mathlib3 documentation


Partial derivatives of polynomials #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

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 #

Notation #

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

noncomputable def mv_polynomial.pderiv {R : Type u} {σ : Type v} [comm_semiring R] (i : σ) :

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

theorem mv_polynomial.pderiv_monomial {R : Type u} {σ : Type v} {a : R} {s : σ →₀ } [comm_semiring R] {i : σ} :
theorem mv_polynomial.pderiv_C {R : Type u} {σ : Type v} {a : R} [comm_semiring R] {i : σ} :
theorem mv_polynomial.pderiv_one {R : Type u} {σ : Type v} [comm_semiring R] {i : σ} :
theorem mv_polynomial.pderiv_X {R : Type u} {σ : Type v} [comm_semiring R] [decidable_eq σ] (i j : σ) :
theorem mv_polynomial.pderiv_X_self {R : Type u} {σ : Type v} [comm_semiring R] (i : σ) :
theorem mv_polynomial.pderiv_X_of_ne {R : Type u} {σ : Type v} [comm_semiring R] {i j : σ} (h : j i) :
theorem mv_polynomial.pderiv_eq_zero_of_not_mem_vars {R : Type u} {σ : Type v} [comm_semiring R] {i : σ} {f : mv_polynomial σ R} (h : i f.vars) :
theorem mv_polynomial.pderiv_mul {R : Type u} {σ : Type v} [comm_semiring R] {i : σ} {f g : mv_polynomial σ R} :