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` #

source

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}.)

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Instances For

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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

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

((monomial s) a).degrees ≤ Finsupp.toMultiset s

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

((monomial s) a).degrees = Finsupp.toMultiset s

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

(C a).degrees = 0

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

(X n).degrees ≤ {n}

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@[simp]

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

(X n).degrees = {n}

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@[simp]

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

degrees 0 = 0

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@[simp]

theorem MvPolynomial.degrees_one {R : Type u} {σ : Type u_1} [CommSemiring R] :

degrees 1 = 0

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theorem MvPolynomial.degrees_add_le {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] {p q : MvPolynomial σ R} :

(p + q).degrees ≤ p.degrees ⊔ q.degrees

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theorem MvPolynomial.degrees_sum_le {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

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theorem MvPolynomial.degrees_mul_le {R : Type u} {σ : Type u_1} [CommSemiring R] {p q : MvPolynomial σ R} :

(p * q).degrees ≤ p.degrees + q.degrees

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theorem MvPolynomial.degrees_prod_le {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

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theorem MvPolynomial.degrees_pow_le {R : Type u} {σ : Type u_1} [CommSemiring R] {p : MvPolynomial σ R} {n : ℕ} :

(p ^ n).degrees ≤ n • p.degrees

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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), coeff d p ≠ 0 ∧ i ∈ d.support

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theorem MvPolynomial.le_degrees_add_left {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

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

q.degrees ≤ (p + q).degrees

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

(p + q).degrees = p.degrees ∪ q.degrees

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theorem MvPolynomial.degrees_map_le {R : Type u} {S : Type v} {σ : Type u_1} [CommSemiring R] {p : MvPolynomial σ R} [CommSemiring S] {f : R →+* S} :

((map f) p).degrees ≤ p.degrees

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

((rename f) φ).degrees ⊆ Multiset.map f φ.degrees

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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) :

((map f) p).degrees = p.degrees

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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) :

((rename f) p).degrees = Multiset.map f p.degrees

`degreeOf` #

source

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

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Instances For

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

degreeOf n p = Multiset.count n p.degrees

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theorem MvPolynomial.degreeOf_eq_sup {R : Type u} {σ : Type u_1} [CommSemiring R] (n : σ) (f : MvPolynomial σ R) :

degreeOf n f = f.support.sup fun (m : σ →₀ ℕ) => m n

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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) :

degreeOf n f < d ↔ ∀ m ∈ f.support, m n < d

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theorem MvPolynomial.degreeOf_le_iff {R : Type u} {σ : Type u_1} [CommSemiring R] {n : σ} {f : MvPolynomial σ R} {d : ℕ} :

degreeOf n f ≤ d ↔ ∀ m ∈ f.support, m n ≤ d

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@[simp]

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

degreeOf n 0 = 0

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@[simp]

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

degreeOf x (C a) = 0

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theorem MvPolynomial.degreeOf_X {R : Type u} {σ : Type u_1} [CommSemiring R] [DecidableEq σ] (i j : σ) [Nontrivial R] :

degreeOf i (X j) = if i = j then 1 else 0

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

degreeOf n (f + g) ≤ max (degreeOf n f) (degreeOf n g)

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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 ≤ degreeOf i f

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

degreeOf i ((monomial s) a) = s i

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

degreeOf i (f * g) ≤ degreeOf i f + degreeOf i g

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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) :

degreeOf i (∑ j ∈ s, f j) ≤ s.sup fun (j : ι) => degreeOf i (f j)

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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) :

degreeOf i (∏ j ∈ s, f j) ≤ ∑ j ∈ s, degreeOf i (f j)

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

degreeOf i (p ^ n) ≤ n * degreeOf i p

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theorem MvPolynomial.degreeOf_mul_X_of_ne {R : Type u} {σ : Type u_1} [CommSemiring R] {i j : σ} (f : MvPolynomial σ R) (h : i ≠ j) :

degreeOf i (f * X j) = degreeOf i f

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theorem MvPolynomial.degreeOf_mul_X_self {R : Type u} {σ : Type u_1} [CommSemiring R] (j : σ) (f : MvPolynomial σ R) :

degreeOf j (f * X j) ≤ degreeOf j f + 1

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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) :

degreeOf j (f * X j) = degreeOf j f + 1 ↔ f ≠ 0

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

degreeOf i (C c * p) ≤ degreeOf i p

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

degreeOf i (p * C c) ≤ degreeOf i p

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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) :

degreeOf (f i) ((rename f) p) = degreeOf i p

`totalDegree` #

source

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

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Instances For

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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) => (Finsupp.toMultiset m).card

