.eval x) f = 0 }
char_dvd_card_solutions_of_fintype_sum_lt 📋 Mathlib.FieldTheory.ChevalleyWarning
{K : Type u_1} {σ : Type u_2} {ι : Type u_3} [Fintype K] [Field K] [Fintype σ] [DecidableEq σ] [DecidableEq K] (p : ℕ) [CharP K p] [Fintype ι] {f : ι → MvPolynomial σ K} (h : ∑ i, (f i).totalDegree < Fintype.card σ) : p ∣ Fintype.card { x // ∀ (i : ι), (MvPolynomial.eval x) (f i) = 0 }
char_dvd_card_solutions_of_sum_lt 📋 Mathlib.FieldTheory.ChevalleyWarning
{K : Type u_1} {σ : Type u_2} {ι : Type u_3} [Fintype K] [Field K] [Fintype σ] [DecidableEq σ] [DecidableEq K] (p : ℕ) [CharP K p] {s : Finset ι} {f : ι → MvPolynomial σ K} (h : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ) : p ∣ Fintype.card { x // ∀ i ∈ s, (MvPolynomial.eval x) (f i) = 0 }
char_dvd_card_solutions_of_add_lt 📋 Mathlib.FieldTheory.ChevalleyWarning
{K : Type u_1} {σ : Type u_2} [Fintype K] [Field K] [Fintype σ] [DecidableEq σ] [DecidableEq K] (p : ℕ) [CharP K p] {f₁ f₂ : MvPolynomial σ K} (h : f₁.totalDegree + f₂.totalDegree < Fintype.card σ) : p ∣ Fintype.card { x // (MvPolynomial.eval x) f₁ = 0 ∧ (MvPolynomial.eval x) f₂ = 0 }
MvPolynomial.eq_zero_of_eval_zero_at_prod_finset 📋 Mathlib.Combinatorics.Nullstellensatz
{R : Type u_1} [CommRing R] {σ : Type u_2} [Finite σ] [IsDomain R] (P : MvPolynomial σ R) (S : σ → Finset R) (Hdeg : ∀ (i : σ), MvPolynomial.degreeOf i P < (S i).card) (Heval : ∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (MvPolynomial.eval x) P = 0) : P = 0
MvPolynomial.combinatorial_nullstellensatz_exists_eval_nonzero 📋 Mathlib.Combinatorics.Nullstellensatz
{R : Type u_1} [CommRing R] {σ : Type u_2} [Finite σ] [IsDomain R] (f : MvPolynomial σ R) (t : σ →₀ ℕ) (ht : f.coeff t ≠ 0) (ht' : f.totalDegree = Finsupp.degree t) (S : σ → Finset R) (htS : ∀ (i : σ), t i < (S i).card) : ∃ s, (∀ (i : σ), s i ∈ S i) ∧ (MvPolynomial.eval s) f ≠ 0
MvPolynomial.combinatorial_nullstellensatz_exists_linearCombination 📋 Mathlib.Combinatorics.Nullstellensatz
{R : Type u_1} [CommRing R] {σ : Type u_2} [Finite σ] [IsDomain R] (S : σ → Finset R) (Sne : ∀ (i : σ), (S i).Nonempty) (f : MvPolynomial σ R) (Heval : ∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (MvPolynomial.eval x) f = 0) : ∃ h, (∀ (i : σ), ((∏ s ∈ S i, (MvPolynomial.X i - MvPolynomial.C s)) * h i).totalDegree ≤ f.totalDegree) ∧ f = (Finsupp.linearCombination (MvPolynomial σ R) fun i => ∏ r ∈ S i, (MvPolynomial.X i - MvPolynomial.C r)) h
FirstOrder.Ring.lift_genericPolyMap 📋 Mathlib.RingTheory.MvPolynomial.FreeCommRing
{ι : Type u_1} {κ : Type u_2} {R : Type u_3} [DecidableEq κ] [CommRing R] [DecidableEq R] (monoms : ι → Finset (κ →₀ ℕ)) (f : (i : ι) × ↥(monoms i) ⊕ κ → R) (i : ι) : (FreeCommRing.lift f) (FirstOrder.Ring.genericPolyMap monoms i) = (MvPolynomial.eval (f ∘ Sum.inr)) (↑((FirstOrder.Ring.mvPolynomialSupportLEEquiv monoms).symm (f ∘ Sum.inl)) i)
