Degree, leading coefficient and leading term 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.leadingCoeff f is the leading coefficient of f for the monomial ordering m.

• m.Monic f asserts that the leading coefficient of f is 1.

• m.leadingTerm f is the leading term of f for the monomial ordering m.

• m.sPolynomial f g is S-polynomial of f and g.

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

• in a field, m.isUnit_leadingCoeff 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.leadingCoeff_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.leadingCoeff_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_mul_leadingCoeff_ne_zero : if the product of the leading coefficients is nonzero, then the degree is the sum of the degrees.

• m.leadingCoeff_mul_of_mul_leadingCoeff_ne_zero : if the product of the leading coefficients is nonzero, then the leading coefficient is that product.

• 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.leadingCoeff_mul_of_isRegular_left, m.leadingCoeff_mul_of_isRegular_right and m.leadingCoeff_mul say that m.leadingCoeff (f * g) = m.leadingCoeff f * m.leadingCoeff g

• m.degree_pow_of_pow_leadingCoeff_ne_zero : is the nth power of the leading coefficient of f is nonzero, then the degree of f ^ n is n • (m.degree f)

• m.leadingCoeff_pow_of_pow_leadingCoeff_ne_zero : is the nth power of the leading coefficient of f is nonzero, then the leading coefficient of f ^ n is that power.

• m.degree_prod_of_regular : the degree of a product of polynomials whose leading coefficients are regular is the sum of their degrees.

• m.leadingCoeff_prod_of_regular : the leading coefficient of a product of polynomials whose leading coefficients are regular is the product of their leading coefficients.

• m.Monic.prod : a product of monic polynomials is monic.

• m.degree_sub_leadingTerm_lt_iff : the degree of f - m.leadingTerm f is smaller than the degree of f if and only if m.degree f ≠ 0.

Reference

[BW93]

def MonomialOrder.degree {σ : Type u_1} (m : MonomialOrder σ) {R : Type u_2} [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.leadingCoeff {σ : Type u_1} (m : MonomialOrder σ) {R : Type u_2} [CommSemiring R] (f : MvPolynomial σ R) : R

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

Equations

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

def MonomialOrder.Monic {σ : Type u_1} (m : MonomialOrder σ) {R : Type u_2} [CommSemiring R] (f : MvPolynomial σ R) : Prop

A multivariate polynomial is Monic with respect to a monomial order if its leading coefficient (for that monomial order) is 1.

Equations

• m.Monic f = (m.leadingCoeff f = 1)

noncomputable def MonomialOrder.leadingTerm {σ : Type u_1} (m : MonomialOrder σ) {R : Type u_2} [CommSemiring R] (f : MvPolynomial σ R) : MvPolynomial σ R

The leading term of of a multivariate polynomial with respect to a monomial ordering.

Equations

• m.leadingTerm f = (MvPolynomial.monomial (m.degree f)) (m.leadingCoeff f)

@[simp]

theorem MonomialOrder.C_mul_leadingCoeff_monomial_degree {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] (p : MvPolynomial σ R) : MvPolynomial.C (m.leadingCoeff p) * (MvPolynomial.monomial (m.degree p)) 1 = m.leadingTerm p

theorem MonomialOrder.Monic.of_subsingleton {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] [Subsingleton R] {f : MvPolynomial σ R} : m.Monic f

instance MonomialOrder.Monic.decidable {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] [DecidableEq R] (f : MvPolynomial σ R) : Decidable (m.Monic f)

Equations

• MonomialOrder.Monic.decidable f = id inferInstance

@[simp]

theorem MonomialOrder.Monic.leadingCoeff_eq_one {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f : MvPolynomial σ R} (hf : m.Monic f) : m.leadingCoeff f = 1

theorem MonomialOrder.Monic.coeff_degree {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f : MvPolynomial σ R} (hf : m.Monic f) : MvPolynomial.coeff (m.degree f) f = 1

@[simp]

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

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

@[simp]

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

@[simp]

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

theorem MonomialOrder.Monic.ne_zero {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] [Nontrivial R] {f : MvPolynomial σ R} (hf : m.Monic f) : f ≠ 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

theorem MonomialOrder.degree_X_le_single {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {s : σ} : m.toSyn (m.degree (MvPolynomial.X s)) ≤ m.toSyn (Finsupp.single s 1)

theorem MonomialOrder.degree_X {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] [Nontrivial R] {s : σ} : m.degree (MvPolynomial.X s) = Finsupp.single s 1

