Degrees of polynomials

This file establishes many results about the degree of a multivariate polynomial.

The degree set of a polynomial $P \in R[X]$ is a Multiset containing, for each $x$ in the variable set, $n$ copies of $x$, where $n$ is the maximum number of copies of $x$ appearing in a monomial of $P$.

Main declarations

• MvPolynomial.degrees p : the multiset of variables representing the union of the multisets corresponding to each non-zero monomial in p. For example if 7 ≠ 0 in R and p = x²y+7y³ then degrees p = {x, x, y, y, y}

• MvPolynomial.degreeOf n p : ℕ : the total degree of p with respect to the variable n. For example if p = x⁴y+yz then degreeOf y p = 1.

• MvPolynomial.totalDegree p : ℕ : the max of the sizes of the multisets s whose monomials X^s occur in p. For example if p = x⁴y+yz then totalDegree p = 5.

Notation

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

• σ τ : Type* (indexing the variables)

• R : Type* [CommSemiring 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

• r : R

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

• p : MvPolynomial σ R

degrees

def MvPolynomial.degrees {R : Type u} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) : Multiset σ

The maximal degrees of each variable in a multi-variable polynomial, expressed as a multiset.

(For example, degrees (x^2 * y + y^3) would be {x, x, y, y, y}.)

Equations

• p.degrees = p.support.sup fun (s : σ →₀ ℕ) => Finsupp.toMultiset s

theorem MvPolynomial.degrees_def {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (p : MvPolynomial σ R) : p.degrees = p.support.sup fun (s : σ →₀ ℕ) => Finsupp.toMultiset s

theorem MvPolynomial.degrees_monomial {R : Type u} {σ : Type u_1} [CommSemiring R] (s : σ →₀ ℕ) (a : R) : ((MvPolynomial.monomial s) a).degrees ≤ Finsupp.toMultiset s

theorem MvPolynomial.degrees_monomial_eq {R : Type u} {σ : Type u_1} [CommSemiring R] (s : σ →₀ ℕ) (a : R) (ha : a ≠ 0) : ((MvPolynomial.monomial s) a).degrees = Finsupp.toMultiset s

theorem MvPolynomial.degrees_C {R : Type u} {σ : Type u_1} [CommSemiring R] (a : R) : (MvPolynomial.C a).degrees = 0

theorem MvPolynomial.degrees_X' {R : Type u} {σ : Type u_1} [CommSemiring R] (n : σ) : (MvPolynomial.X n).degrees ≤ {n}

@[simp]

theorem MvPolynomial.degrees_X {R : Type u} {σ : Type u_1} [CommSemiring R] [Nontrivial R] (n : σ) : (MvPolynomial.X n).degrees = {n}

@[simp]

theorem MvPolynomial.degrees_zero {R : Type u} {σ : Type u_1} [CommSemiring R] : MvPolynomial.degrees 0 = 0

@[simp]

theorem MvPolynomial.degrees_one {R : Type u} {σ : Type u_1} [CommSemiring R] : MvPolynomial.degrees 1 = 0

theorem MvPolynomial.degrees_add {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (p q : MvPolynomial σ R) : (p + q).degrees ≤ p.degrees ⊔ q.degrees

theorem MvPolynomial.degrees_sum {R : Type u} {σ : Type u_1} [CommSemiring R] {ι : Type u_3} [DecidableEq σ] (s : Finset ι) (f : ι → MvPolynomial σ R) : (∑ i ∈ s, f i).degrees ≤ s.sup fun (i : ι) => (f i).degrees

theorem MvPolynomial.degrees_mul {R : Type u} {σ : Type u_1} [CommSemiring R] (p q : MvPolynomial σ R) : (p * q).degrees ≤ p.degrees + q.degrees

theorem MvPolynomial.degrees_prod {R : Type u} {σ : Type u_1} [CommSemiring R] {ι : Type u_3} (s : Finset ι) (f : ι → MvPolynomial σ R) : (∏ i ∈ s, f i).degrees ≤ ∑ i ∈ s, (f i).degrees

theorem MvPolynomial.degrees_pow {R : Type u} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) (n : ℕ) : (p ^ n).degrees ≤ n • p.degrees

theorem MvPolynomial.mem_degrees {R : Type u} {σ : Type u_1} [CommSemiring R] {p : MvPolynomial σ R} {i : σ} : i ∈ p.degrees ↔ ∃ (d : σ →₀ ℕ), MvPolynomial.coeff d p ≠ 0 ∧ i ∈ d.support

theorem MvPolynomial.le_degrees_add {R : Type u} {σ : Type u_1} [CommSemiring R] {p q : MvPolynomial σ R} (h : Disjoint p.degrees q.degrees) : p.degrees ≤ (p + q).degrees