ax_grothendieck_univ 📋 Mathlib.FieldTheory.AxGrothendieck
{K : Type u_1} {ι : Type u_2} [Field K] [IsAlgClosed K] [Finite ι] (p : ι → MvPolynomial ι K) : (Function.Injective fun v i => (MvPolynomial.eval v) (p i)) → Function.Surjective fun v i => (MvPolynomial.eval v) (p i)
ax_grothendieck_of_locally_finite 📋 Mathlib.FieldTheory.AxGrothendieck
{ι : Type u_1} {K : Type u_2} {R : Type u_3} [Field K] [Finite K] [CommRing R] [Finite ι] [Algebra K R] [alg : Algebra.IsAlgebraic K R] (ps : ι → MvPolynomial ι R) (S : Set (ι → R)) (hm : Set.MapsTo (fun v i => (MvPolynomial.eval v) (ps i)) S S) (hinj : Set.InjOn (fun v i => (MvPolynomial.eval v) (ps i)) S) : Set.SurjOn (fun v i => (MvPolynomial.eval v) (ps i)) S S
ax_grothendieck_zeroLocus 📋 Mathlib.FieldTheory.AxGrothendieck
{K : Type u_1} {ι : Type u_2} [Field K] [IsAlgClosed K] [Finite ι] (I : Ideal (MvPolynomial ι K)) (p : ι → MvPolynomial ι K) : have S := MvPolynomial.zeroLocus K I; Set.MapsTo (fun v i => (MvPolynomial.eval v) (p i)) S S → Set.InjOn (fun v i => (MvPolynomial.eval v) (p i)) S → Set.SurjOn (fun v i => (MvPolynomial.eval v) (p i)) S S
FirstOrder.realize_genericPolyMapSurjOnOfInjOn 📋 Mathlib.FieldTheory.AxGrothendieck
{ι : Type u_1} {α : Type u_2} [Finite α] {K : Type u_3} [Field K] [FirstOrder.Ring.CompatibleRing K] [Finite ι] (φ : FirstOrder.Language.ring.Formula (α ⊕ ι)) (mons : ι → Finset (ι →₀ ℕ)) : K ⊨ FirstOrder.genericPolyMapSurjOnOfInjOn φ mons ↔ ∀ (v : α → K) (p : { p // ∀ (i : ι), (p i).support ⊆ mons i }), have f := fun v i => (MvPolynomial.eval v) (↑p i); have S := {x | φ.Realize (Sum.elim v x)}; Set.MapsTo f S S → Set.InjOn f S → Set.SurjOn f S S
ax_grothendieck_of_definable 📋 Mathlib.FieldTheory.AxGrothendieck
{K : Type u_1} {ι : Type u_2} [Field K] [IsAlgClosed K] [Finite ι] [FirstOrder.Ring.CompatibleRing K] {c : Set K} (S : Set (ι → K)) (hS : c.Definable FirstOrder.Language.ring S) (ps : ι → MvPolynomial ι K) : Set.MapsTo (fun v i => (MvPolynomial.eval v) (ps i)) S S → Set.InjOn (fun v i => (MvPolynomial.eval v) (ps i)) S → Set.SurjOn (fun v i => (MvPolynomial.eval v) (ps i)) S S
MvPolynomial.eval_indicator_apply_eq_one 📋 Mathlib.FieldTheory.Finite.Polynomial
{K : Type u_1} {σ : Type u_2} [Fintype K] [Fintype σ] [CommRing K] (a : σ → K) : (MvPolynomial.eval a) (MvPolynomial.indicator a) = 1
MvPolynomial.eval_indicator_apply_eq_zero 📋 Mathlib.FieldTheory.Finite.Polynomial
{K : Type u_1} {σ : Type u_2} [Fintype K] [Fintype σ] [Field K] (a b : σ → K) (h : a ≠ b) : (MvPolynomial.eval a) (MvPolynomial.indicator b) = 0
MvPolynomial.evalₗ_apply 📋 Mathlib.FieldTheory.Finite.Polynomial
(K : Type u_1) (σ : Type u_2) [CommSemiring K] (p : MvPolynomial σ K) (e : σ → K) : (MvPolynomial.evalₗ K σ) p e = (MvPolynomial.eval e) p
MvPolynomial.eq_zero_of_eval_eq_zero 📋 Mathlib.FieldTheory.Finite.Polynomial
