Documentation

Mathlib.Data.Polynomial.Coeff

Theory of univariate polynomials #

The theorems include formulas for computing coefficients, such as coeff_add, coeff_sum, coeff_mul

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theorem Polynomial.coeff_smul {R : Type u} {S : Type v} [Semiring R] [SMulZeroClass S R] (r : S) (p : Polynomial R) (n : ) :
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theorem Polynomial.lsum_apply {R : Type u_1} {A : Type u_2} {M : Type u_3} [Semiring R] [Semiring A] [AddCommMonoid M] [Module R A] [Module R M] (f : A →ₗ[R] M) (p : Polynomial A) :
(Polynomial.lsum f) p = Polynomial.sum p fun (x : ) (x_1 : A) => (f x) x_1
def Polynomial.lsum {R : Type u_1} {A : Type u_2} {M : Type u_3} [Semiring R] [Semiring A] [AddCommMonoid M] [Module R A] [Module R M] (f : A →ₗ[R] M) :

Polynomial.sum as a linear map.

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    The nth coefficient, as a linear map.

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      theorem Polynomial.finset_sum_coeff {R : Type u} [Semiring R] {ι : Type u_1} (s : Finset ι) (f : ιPolynomial R) (n : ) :
      Polynomial.coeff (Finset.sum s fun (b : ι) => f b) n = Finset.sum s fun (b : ι) => Polynomial.coeff (f b) n
      theorem Polynomial.coeff_sum {R : Type u} {S : Type v} [Semiring R] {p : Polynomial R} [Semiring S] (n : ) (f : RPolynomial S) :
      Polynomial.coeff (Polynomial.sum p f) n = Polynomial.sum p fun (a : ) (b : R) => Polynomial.coeff (f a b) n
      theorem Polynomial.coeff_mul {R : Type u} [Semiring R] (p : Polynomial R) (q : Polynomial R) (n : ) :

      Decomposes the coefficient of the product p * q as a sum over antidiagonal. A version which sums over range (n + 1) can be obtained by using Finset.Nat.sum_antidiagonal_eq_sum_range_succ.

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      theorem Polynomial.constantCoeff_apply {R : Type u} [Semiring R] (p : Polynomial R) :
      Polynomial.constantCoeff p = Polynomial.coeff p 0

      constantCoeff p returns the constant term of the polynomial p, defined as coeff p 0. This is a ring homomorphism.

