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.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.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 non zero, 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 non zero, 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 non zero, 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 non zero, 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.

Reference

[Becker-Weispfenning1993]

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

@[deprecated MonomialOrder.leadingCoeff (since := "2025-01-31")]

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

Alias of MonomialOrder.leadingCoeff.

---

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

Equations

• @MonomialOrder.lCoeff = @MonomialOrder.leadingCoeff

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)

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

@[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

@[deprecated MonomialOrder.leadingCoeff_zero (since := "2025-01-31")]

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

Alias of MonomialOrder.leadingCoeff_zero.

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

@[deprecated MonomialOrder.leadingCoeff_monomial (since := "2025-01-31")]

theorem MonomialOrder.lCoeff_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

Alias of MonomialOrder.leadingCoeff_monomial.

@[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

@[deprecated MonomialOrder.leadingCoeff_ne_zero_iff (since := "2025-01-31")]

theorem MonomialOrder.lCoeff_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

Alias of MonomialOrder.leadingCoeff_ne_zero_iff.

@[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

@[deprecated MonomialOrder.leadingCoeff_eq_zero_iff (since := "2025-01-31")]

theorem MonomialOrder.lCoeff_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

Alias of MonomialOrder.leadingCoeff_eq_zero_iff.

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_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

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_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_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.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

@[deprecated MonomialOrder.leadingCoeff_add_of_lt (since := "2025-01-31")]

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.leadingCoeff (f + g) = m.leadingCoeff f

Alias of MonomialOrder.leadingCoeff_add_of_lt.

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

@[deprecated MonomialOrder.leadingCoeff_mul_of_isRegular_left (since := "2025-01-31")]

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.leadingCoeff f)) (hg : g ≠ 0) : m.leadingCoeff (f * g) = m.leadingCoeff f * m.leadingCoeff g

Alias of MonomialOrder.leadingCoeff_mul_of_isRegular_left.

---

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

@[deprecated MonomialOrder.leadingCoeff_mul_of_isRegular_right (since := "2025-01-31")]

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.leadingCoeff g)) : m.leadingCoeff (f * g) = m.leadingCoeff f * m.leadingCoeff g

Alias of MonomialOrder.leadingCoeff_mul_of_isRegular_right.

---

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

@[deprecated MonomialOrder.leadingCoeff_mul (since := "2025-01-31")]

theorem MonomialOrder.lCoeff_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

Alias of MonomialOrder.leadingCoeff_mul.

---

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.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.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

@[deprecated MonomialOrder.leadingCoeff_sub_of_lt (since := "2025-01-31")]

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.leadingCoeff (f - g) = m.leadingCoeff f

Alias of MonomialOrder.leadingCoeff_sub_of_lt.

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

@[deprecated MonomialOrder.isUnit_leadingCoeff (since := "2025-01-31")]

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.leadingCoeff f) ↔ f ≠ 0

Alias of MonomialOrder.isUnit_leadingCoeff.

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)