(σ K : Type u) [Fintype K] [Field K] [Finite σ] (p : MvPolynomial σ K) (h : ∀ (v : σ → K), (MvPolynomial.eval v) p = 0) (hp : p ∈ MvPolynomial.restrictDegree σ K (Fintype.card K - 1)) : p = 0
MvPolynomial.eq_of_eval_eq_on_gl 📋 Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.MvPolynomial
{m : Type u_1} {k : Type u_2} [Fintype m] [DecidableEq m] [Field k] [Infinite k] {p q : MvPolynomial (m × m) k} (h : ∀ (g : GL m k), (MvPolynomial.eval fun ij => ↑g ij.1 ij.2) p = (MvPolynomial.eval fun ij => ↑g ij.1 ij.2) q) : p = q
Height.logHeight_eval_le' 📋 Mathlib.NumberTheory.Height.MvPolynomial
{K : Type u_4} [Field K] {ι : Type u_5} {ι' : Type u_6} [Height.AdmissibleAbsValues K] [Finite ι'] [Finite ι] {N : ℕ} {p : ι' → MvPolynomial ι K} (hp : ∀ (i : ι'), (p i).IsHomogeneous N) : ∃ C, ∀ (x : ι → K), (Height.logHeight fun j => (MvPolynomial.eval x) (p j)) ≤ C + ↑N * Height.logHeight x
Height.logHeight_eval_le 📋 Mathlib.NumberTheory.Height.MvPolynomial
{K : Type u_4} [Field K] {ι : Type u_5} {ι' : Type u_6} [Height.AdmissibleAbsValues K] [Finite ι'] [Finite ι] {N : ℕ} {p : ι' → MvPolynomial ι K} (hp : ∀ (i : ι'), (p i).IsHomogeneous N) (x : ι → K) : (Height.logHeight fun j => (MvPolynomial.eval x) (p j)) ≤ Real.log (max (Height.mulHeightBound p) 1) + ↑N * Height.logHeight x
Height.mulHeight_eval_le' 📋 Mathlib.NumberTheory.Height.MvPolynomial
{K : Type u_4} [Field K] {ι : Type u_5} {ι' : Type u_6} [Height.AdmissibleAbsValues K] [Finite ι'] [Finite ι] {N : ℕ} {p : ι' → MvPolynomial ι K} (hp : ∀ (i : ι'), (p i).IsHomogeneous N) : ∃ C > 0, ∀ (x : ι → K), (Height.mulHeight fun j => (MvPolynomial.eval x) (p j)) ≤ C * Height.mulHeight x ^ N
Height.mulHeight_eval_le 📋 Mathlib.NumberTheory.Height.MvPolynomial
{K : Type u_4} [Field K] {ι : Type u_5} {ι' : Type u_6} [Height.AdmissibleAbsValues K] [Finite ι'] [Finite ι] {N : ℕ} {p : ι' → MvPolynomial ι K} (hp : ∀ (i : ι'), (p i).IsHomogeneous N) (x : ι → K) : (Height.mulHeight fun j => (MvPolynomial.eval x) (p j)) ≤ max (Height.mulHeightBound p) 1 * Height.mulHeight x ^ N
AbsoluteValue.eval_mvPolynomial_le 📋 Mathlib.NumberTheory.Height.MvPolynomial
{K : Type u_4} [Field K] {ι : Type u_5} [Finite ι] (v : AbsoluteValue K ℝ) {p : MvPolynomial ι K} {N : ℕ} (hp : p.IsHomogeneous N) (x : ι → K) : v ((MvPolynomial.eval x) p) ≤ (p.coeff.sum fun x c => v c) * (⨆ i, v (x i)) ^ N
Height.logHeight_eval_ge' 📋 Mathlib.NumberTheory.Height.MvPolynomial
{K : Type u_6} [Field K] {ι : Type u_7} {ι' : Type u_8} [Fintype ι'] [Height.AdmissibleAbsValues K] [Finite ι] {M N : ℕ} {q : ι × ι' → MvPolynomial ι K} (hq : ∀ (a : ι × ι'), (q a).IsHomogeneous M) : ∃ C, ∀ (p : ι' → MvPolynomial ι K) {x : ι → K}, (∀ (k : ι), ∑ j, (MvPolynomial.eval x) (q (k, j)) * (MvPolynomial.eval x) (p j) = x k ^ (M + N)) → C + ↑N * Height.logHeight x ≤ Height.logHeight fun j => (MvPolynomial.eval x) (p j)