@[simp]

theorem MonomialOrder.degree_one {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] : m.degree 1 = 0

@[simp]

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

@[simp]

theorem MonomialOrder.monic_monomial_one {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {d : σ →₀ ℕ} : m.Monic ((MvPolynomial.monomial d) 1)

theorem MonomialOrder.monic_monomial {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {d : σ →₀ ℕ} {c : R} : m.Monic ((MvPolynomial.monomial d) c) ↔ c = 1

theorem MonomialOrder.leadingCoeff_X {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {s : σ} : m.leadingCoeff (MvPolynomial.X s) = 1

@[simp]

theorem MonomialOrder.monic_X {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {s : σ} : m.Monic (MvPolynomial.X s)

theorem MonomialOrder.leadingCoeff_one {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] : m.leadingCoeff 1 = 1

theorem MonomialOrder.monic_one {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] : m.Monic (MvPolynomial.C 1)

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.leadingCoeff_ne_zero_iff {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f : MvPolynomial σ R} : m.leadingCoeff f ≠ 0 ↔ f ≠ 0

@[simp]

theorem MonomialOrder.leadingCoeff_eq_zero_iff {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f : MvPolynomial σ R} : m.leadingCoeff 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

theorem MonomialOrder.degree_mem_support_iff {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] (f : MvPolynomial σ R) : m.degree f ∈ f.support ↔ 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_mem_support {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {p : MvPolynomial σ R} (hp : p ≠ 0) : m.degree p ∈ p.support

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

@[simp]

theorem MonomialOrder.leadingCoeff_C {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] (c : R) : m.leadingCoeff (MvPolynomial.C c) = c

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.leadingCoeff f)

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

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)) ≤ max (m.toSyn (m.degree f)) (m.toSyn (m.degree g))

theorem MonomialOrder.degree_sum_le {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {α : Type u_3} {s : Finset α} {f : α → MvPolynomial σ R} : m.toSyn (m.degree (∑ x ∈ s, f x)) ≤ s.sup fun (x : α) => m.toSyn (m.degree (f x))

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.degree_add_eq_right_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 f) < m.toSyn (m.degree g)) : m.degree (f + g) = m.degree g

theorem MonomialOrder.leadingCoeff_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.leadingCoeff (f + g) = m.leadingCoeff f

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

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)) = max (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_add_of_degree_le {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f g : MvPolynomial σ R} {a b : σ →₀ ℕ} (ha : m.toSyn (m.degree f) ≤ m.toSyn a) (hb : m.toSyn (m.degree g) ≤ m.toSyn b) : MvPolynomial.coeff (a + b) (f * g) = MvPolynomial.coeff a f * MvPolynomial.coeff b g

Multiplicativity of leading coefficients

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.leadingCoeff f * m.leadingCoeff g

Multiplicativity of leading coefficients

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

Monomial degree of product

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

Monomial degree of product

theorem MonomialOrder.leadingCoeff_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.leadingCoeff f)) (hg : g ≠ 0) : m.leadingCoeff (f * g) = m.leadingCoeff f * m.leadingCoeff 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.leadingCoeff g)) : m.degree (f * g) = m.degree f + m.degree g

Monomial degree of product

theorem MonomialOrder.leadingCoeff_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.leadingCoeff g)) : m.leadingCoeff (f * g) = m.leadingCoeff f * m.leadingCoeff g

Multiplicativity of leading coefficients

theorem MonomialOrder.Monic.mul {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f g : MvPolynomial σ R} (hf : m.Monic f) (hg : m.Monic g) : m.Monic (f * g)

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

Monomial degree of product

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

Multiplicativity of leading coefficients

theorem MonomialOrder.degree_pow_le {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f : MvPolynomial σ R} (n : ℕ) : m.toSyn (m.degree (f ^ n)) ≤ m.toSyn (n • m.degree f)

Monomial degree of powers

theorem MonomialOrder.coeff_pow_nsmul_degree {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] (f : MvPolynomial σ R) (n : ℕ) : MvPolynomial.coeff (n • m.degree f) (f ^ n) = m.leadingCoeff f ^ n

theorem MonomialOrder.degree_pow_of_pow_leadingCoeff_ne_zero {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f : MvPolynomial σ R} {n : ℕ} (hf : m.leadingCoeff f ^ n ≠ 0) : m.degree (f ^ n) = n • m.degree f