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        theorem Polynomial.isUnit_C {R : Type u} [Semiring R] {x : R} :
        IsUnit (Polynomial.C x) IsUnit x
        theorem Polynomial.coeff_mul_X_zero {R : Type u} [Semiring R] (p : Polynomial R) :
        Polynomial.coeff (p * Polynomial.X) 0 = 0
        theorem Polynomial.coeff_X_mul_zero {R : Type u} [Semiring R] (p : Polynomial R) :
        Polynomial.coeff (Polynomial.X * p) 0 = 0
        theorem Polynomial.coeff_C_mul_X_pow {R : Type u} [Semiring R] (x : R) (k : ) (n : ) :
        Polynomial.coeff (Polynomial.C x * Polynomial.X ^ k) n = if n = k then x else 0
        theorem Polynomial.coeff_C_mul_X {R : Type u} [Semiring R] (x : R) (n : ) :
        Polynomial.coeff (Polynomial.C x * Polynomial.X) n = if n = 1 then x else 0
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        theorem Polynomial.coeff_C_mul {R : Type u} {a : R} {n : } [Semiring R] (p : Polynomial R) :
        Polynomial.coeff (Polynomial.C a * p) n = a * Polynomial.coeff p n
        theorem Polynomial.C_mul' {R : Type u} [Semiring R] (a : R) (f : Polynomial R) :
        Polynomial.C a * f = a f
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        theorem Polynomial.coeff_mul_C {R : Type u} [Semiring R] (p : Polynomial R) (n : ) (a : R) :
        Polynomial.coeff (p * Polynomial.C a) n = Polynomial.coeff p n * a
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        theorem Polynomial.coeff_mul_natCast {R : Type u} [Semiring R] {p : Polynomial R} {a : } {k : } :
        Polynomial.coeff (p * a) k = Polynomial.coeff p k * a
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        theorem Polynomial.coeff_natCast_mul {R : Type u} [Semiring R] {p : Polynomial R} {a : } {k : } :
        Polynomial.coeff (a * p) k = a * Polynomial.coeff p k
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        theorem Polynomial.coeff_mul_intCast {S : Type v} [Ring S] {p : Polynomial S} {a : } {k : } :
        Polynomial.coeff (p * a) k = Polynomial.coeff p k * a
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        theorem Polynomial.coeff_intCast_mul {S : Type v} [Ring S] {p : Polynomial S} {a : } {k : } :
        Polynomial.coeff (a * p) k = a * Polynomial.coeff p k
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        theorem Polynomial.coeff_X_pow {R : Type u} [Semiring R] (k : ) (n : ) :
        Polynomial.coeff (Polynomial.X ^ k) n = if n = k then 1 else 0
        theorem Polynomial.coeff_X_pow_self {R : Type u} [Semiring R] (n : ) :
        Polynomial.coeff (Polynomial.X ^ n) n = 1
        theorem Polynomial.support_binomial {R : Type u} [Semiring R] {k : } {m : } (hkm : k m) {x : R} {y : R} (hx : x 0) (hy : y 0) :
        Polynomial.support (Polynomial.C x * Polynomial.X ^ k + Polynomial.C y * Polynomial.X ^ m) = {k, m}
        theorem Polynomial.support_trinomial {R : Type u} [Semiring R] {k : } {m : } {n : } (hkm : k < m) (hmn : m < n) {x : R} {y : R} {z : R} (hx : x 0) (hy : y 0) (hz : z 0) :
        Polynomial.support (Polynomial.C x * Polynomial.X ^ k + Polynomial.C y * Polynomial.X ^ m + Polynomial.C z * Polynomial.X ^ n) = {k, m, n}
        theorem Polynomial.card_support_binomial {R : Type u} [Semiring R] {k : } {m : } (h : k m) {x : R} {y : R} (hx : x 0) (hy : y 0) :
        (Polynomial.support (Polynomial.C x * Polynomial.X ^ k + Polynomial.C y * Polynomial.X ^ m)).card = 2
        theorem Polynomial.card_support_trinomial {R : Type u} [Semiring R] {k : } {m : } {n : } (hkm : k < m) (hmn : m < n) {x : R} {y : R} {z : R} (hx : x 0) (hy : y 0) (hz : z 0) :
        (Polynomial.support (Polynomial.C x * Polynomial.X ^ k + Polynomial.C y * Polynomial.X ^ m + Polynomial.C z * Polynomial.X ^ n)).card = 3
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        theorem Polynomial.coeff_mul_X_pow {R : Type u} [Semiring R] (p : Polynomial R) (n : ) (d : ) :
        Polynomial.coeff (p * Polynomial.X ^ n) (d + n) = Polynomial.coeff p d
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        theorem Polynomial.coeff_X_pow_mul {R : Type u} [Semiring R] (p : Polynomial R) (n : ) (d : ) :
        Polynomial.coeff (Polynomial.X ^ n * p) (d + n) = Polynomial.coeff p d
        theorem Polynomial.coeff_mul_X_pow' {R : Type u} [Semiring R] (p : Polynomial R) (n : ) (d : ) :
        Polynomial.coeff (p * Polynomial.X ^ n) d = if n d then Polynomial.coeff p (d - n) else 0
        theorem Polynomial.coeff_X_pow_mul' {R : Type u} [Semiring R] (p : Polynomial R) (n : ) (d : ) :
        Polynomial.coeff (Polynomial.X ^ n * p) d = if n d then Polynomial.coeff p (d - n) else 0
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        theorem Polynomial.coeff_mul_X {R : Type u} [Semiring R] (p : Polynomial R) (n : ) :
        Polynomial.coeff (p * Polynomial.X) (n + 1) = Polynomial.coeff p n
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        theorem Polynomial.coeff_X_mul {R : Type u} [Semiring R] (p : Polynomial R) (n : ) :
        Polynomial.coeff (Polynomial.X * p) (n + 1) = Polynomial.coeff p n
        theorem Polynomial.coeff_mul_monomial {R : Type u} [Semiring R] (p : Polynomial R) (n : ) (d : ) (r : R) :
        theorem Polynomial.coeff_monomial_mul {R : Type u} [Semiring R] (p : Polynomial R) (n : ) (d : ) (r : R) :
        theorem Polynomial.mul_X_pow_eq_zero {R : Type u} [Semiring R] {p : Polynomial R} {n : } (H : p * Polynomial.X ^ n = 0) :
        p = 0
        theorem Polynomial.isRegular_X_pow {R : Type u} [Semiring R] (n : ) :
        IsRegular (Polynomial.X ^ n)
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        theorem Polynomial.isRegular_X {R : Type u} [Semiring R] :
        IsRegular Polynomial.X
        theorem Polynomial.coeff_X_add_C_pow {R : Type u} [Semiring R] (r : R) (n : ) (k : ) :
        Polynomial.coeff ((Polynomial.X + Polynomial.C r) ^ n) k = r ^ (n - k) * (Nat.choose n k)
        theorem Polynomial.coeff_X_add_one_pow (R : Type u_1) [Semiring R] (n : ) (k : ) :
        Polynomial.coeff ((Polynomial.X + 1) ^ n) k = (Nat.choose n k)
        theorem Polynomial.coeff_one_add_X_pow (R : Type u_1) [Semiring R] (n : ) (k : ) :
        Polynomial.coeff ((1 + Polynomial.X) ^ n) k = (Nat.choose n k)
        theorem Polynomial.C_dvd_iff_dvd_coeff {R : Type u} [Semiring R] (r : R) (φ : Polynomial R) :
        Polynomial.C r φ ∀ (i : ), r Polynomial.coeff φ i
        theorem Polynomial.coeff_bit0_mul {R : Type u} [Semiring R] (P : Polynomial R) (Q : Polynomial R) (n : ) :
        theorem Polynomial.smul_eq_C_mul {R : Type u} [Semiring R] {p : Polynomial R} (a : R) :
        a p = Polynomial.C a * p
        theorem Polynomial.update_eq_add_sub_coeff {R : Type u_1} [Ring R] (p : Polynomial R) (n : ) (a : R) :
        Polynomial.update p n a = p + Polynomial.C (a - Polynomial.coeff p n) * Polynomial.X ^ n
        theorem Polynomial.nat_cast_coeff_zero {n : } {R : Type u_1} [Semiring R] :
        Polynomial.coeff (n) 0 = n
        theorem Polynomial.nat_cast_inj {m : } {n : } {R : Type u_1} [Semiring R] [CharZero R] :
        m = n m = n
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        theorem Polynomial.int_cast_coeff_zero {i : } {R : Type u_1} [Ring R] :
        Polynomial.coeff (i) 0 = i
        theorem Polynomial.int_cast_inj {m : } {n : } {R : Type u_1} [Ring R] [CharZero R] :
        m = n m = n
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