Height.mulHeight_eval_ge' 📋 Mathlib.NumberTheory.Height.MvPolynomial
{K : Type u_6} [Field K] {ι : Type u_7} {ι' : Type u_8} [Fintype ι'] [Height.AdmissibleAbsValues K] [Finite ι] {M N : ℕ} {q : ι × ι' → MvPolynomial ι K} (hq : ∀ (a : ι × ι'), (q a).IsHomogeneous M) : ∃ C > 0, ∀ (p : ι' → MvPolynomial ι K) {x : ι → K}, (∀ (k : ι), ∑ j, (MvPolynomial.eval x) (q (k, j)) * (MvPolynomial.eval x) (p j) = x k ^ (M + N)) → C * Height.mulHeight x ^ N ≤ Height.mulHeight fun j => (MvPolynomial.eval x) (p j)
Height.logHeight_eval_ge 📋 Mathlib.NumberTheory.Height.MvPolynomial
{K : Type u_6} [Field K] {ι : Type u_7} {ι' : Type u_8} [Fintype ι'] [Height.AdmissibleAbsValues K] [Finite ι] {M N : ℕ} {q : ι × ι' → MvPolynomial ι K} (hq : ∀ (a : ι × ι'), (q a).IsHomogeneous M) (p : ι' → MvPolynomial ι K) {x : ι → K} (h : ∀ (k : ι), ∑ j, (MvPolynomial.eval x) (q (k, j)) * (MvPolynomial.eval x) (p j) = x k ^ (M + N)) : -Real.log (↑(Nat.card ι') ^ Height.totalWeight K * max (Height.mulHeightBound q) 1) + ↑N * Height.logHeight x ≤ Height.logHeight fun j => (MvPolynomial.eval x) (p j)
Height.mulHeight_eval_ge 📋 Mathlib.NumberTheory.Height.MvPolynomial
{K : Type u_6} [Field K] {ι : Type u_7} {ι' : Type u_8} [Fintype ι'] [Height.AdmissibleAbsValues K] [Finite ι] {M N : ℕ} {q : ι × ι' → MvPolynomial ι K} (hq : ∀ (a : ι × ι'), (q a).IsHomogeneous M) (p : ι' → MvPolynomial ι K) {x : ι → K} (h : ∀ (k : ι), ∑ j, (MvPolynomial.eval x) (q (k, j)) * (MvPolynomial.eval x) (p j) = x k ^ (M + N)) : (↑(Nat.card ι') ^ Height.totalWeight K * max (Height.mulHeightBound q) 1)⁻¹ * Height.mulHeight x ^ N ≤ Height.mulHeight fun j => (MvPolynomial.eval x) (p j)
IsNonarchimedean.eval_mvPolynomial_le 📋 Mathlib.NumberTheory.Height.MvPolynomial
{K : Type u_4} [Field K] {ι : Type u_5} [Finite ι] {v : AbsoluteValue K ℝ} (hv : IsNonarchimedean ⇑v) {p : MvPolynomial ι K} {N : ℕ} (hp : p.IsHomogeneous N) (x : ι → K) : v ((MvPolynomial.eval x) p) ≤ (⨆ s, v (p.coeff ↑s)) * (⨆ i, v (x i)) ^ N
WeierstrassCurve.abs_logHeight_addSubMap_sub_two_mul_logHeight_le 📋 Mathlib.NumberTheory.Height.EllipticCurve
{K : Type u_1} [Field K] [Height.AdmissibleAbsValues K] (W : WeierstrassCurve K) [W.IsElliptic] : ∃ C, ∀ (x : Fin 3 → K), |(Height.logHeight fun i => (MvPolynomial.eval x) (W.addSubMap i)) - 2 * Height.logHeight x| ≤ C
MvPolynomial.continuous_eval 📋 Mathlib.Topology.Algebra.MvPolynomial
{X : Type u_1} {σ : Type u_2} [TopologicalSpace X] [CommSemiring X] [IsTopologicalSemiring X] (p : MvPolynomial σ X) : Continuous fun x => (MvPolynomial.eval x) p

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. 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.
You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

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. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c