Degree and leading coefficient of polynomials with respect to a monomial order

We consider a type σ of indeterminates and a commutative semiring R and a monomial order m : MonomialOrder σ.

• m.degree f is the degree of f for the monomial ordering m

• m.lCoeff f is the leading coefficient of f for the monomial ordering m

• m.lCoeff_ne_zero_iff f asserts that this coefficient is nonzero iff f ≠ 0.

• in a field, m.lCoeff_is_unit_iff f asserts that this coefficient is a unit iff f ≠ 0.

• m.degree_add_le : the m.degree of f + g is smaller than or equal to the supremum of those of f and g

• m.degree_add_of_lt h : the m.degree of f + g is equal to that of f if the m.degree of g is strictly smaller than that f

• m.lCoeff_add_of_lt h: then, the leading coefficient of f + g is that of f .

• m.degree_add_of_ne h : the m.degree of f + g is equal to that the supremum of those of f and g if they are distinct

• m.degree_sub_le : the m.degree of f - g is smaller than or equal to the supremum of those of f and g

• m.degree_sub_of_lt h : the m.degree of f - g is equal to that of f if the m.degree of g is strictly smaller than that f

• m.lCoeff_sub_of_lt h: then, the leading coefficient of f - g is that of f .

• m.degree_mul_le: the m.degree of f * g is smaller than or equal to the sum of those of f and g.

• m.degree_mul_of_isRegular_left, m.degree_mul_of_isRegular_right and m.degree_mul assert the equality when the leading coefficient of f or g is regular, or when R is a domain and f and g are nonzero.

• m.lCoeff_mul_of_isRegular_left, m.lCoeff_mul_of_isRegular_right and m.lCoeff_mul say that m.lCoeff (f * g) = m.lCoeff f * m.lCoeff g

Reference

[BW93]

def MonomialOrder.degree {σ : Type u_1} (m : MonomialOrder σ) {R : Type u_3} [CommSemiring R] (f : MvPolynomial σ R) : σ →₀ ℕ

the degree of a multivariate polynomial with respect to a monomial ordering

Equations

• m.degree f = m.toSyn.symm (f.support.sup ⇑m.toSyn)

def MonomialOrder.lCoeff {σ : Type u_1} (m : MonomialOrder σ) {R : Type u_3} [CommSemiring R] (f : MvPolynomial σ R) : R

the leading coefficient of a multivariate polynomial with respect to a monomial ordering

Equations

• m.lCoeff f = MvPolynomial.coeff (m.degree f) f

@[simp]

theorem MonomialOrder.degree_zero {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] : m.degree 0 = 0

@[simp]

theorem MonomialOrder.lCoeff_zero {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] : m.lCoeff 0 = 0

theorem MonomialOrder.degree_monomial_le {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {d : σ →₀ ℕ} (c : R) : m.toSyn (m.degree ((MvPolynomial.monomial d) c)) ≤ m.toSyn d

theorem MonomialOrder.degree_monomial {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {d : σ →₀ ℕ} (c : R) [Decidable (c = 0)] : m.degree ((MvPolynomial.monomial d) c) = if c = 0 then 0 else d

@[simp]

theorem MonomialOrder.lCoeff_monomial {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {d : σ →₀ ℕ} (c : R) : m.lCoeff ((MvPolynomial.monomial d) c) = c

theorem MonomialOrder.degree_le_iff {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f : MvPolynomial σ R} {d : σ →₀ ℕ} : m.toSyn (m.degree f) ≤ m.toSyn d ↔ ∀ c ∈ f.support, m.toSyn c ≤ m.toSyn d

theorem MonomialOrder.degree_lt_iff {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f : MvPolynomial σ R} {d : σ →₀ ℕ} (hd : m.toSyn 0 < m.toSyn d) : m.toSyn (m.degree f) < m.toSyn d ↔ ∀ c ∈ f.support, m.toSyn c < m.toSyn d

theorem MonomialOrder.le_degree {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f : MvPolynomial σ R} {d : σ →₀ ℕ} (hd : d ∈ f.support) : m.toSyn d ≤ m.toSyn (m.degree f)

theorem MonomialOrder.coeff_eq_zero_of_lt {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f : MvPolynomial σ R} {d : σ →₀ ℕ} (hd : m.toSyn (m.degree f) < m.toSyn d) : MvPolynomial.coeff d f = 0

theorem MonomialOrder.lCoeff_ne_zero_iff {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f : MvPolynomial σ R} : m.lCoeff f ≠ 0 ↔ f ≠ 0

@[simp]

theorem MonomialOrder.lCoeff_eq_zero_iff {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f : MvPolynomial σ R} : m.lCoeff f = 0 ↔ f = 0

theorem MonomialOrder.coeff_degree_ne_zero_iff {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f : MvPolynomial σ R} : MvPolynomial.coeff (m.degree f) f ≠ 0 ↔ f ≠ 0

@[simp]

theorem MonomialOrder.coeff_degree_eq_zero_iff {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f : MvPolynomial σ R} : MvPolynomial.coeff (m.degree f) f = 0 ↔ f = 0

theorem MonomialOrder.degree_eq_zero_iff_totalDegree_eq_zero {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f : MvPolynomial σ R} : m.degree f = 0 ↔ f.totalDegree = 0