theorem MvPolynomial.degrees_add_of_disjoint {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] {p q : MvPolynomial σ R} (h : Disjoint p.degrees q.degrees) : (p + q).degrees = p.degrees ∪ q.degrees

theorem MvPolynomial.degrees_map {R : Type u} {S : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S] (p : MvPolynomial σ R) (f : R →+* S) : ((MvPolynomial.map f) p).degrees ⊆ p.degrees

theorem MvPolynomial.degrees_rename {R : Type u} {σ : Type u_1} {τ : Type u_2} [CommSemiring R] (f : σ → τ) (φ : MvPolynomial σ R) : ((MvPolynomial.rename f) φ).degrees ⊆ Multiset.map f φ.degrees

theorem MvPolynomial.degrees_map_of_injective {R : Type u} {S : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S] (p : MvPolynomial σ R) {f : R →+* S} (hf : Function.Injective ⇑f) : ((MvPolynomial.map f) p).degrees = p.degrees

theorem MvPolynomial.degrees_rename_of_injective {R : Type u} {σ : Type u_1} {τ : Type u_2} [CommSemiring R] {p : MvPolynomial σ R} {f : σ → τ} (h : Function.Injective f) : ((MvPolynomial.rename f) p).degrees = Multiset.map f p.degrees

degreeOf

def MvPolynomial.degreeOf {R : Type u} {σ : Type u_1} [CommSemiring R] (n : σ) (p : MvPolynomial σ R) : ℕ

degreeOf n p gives the highest power of X_n that appears in p

Equations

• MvPolynomial.degreeOf n p = Multiset.count n p.degrees

theorem MvPolynomial.degreeOf_def {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (n : σ) (p : MvPolynomial σ R) : MvPolynomial.degreeOf n p = Multiset.count n p.degrees

theorem MvPolynomial.degreeOf_eq_sup {R : Type u} {σ : Type u_1} [CommSemiring R] (n : σ) (f : MvPolynomial σ R) : MvPolynomial.degreeOf n f = f.support.sup fun (m : σ →₀ ℕ) => m n

theorem MvPolynomial.degreeOf_lt_iff {R : Type u} {σ : Type u_1} [CommSemiring R] {n : σ} {f : MvPolynomial σ R} {d : ℕ} (h : 0 < d) : MvPolynomial.degreeOf n f < d ↔ ∀ m ∈ f.support, m n < d

theorem MvPolynomial.degreeOf_le_iff {R : Type u} {σ : Type u_1} [CommSemiring R] {n : σ} {f : MvPolynomial σ R} {d : ℕ} : MvPolynomial.degreeOf n f ≤ d ↔ ∀ m ∈ f.support, m n ≤ d

@[simp]

theorem MvPolynomial.degreeOf_zero {R : Type u} {σ : Type u_1} [CommSemiring R] (n : σ) : MvPolynomial.degreeOf n 0 = 0

@[simp]

theorem MvPolynomial.degreeOf_C {R : Type u} {σ : Type u_1} [CommSemiring R] (a : R) (x : σ) : MvPolynomial.degreeOf x (MvPolynomial.C a) = 0

theorem MvPolynomial.degreeOf_X {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (i j : σ) [Nontrivial R] : MvPolynomial.degreeOf i (MvPolynomial.X j) = if i = j then 1 else 0

theorem MvPolynomial.degreeOf_add_le {R : Type u} {σ : Type u_1} [CommSemiring R] (n : σ) (f g : MvPolynomial σ R) : MvPolynomial.degreeOf n (f + g) ≤ MvPolynomial.degreeOf n f ⊔ MvPolynomial.degreeOf n g

theorem MvPolynomial.monomial_le_degreeOf {R : Type u} {σ : Type u_1} [CommSemiring R] (i : σ) {f : MvPolynomial σ R} {m : σ →₀ ℕ} (h_m : m ∈ f.support) : m i ≤ MvPolynomial.degreeOf i f

theorem MvPolynomial.degreeOf_monomial_eq {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (s : σ →₀ ℕ) (i : σ) {a : R} (ha : a ≠ 0) : MvPolynomial.degreeOf i ((MvPolynomial.monomial s) a) = s i

theorem MvPolynomial.degreeOf_mul_le {R : Type u} {σ : Type u_1} [CommSemiring R] (i : σ) (f g : MvPolynomial σ R) : MvPolynomial.degreeOf i (f * g) ≤ MvPolynomial.degreeOf i f + MvPolynomial.degreeOf i g

theorem MvPolynomial.degreeOf_sum_le {R : Type u} {σ : Type u_1} [CommSemiring R] {ι : Type u_3} (i : σ) (s : Finset ι) (f : ι → MvPolynomial σ R) : MvPolynomial.degreeOf i (∑ j ∈ s, f j) ≤ s.sup fun (j : ι) => MvPolynomial.degreeOf i (f j)