Monomial degree of powers

theorem MonomialOrder.leadingCoeff_pow_of_pow_leadingCoeff_ne_zero {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f : MvPolynomial σ R} {n : ℕ} (hf : m.leadingCoeff f ^ n ≠ 0) : m.leadingCoeff (f ^ n) = m.leadingCoeff f ^ n

Leading coefficient of powers

theorem MonomialOrder.Monic.pow {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {f : MvPolynomial σ R} {n : ℕ} (hf : m.Monic f) : m.Monic (f ^ n)

theorem MonomialOrder.degree_pow {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] [IsReduced R] (f : MvPolynomial σ R) (n : ℕ) : m.degree (f ^ n) = n • m.degree f

Monomial degree of powers (in a reduced ring)

theorem MonomialOrder.leadingCoeff_pow {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] [IsReduced R] (f : MvPolynomial σ R) (n : ℕ) : m.leadingCoeff (f ^ n) = m.leadingCoeff f ^ n

Leading coefficient of powers (in a reduced ring)

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.degree_prod_le {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {ι : Type u_3} {P : ι → MvPolynomial σ R} {s : Finset ι} : m.toSyn (m.degree (∏ i ∈ s, P i)) ≤ m.toSyn (∑ i ∈ s, m.degree (P i))

theorem MonomialOrder.coeff_prod_sum_degree {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {ι : Type u_3} (P : ι → MvPolynomial σ R) (s : Finset ι) : MvPolynomial.coeff (∑ i ∈ s, m.degree (P i)) (∏ i ∈ s, P i) = ∏ i ∈ s, m.leadingCoeff (P i)

theorem MonomialOrder.degree_prod_of_regular {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {ι : Type u_3} {P : ι → MvPolynomial σ R} {s : Finset ι} (H : ∀ i ∈ s, IsRegular (m.leadingCoeff (P i))) : m.degree (∏ i ∈ s, P i) = ∑ i ∈ s, m.degree (P i)

theorem MonomialOrder.degree_prod {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] [IsCancelMulZero R] {ι : Type u_3} {P : ι → MvPolynomial σ R} {s : Finset ι} (H : ∀ i ∈ s, P i ≠ 0) : m.degree (∏ i ∈ s, P i) = ∑ i ∈ s, m.degree (P i)

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

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

theorem MonomialOrder.leadingCoeff_prod_of_regular {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {ι : Type u_3} {P : ι → MvPolynomial σ R} {s : Finset ι} (H : ∀ i ∈ s, IsRegular (m.leadingCoeff (P i))) : m.leadingCoeff (∏ i ∈ s, P i) = ∏ i ∈ s, m.leadingCoeff (P i)

theorem MonomialOrder.Monic.prod {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] {ι : Type u_3} {P : ι → MvPolynomial σ R} {s : Finset ι} (H : ∀ i ∈ s, m.Monic (P i)) : m.Monic (∏ i ∈ s, P i)

A product of monic polynomials is monic

@[simp]

theorem MonomialOrder.leadingTerm_eq_zero_iff {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] (p : MvPolynomial σ R) : m.leadingTerm p = 0 ↔ p = 0

The leading term in a multivariate polynomial is zero if and only if this polynomial is zero.

@[simp]

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

The leading term of the zero polynomial is zero

theorem MonomialOrder.image_leadingTerm_sdiff_singleton_zero {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] (B : Set (MvPolynomial σ R)) : m.leadingTerm '' (B \ {0}) = m.leadingTerm '' B \ {0}

The set of leading terms of non-zero polynomials within a set B is equal to the set of leading terms of all polynomials within B, excluding zero.

theorem MonomialOrder.image_leadingTerm_insert_zero {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] (B : Set (MvPolynomial σ R)) : m.leadingTerm '' insert 0 B = insert 0 (m.leadingTerm '' B)

The set of leading terms of zero and polynomials within a set B is equal to the set of zero and leading terms of polynomials within B.