@[simp]

theorem MonomialOrder.degree_C {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] (r : R) : m.degree (MvPolynomial.C r) = 0

theorem MonomialOrder.degree_add_le {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f g : MvPolynomial σ R} : m.toSyn (m.degree (f + g)) ≤ m.toSyn (m.degree f) ⊔ m.toSyn (m.degree g)

theorem MonomialOrder.degree_add_of_lt {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f g : MvPolynomial σ R} (h : m.toSyn (m.degree g) < m.toSyn (m.degree f)) : m.degree (f + g) = m.degree f

theorem MonomialOrder.lCoeff_add_of_lt {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f g : MvPolynomial σ R} (h : m.toSyn (m.degree g) < m.toSyn (m.degree f)) : m.lCoeff (f + g) = m.lCoeff f

theorem MonomialOrder.degree_add_of_ne {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f g : MvPolynomial σ R} (h : m.degree f ≠ m.degree g) : m.toSyn (m.degree (f + g)) = m.toSyn (m.degree f) ⊔ m.toSyn (m.degree g)

theorem MonomialOrder.degree_mul_le {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f g : MvPolynomial σ R} : m.toSyn (m.degree (f * g)) ≤ m.toSyn (m.degree f + m.degree g)

theorem MonomialOrder.coeff_mul_of_degree_add {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f g : MvPolynomial σ R} : MvPolynomial.coeff (m.degree f + m.degree g) (f * g) = m.lCoeff f * m.lCoeff g

Multiplicativity of leading coefficients

theorem MonomialOrder.degree_mul_of_isRegular_left {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f g : MvPolynomial σ R} (hf : IsRegular (m.lCoeff f)) (hg : g ≠ 0) : m.degree (f * g) = m.degree f + m.degree g

Multiplicativity of leading coefficients

theorem MonomialOrder.lCoeff_mul_of_isRegular_left {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f g : MvPolynomial σ R} (hf : IsRegular (m.lCoeff f)) (hg : g ≠ 0) : m.lCoeff (f * g) = m.lCoeff f * m.lCoeff g

Multiplicativity of leading coefficients

theorem MonomialOrder.degree_mul_of_isRegular_right {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f g : MvPolynomial σ R} (hf : f ≠ 0) (hg : IsRegular (m.lCoeff g)) : m.degree (f * g) = m.degree f + m.degree g

Multiplicativity of leading coefficients

theorem MonomialOrder.lCoeff_mul_of_isRegular_right {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f g : MvPolynomial σ R} (hf : f ≠ 0) (hg : IsRegular (m.lCoeff g)) : m.lCoeff (f * g) = m.lCoeff f * m.lCoeff g

Multiplicativity of leading coefficients

theorem MonomialOrder.degree_mul {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] [IsDomain R] {f g : MvPolynomial σ R} (hf : f ≠ 0) (hg : g ≠ 0) : m.degree (f * g) = m.degree f + m.degree g

Degree of product

theorem MonomialOrder.degree_mul_of_nonzero_mul {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] [IsDomain R] {f g : MvPolynomial σ R} (hfg : f * g ≠ 0) : m.degree (f * g) = m.degree f + m.degree g

Degree of of product

theorem MonomialOrder.lCoeff_mul {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] [IsDomain R] {f g : MvPolynomial σ R} (hf : f ≠ 0) (hg : g ≠ 0) : m.lCoeff (f * g) = m.lCoeff f * m.lCoeff g

Multiplicativity of leading coefficients

theorem MonomialOrder.degree_smul_le {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {r : R} {f : MvPolynomial σ R} : m.toSyn (m.degree (r • f)) ≤ m.toSyn (m.degree f)

theorem MonomialOrder.degree_smul {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {r : R} (hr : IsRegular r) {f : MvPolynomial σ R} : m.degree (r • f) = m.degree f

theorem MonomialOrder.eq_C_of_degree_eq_zero {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f : MvPolynomial σ R} (hf : m.degree f = 0) : f = MvPolynomial.C (m.lCoeff f)

@[simp]

theorem MonomialOrder.degree_neg {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommRing R] {f : MvPolynomial σ R} : m.degree (-f) = m.degree f

theorem MonomialOrder.degree_sub_le {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommRing R] {f g : MvPolynomial σ R} : m.toSyn (m.degree (f - g)) ≤ m.toSyn (m.degree f) ⊔ m.toSyn (m.degree g)

theorem MonomialOrder.degree_sub_of_lt {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommRing R] {f g : MvPolynomial σ R} (h : m.toSyn (m.degree g) < m.toSyn (m.degree f)) : m.degree (f - g) = m.degree f

theorem MonomialOrder.lCoeff_sub_of_lt {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommRing R] {f g : MvPolynomial σ R} (h : m.toSyn (m.degree g) < m.toSyn (m.degree f)) : m.lCoeff (f - g) = m.lCoeff f

theorem MonomialOrder.lCoeff_is_unit_iff {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [Field R] {f : MvPolynomial σ R} : IsUnit (m.lCoeff f) ↔ f ≠ 0