theorem MvPolynomial.degreeOf_prod_le {R : Type u} {σ : Type u_1} [CommSemiring R] {ι : Type u_3} (i : σ) (s : Finset ι) (f : ι → MvPolynomial σ R) : MvPolynomial.degreeOf i (∏ j ∈ s, f j) ≤ ∑ j ∈ s, MvPolynomial.degreeOf i (f j)

theorem MvPolynomial.degreeOf_pow_le {R : Type u} {σ : Type u_1} [CommSemiring R] (i : σ) (p : MvPolynomial σ R) (n : ℕ) : MvPolynomial.degreeOf i (p ^ n) ≤ n * MvPolynomial.degreeOf i p

theorem MvPolynomial.degreeOf_mul_X_ne {R : Type u} {σ : Type u_1} [CommSemiring R] {i j : σ} (f : MvPolynomial σ R) (h : i ≠ j) : MvPolynomial.degreeOf i (f * MvPolynomial.X j) = MvPolynomial.degreeOf i f

theorem MvPolynomial.degreeOf_mul_X_eq {R : Type u} {σ : Type u_1} [CommSemiring R] (j : σ) (f : MvPolynomial σ R) : MvPolynomial.degreeOf j (f * MvPolynomial.X j) ≤ MvPolynomial.degreeOf j f + 1

theorem MvPolynomial.degreeOf_mul_X_eq_degreeOf_add_one_iff {R : Type u} {σ : Type u_1} [CommSemiring R] (j : σ) (f : MvPolynomial σ R) : MvPolynomial.degreeOf j (f * MvPolynomial.X j) = MvPolynomial.degreeOf j f + 1 ↔ f ≠ 0

theorem MvPolynomial.degreeOf_C_mul_le {R : Type u} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) (i : σ) (c : R) : MvPolynomial.degreeOf i (MvPolynomial.C c * p) ≤ MvPolynomial.degreeOf i p

theorem MvPolynomial.degreeOf_mul_C_le {R : Type u} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) (i : σ) (c : R) : MvPolynomial.degreeOf i (p * MvPolynomial.C c) ≤ MvPolynomial.degreeOf i p

theorem MvPolynomial.degreeOf_rename_of_injective {R : Type u} {σ : Type u_1} {τ : Type u_2} [CommSemiring R] {p : MvPolynomial σ R} {f : σ → τ} (h : Function.Injective f) (i : σ) : MvPolynomial.degreeOf (f i) ((MvPolynomial.rename f) p) = MvPolynomial.degreeOf i p

totalDegree

def MvPolynomial.totalDegree {R : Type u} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) : ℕ

totalDegree p gives the maximum |s| over the monomials X^s in p

Equations

• p.totalDegree = p.support.sup fun (s : σ →₀ ℕ) => s.sum fun (x : σ) (e : ℕ) => e

theorem MvPolynomial.totalDegree_eq {R : Type u} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) : p.totalDegree = p.support.sup fun (m : σ →₀ ℕ) => Multiset.card (Finsupp.toMultiset m)

theorem MvPolynomial.le_totalDegree {R : Type u} {σ : Type u_1} [CommSemiring R] {p : MvPolynomial σ R} {s : σ →₀ ℕ} (h : s ∈ p.support) : (s.sum fun (x : σ) (e : ℕ) => e) ≤ p.totalDegree

theorem MvPolynomial.totalDegree_le_degrees_card {R : Type u} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) : p.totalDegree ≤ Multiset.card p.degrees

theorem MvPolynomial.totalDegree_le_of_support_subset {R : Type u} {σ : Type u_1} [CommSemiring R] {p q : MvPolynomial σ R} (h : p.support ⊆ q.support) : p.totalDegree ≤ q.totalDegree

@[simp]

theorem MvPolynomial.totalDegree_C {R : Type u} {σ : Type u_1} [CommSemiring R] (a : R) : (MvPolynomial.C a).totalDegree = 0

@[simp]

theorem MvPolynomial.totalDegree_zero {R : Type u} {σ : Type u_1} [CommSemiring R] : MvPolynomial.totalDegree 0 = 0

@[simp]

theorem MvPolynomial.totalDegree_one {R : Type u} {σ : Type u_1} [CommSemiring R] : MvPolynomial.totalDegree 1 = 0

