Partial derivatives of polynomials

This file defines the notion of the formal partial derivative of a polynomial, the derivative with respect to a single variable. This derivative is not connected to the notion of derivative from analysis. It is based purely on the polynomial exponents and coefficients.

Main declarations

• MvPolynomial.pderiv i p : the partial derivative of p with respect to i, as a bundled derivation of MvPolynomial σ R.

Notation

As in other polynomial files, we typically use the notation:

• σ : Type* (indexing the variables)

• R : Type* [CommRing R] (the coefficients)

• s : σ →₀ ℕ, a function from σ to ℕ which is zero away from a finite set. This will give rise to a monomial in MvPolynomial σ R which mathematicians might call X^s

• a : R

• i : σ, with corresponding monomial X i, often denoted X_i by mathematicians

• p : MvPolynomial σ R

def MvPolynomial.pderiv {R : Type u} {σ : Type v} [CommSemiring R] (i : σ) : Derivation R (MvPolynomial σ R) (MvPolynomial σ R)

pderiv i p is the partial derivative of p with respect to i

theorem MvPolynomial.pderiv_def {R : Type u} {σ : Type v} [CommSemiring R] [DecidableEq σ] (i : σ) : MvPolynomial.pderiv i = MvPolynomial.mkDerivation R (Pi.single i 1)

@[simp]

theorem MvPolynomial.pderiv_monomial {R : Type u} {σ : Type v} {a : R} {s : σ →₀ ℕ} [CommSemiring R] {i : σ} : ↑(MvPolynomial.pderiv i) (↑(MvPolynomial.monomial s) a) = ↑(MvPolynomial.monomial (s - fun₀ | i => 1)) (a * ↑(↑s i))

theorem MvPolynomial.pderiv_C {R : Type u} {σ : Type v} {a : R} [CommSemiring R] {i : σ} : ↑(MvPolynomial.pderiv i) (↑MvPolynomial.C a) = 0

theorem MvPolynomial.pderiv_one {R : Type u} {σ : Type v} [CommSemiring R] {i : σ} : ↑(MvPolynomial.pderiv i) 1 = 0

@[simp]

theorem MvPolynomial.pderiv_X {R : Type u} {σ : Type v} [CommSemiring R] [DecidableEq σ] (i : σ) (j : σ) : ↑(MvPolynomial.pderiv i) (MvPolynomial.X j) = Pi.single i 1 j

@[simp]

theorem MvPolynomial.pderiv_X_self {R : Type u} {σ : Type v} [CommSemiring R] (i : σ) : ↑(MvPolynomial.pderiv i) (MvPolynomial.X i) = 1

@[simp]

theorem MvPolynomial.pderiv_X_of_ne {R : Type u} {σ : Type v} [CommSemiring R] {i : σ} {j : σ} (h : j ≠ i) : ↑(MvPolynomial.pderiv i) (MvPolynomial.X j) = 0

theorem MvPolynomial.pderiv_eq_zero_of_not_mem_vars {R : Type u} {σ : Type v} [CommSemiring R] {i : σ} {f : MvPolynomial σ R} (h : ¬i ∈ MvPolynomial.vars f) : ↑(MvPolynomial.pderiv i) f = 0

theorem MvPolynomial.pderiv_monomial_single {R : Type u} {σ : Type v} {a : R} [CommSemiring R] {i : σ} {n : ℕ} : ↑(MvPolynomial.pderiv i) (↑(MvPolynomial.monomial fun₀ | i => n) a) = ↑(MvPolynomial.monomial fun₀ | i => n - 1) (a * ↑n)

theorem MvPolynomial.pderiv_mul {R : Type u} {σ : Type v} [CommSemiring R] {i : σ} {f : MvPolynomial σ R} {g : MvPolynomial σ R} : ↑(MvPolynomial.pderiv i) (f * g) = ↑(MvPolynomial.pderiv i) f * g + f * ↑(MvPolynomial.pderiv i) g

theorem MvPolynomial.pderiv_C_mul {R : Type u} {σ : Type v} {a : R} [CommSemiring R] {f : MvPolynomial σ R} {i : σ} : ↑(MvPolynomial.pderiv i) (↑MvPolynomial.C a * f) = ↑MvPolynomial.C a * ↑(MvPolynomial.pderiv i) f