@[simp]

theorem MonomialOrder.degree_leadingTerm {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] (f : MvPolynomial σ R) : m.degree (m.leadingTerm f) = m.degree f

The degree of f equals to the degree of leadingTerm f

@[simp]

theorem MonomialOrder.leadingCoeff_leadingTerm {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] (f : MvPolynomial σ R) : m.leadingCoeff (m.leadingTerm f) = m.leadingCoeff f

@[simp]

theorem MonomialOrder.leadingTerm_leadingTerm {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] (f : MvPolynomial σ R) : m.leadingTerm (m.leadingTerm f) = m.leadingTerm f

@[simp]

theorem MonomialOrder.leadingTerm_C {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommSemiring R] (c : R) : m.leadingTerm (MvPolynomial.C c) = MvPolynomial.C c

noncomputable def MonomialOrder.sPolynomial {σ : Type u_1} (m : MonomialOrder σ) {R : Type u_2} [CommRing R] (f g : MvPolynomial σ R) : MvPolynomial σ R

The S-polynomial of two polynomials.

Denoting

• the leading monomial of polynomial $f$ and $g$ as $lm(f)$ and $lm(g)$,
• the leading coefficient of $f$ and $g$ as $lc(f)$ and $lc(g)$ (formalized as m.leadingCoeff f and m.leadingCoeff g), and
• the least common multiple of $lm(f)$ and $lm(g)$ as $lcm(lm(f),lm(g))$,

the S-polynomial of $f$ and $g$ is defined as $$sPoly(f,g) := (lcm(lm(f),lm(g)) / lm(f)) * lc(g) * f - (lcm(lm(f),lm(g)) / lm(g)) * lc(f) * g.$$

$(lcm(lm(f),lm(g)) / lm(f))$ and $lcm(lm(f),lm(g)) / lm(g)$ is formalized as monomial (m.degree g - m.degree f) 1 and monomial (m.degree g - m.degree f) 1, while there is also another more direct formalization in sPolynomial_def.

Notice that, when the polynomial ring is over a field, S-polynomial is usually defined as $$sPoly'(f,g) := (lcm(lm(f),lm(g)) / (lm(f) * lc(f))) * f - (lcm(lm(f),lm(g)) / (lm(g) * lc(g))) * g,$$ while we avoid inverting $lc(f)$ and $lc(g)$ in this formalization so that it doesn't require a field or units (IsUnit) over ring.

An equality between these two versions holds: $$sPoly(f,g) = lc(f) * lc(g) * sPoly'(f,g).$$

Equations

• m.sPolynomial f g = (MvPolynomial.monomial (m.degree g - m.degree f)) (m.leadingCoeff g) * f - (MvPolynomial.monomial (m.degree f - m.degree g)) (m.leadingCoeff f) * g

theorem MonomialOrder.sPolynomial_def {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommRing R] (f g : MvPolynomial σ R) : m.sPolynomial f g = (MvPolynomial.monomial (m.degree f ⊔ m.degree g - m.degree f)) (m.leadingCoeff g) * f - (MvPolynomial.monomial (m.degree f ⊔ m.degree g - m.degree g)) (m.leadingCoeff f) * g

theorem MonomialOrder.degree_ne_zero_of_sub_leadingTerm_ne_zero {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommRing R] {f : MvPolynomial σ R} (h : f - m.leadingTerm f ≠ 0) : m.degree f ≠ 0

@[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

@[simp]

theorem MonomialOrder.leadingCoeff_neg {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommRing R] {f : MvPolynomial σ R} : m.leadingCoeff (-f) = -m.leadingCoeff 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)) ≤ max (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.leadingCoeff_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.leadingCoeff (f - g) = m.leadingCoeff f

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

theorem MonomialOrder.degree_sub_leadingTerm_lt_degree {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommRing R] {f : MvPolynomial σ R} (h : m.degree f ≠ 0) : m.toSyn (m.degree (f - m.leadingTerm f)) < m.toSyn (m.degree f)

theorem MonomialOrder.degree_sub_leadingTerm_lt_iff {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommRing R] {f : MvPolynomial σ R} : m.toSyn (m.degree (f - m.leadingTerm f)) < m.toSyn (m.degree f) ↔ m.degree f ≠ 0

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

@[simp]

theorem MonomialOrder.sPolynomial_left_zero {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommRing R] (g : MvPolynomial σ R) : m.sPolynomial 0 g = 0

@[simp]

theorem MonomialOrder.sPolynomial_right_zero {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommRing R] (f : MvPolynomial σ R) : m.sPolynomial f 0 = 0