@[simp]

theorem MvPolynomial.totalDegree_X {σ : Type u_1} {R : Type u_3} [CommSemiring R] [Nontrivial R] (s : σ) : (MvPolynomial.X s).totalDegree = 1

theorem MvPolynomial.totalDegree_add {R : Type u} {σ : Type u_1} [CommSemiring R] (a b : MvPolynomial σ R) : (a + b).totalDegree ≤ a.totalDegree ⊔ b.totalDegree

theorem MvPolynomial.totalDegree_add_eq_left_of_totalDegree_lt {R : Type u} {σ : Type u_1} [CommSemiring R] {p q : MvPolynomial σ R} (h : q.totalDegree < p.totalDegree) : (p + q).totalDegree = p.totalDegree

theorem MvPolynomial.totalDegree_add_eq_right_of_totalDegree_lt {R : Type u} {σ : Type u_1} [CommSemiring R] {p q : MvPolynomial σ R} (h : q.totalDegree < p.totalDegree) : (q + p).totalDegree = p.totalDegree

theorem MvPolynomial.totalDegree_mul {R : Type u} {σ : Type u_1} [CommSemiring R] (a b : MvPolynomial σ R) : (a * b).totalDegree ≤ a.totalDegree + b.totalDegree

theorem MvPolynomial.totalDegree_smul_le {R : Type u} {S : Type v} {σ : Type u_1} [CommSemiring R] [CommSemiring S] [DistribMulAction R S] (a : R) (f : MvPolynomial σ S) : (a • f).totalDegree ≤ f.totalDegree

theorem MvPolynomial.totalDegree_pow {R : Type u} {σ : Type u_1} [CommSemiring R] (a : MvPolynomial σ R) (n : ℕ) : (a ^ n).totalDegree ≤ n * a.totalDegree

@[simp]

theorem MvPolynomial.totalDegree_monomial {R : Type u} {σ : Type u_1} [CommSemiring R] (s : σ →₀ ℕ) {c : R} (hc : c ≠ 0) : ((MvPolynomial.monomial s) c).totalDegree = s.sum fun (x : σ) (e : ℕ) => e

theorem MvPolynomial.totalDegree_monomial_le {R : Type u} {σ : Type u_1} [CommSemiring R] (s : σ →₀ ℕ) (c : R) : ((MvPolynomial.monomial s) c).totalDegree ≤ s.sum fun (x : σ) => id

@[simp]

theorem MvPolynomial.totalDegree_X_pow {R : Type u} {σ : Type u_1} [CommSemiring R] [Nontrivial R] (s : σ) (n : ℕ) : (MvPolynomial.X s ^ n).totalDegree = n

theorem MvPolynomial.totalDegree_list_prod {R : Type u} {σ : Type u_1} [CommSemiring R] (s : List (MvPolynomial σ R)) : s.prod.totalDegree ≤ (List.map MvPolynomial.totalDegree s).sum

theorem MvPolynomial.totalDegree_multiset_prod {R : Type u} {σ : Type u_1} [CommSemiring R] (s : Multiset (MvPolynomial σ R)) : s.prod.totalDegree ≤ (Multiset.map MvPolynomial.totalDegree s).sum

theorem MvPolynomial.totalDegree_finset_prod {R : Type u} {σ : Type u_1} [CommSemiring R] {ι : Type u_3} (s : Finset ι) (f : ι → MvPolynomial σ R) : (s.prod f).totalDegree ≤ ∑ i ∈ s, (f i).totalDegree

theorem MvPolynomial.totalDegree_finset_sum {R : Type u} {σ : Type u_1} [CommSemiring R] {ι : Type u_3} (s : Finset ι) (f : ι → MvPolynomial σ R) : (s.sum f).totalDegree ≤ s.sup fun (i : ι) => (f i).totalDegree

theorem MvPolynomial.totalDegree_finsetSum_le {R : Type u} {σ : Type u_1} [CommSemiring R] {ι : Type u_3} {s : Finset ι} {f : ι → MvPolynomial σ R} {d : ℕ} (hf : ∀ i ∈ s, (f i).totalDegree ≤ d) : (s.sum f).totalDegree ≤ d

theorem MvPolynomial.degreeOf_le_totalDegree {R : Type u} {σ : Type u_1} [CommSemiring R] (f : MvPolynomial σ R) (i : σ) : MvPolynomial.degreeOf i f ≤ f.totalDegree

theorem MvPolynomial.exists_degree_lt {R : Type u} {σ : Type u_1} [CommSemiring R] [Fintype σ] (f : MvPolynomial σ R) (n : ℕ) (h : f.totalDegree < n * Fintype.card σ) {d : σ →₀ ℕ} (hd : d ∈ f.support) : ∃ (i : σ), d i < n

theorem MvPolynomial.coeff_eq_zero_of_totalDegree_lt {R : Type u} {σ : Type u_1} [CommSemiring R] {f : MvPolynomial σ R} {d : σ →₀ ℕ} (h : f.totalDegree < ∑ i ∈ d.support, d i) : MvPolynomial.coeff d f = 0

theorem MvPolynomial.totalDegree_rename_le {R : Type u} {σ : Type u_1} {τ : Type u_2} [CommSemiring R] (f : σ → τ) (p : MvPolynomial σ R) : ((MvPolynomial.rename f) p).totalDegree ≤ p.totalDegree