@[simp]

theorem MonomialOrder.sPolynomial_self {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommRing R] (f : MvPolynomial σ R) : m.sPolynomial f f = 0

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

theorem MonomialOrder.coeff_sPolynomial_sup_eq_zero {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommRing R] (f g : MvPolynomial σ R) : MvPolynomial.coeff (m.degree f ⊔ m.degree g) (m.sPolynomial f g) = 0

theorem MonomialOrder.degree_sPolynomial {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommRing R] (f g : MvPolynomial σ R) : m.toSyn (m.degree (m.sPolynomial f g)) < m.toSyn (m.degree f ⊔ m.degree g) ∨ m.sPolynomial f g = 0

theorem MonomialOrder.degree_sPolynomial_lt_sup_degree {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommRing R] {f g : MvPolynomial σ R} (h : m.sPolynomial f g ≠ 0) : m.toSyn (m.degree (m.sPolynomial f g)) < m.toSyn (m.degree f ⊔ m.degree g)

theorem MonomialOrder.sPolynomial_lt_of_degree_ne_zero_of_degree_eq {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommRing R] {f g : MvPolynomial σ R} (h : m.degree f = m.degree g) (hs : m.sPolynomial f g ≠ 0) : m.toSyn (m.degree (m.sPolynomial f g)) < m.toSyn (m.degree f)

theorem MonomialOrder.sPolynomial_mul_monomial {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommRing R] [IsCancelMulZero R] (p₁ p₂ : MvPolynomial σ R) (d₁ d₂ : σ →₀ ℕ) (c₁ c₂ : R) : m.sPolynomial ((MvPolynomial.monomial d₁) c₁ * p₁) ((MvPolynomial.monomial d₂) c₂ * p₂) = (MvPolynomial.monomial ((d₁ + m.degree p₁) ⊔ (d₂ + m.degree p₂) - m.degree p₁ ⊔ m.degree p₂)) (c₁ * c₂) * m.sPolynomial p₁ p₂

theorem MonomialOrder.sPolynomial_decomposition {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [CommRing R] {d : m.syn} {ι : Type u_3} {B : Finset ι} {g : ι → MvPolynomial σ R} (hd : ∀ b ∈ B, m.toSyn (m.degree (g b)) = d ∧ IsUnit (m.leadingCoeff (g b)) ∨ g b = 0) (hfd : m.toSyn (m.degree (∑ b ∈ B, g b)) < d) : ∃ (c : ι → ι → R), ∑ b ∈ B, g b = ∑ b₁ ∈ B, ∑ b₂ ∈ B, c b₁ b₂ • m.sPolynomial (g b₁) (g b₂)

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

theorem MonomialOrder.sPolynomial_decomposition' {σ : Type u_1} {m : MonomialOrder σ} {R : Type u_2} [Field R] {d : m.syn} {ι : Type u_3} {B : Finset ι} (g : ι → MvPolynomial σ R) (hd : ∀ b ∈ B, m.toSyn (m.degree (g b)) = d ∨ g b = 0) (hfd : m.toSyn (m.degree (∑ b ∈ B, g b)) < d) : ∃ (c : ι → ι → R), ∑ b ∈ B, g b = ∑ b₁ ∈ B, ∑ b₂ ∈ B, c b₁ b₂ • m.sPolynomial (g b₁) (g b₂)

theorem MonomialOrder.degree_X_add_C {R : Type u_2} [CommRing R] [Nontrivial R] {ι : Type u_3} (m : MonomialOrder ι) (i : ι) (r : R) : m.degree (MvPolynomial.X i + MvPolynomial.C r) = Finsupp.single i 1

theorem MonomialOrder.degree_X_sub_C {R : Type u_2} [CommRing R] [Nontrivial R] {ι : Type u_3} (m : MonomialOrder ι) (i : ι) (r : R) : m.degree (MvPolynomial.X i - MvPolynomial.C r) = Finsupp.single i 1

theorem MonomialOrder.monic_X_add_C {R : Type u_2} [CommRing R] {ι : Type u_3} (m : MonomialOrder ι) (i : ι) (r : R) : m.Monic (MvPolynomial.X i + MvPolynomial.C r)

theorem MonomialOrder.monic_X_sub_C {R : Type u_2} [CommRing R] {ι : Type u_3} (m : MonomialOrder ι) (i : ι) (r : R) : m.Monic (MvPolynomial.X i - MvPolynomial.C r)