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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

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theorem MvPolynomial.totalDegree_le_degrees_card {R : Type u} {σ : Type u_1} [CommSemiring R] (p : MvPolynomial σ R) :

p.totalDegree ≤ p.degrees.card

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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

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@[simp]

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

(C a).totalDegree = 0

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@[simp]

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

totalDegree 0 = 0

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@[simp]

theorem MvPolynomial.totalDegree_one {R : Type u} {σ : Type u_1} [CommSemiring R] :

totalDegree 1 = 0

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@[simp]

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

(X s).totalDegree = 1

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

(a + b).totalDegree ≤ max a.totalDegree b.totalDegree

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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

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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

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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

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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

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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

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@[simp]

theorem MvPolynomial.totalDegree_monomial {R : Type u} {σ : Type u_1} [CommSemiring R] (s : σ →₀ ℕ) {c : R} (hc : c ≠ 0) :

((monomial s) c).totalDegree = s.sum fun (x : σ) (e : ℕ) => e

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theorem MvPolynomial.totalDegree_monomial_le {R : Type u} {σ : Type u_1} [CommSemiring R] (s : σ →₀ ℕ) (c : R) :

((monomial s) c).totalDegree ≤ s.sum fun (x : σ) => id

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@[simp]

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

(X s ^ n).totalDegree = n

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theorem MvPolynomial.totalDegree_list_prod {R : Type u} {σ : Type u_1} [CommSemiring R] (s : List (MvPolynomial σ R)) :

s.prod.totalDegree ≤ (List.map totalDegree s).sum

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theorem MvPolynomial.totalDegree_multiset_prod {R : Type u} {σ : Type u_1} [CommSemiring R] (s : Multiset (MvPolynomial σ R)) :

s.prod.totalDegree ≤ (Multiset.map totalDegree s).sum

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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

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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

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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

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

degreeOf i f ≤ f.totalDegree

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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

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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) :

coeff d f = 0

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theorem MvPolynomial.totalDegree_eq_zero_iff_eq_C {R : Type u} {σ : Type u_1} [CommSemiring R] {p : MvPolynomial σ R} :

p.totalDegree = 0 ↔ p = C (coeff 0 p)

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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) :

((rename f) p).totalDegree ≤ p.totalDegree

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theorem MvPolynomial.totalDegree_renameEquiv {R : Type u} {σ : Type u_1} {τ : Type u_2} [CommSemiring R] (f : σ ≃ τ) (p : MvPolynomial σ R) :

((renameEquiv R f) p).totalDegree = p.totalDegree

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def MvPolynomial.degreesLE (R : Type u) (σ : Type u_1) [CommSemiring R] (s : Multiset σ) :

Submodule R (MvPolynomial σ R)

The submodule of multivariate polynomials of degrees bounded by a monomial s.

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Instances For

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@[simp]

theorem MvPolynomial.mem_degreesLE {R : Type u} {σ : Type u_1} [CommSemiring R] {p : MvPolynomial σ R} {s : Multiset σ} :

p ∈ degreesLE R σ s ↔ p.degrees ≤ s

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theorem MvPolynomial.degreesLE_add {R : Type u} {σ : Type u_1} [CommSemiring R] (s t : Multiset σ) :

degreesLE R σ (s + t) = degreesLE R σ s * degreesLE R σ t

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@[simp]

theorem MvPolynomial.degreesLE_zero {R : Type u} {σ : Type u_1} [CommSemiring R] :

degreesLE R σ 0 = 1

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theorem MvPolynomial.degreesLE_nsmul {R : Type u} {σ : Type u_1} [CommSemiring R] (s : Multiset σ) (n : ℕ) :

degreesLE R σ (n • s) = degreesLE R σ s ^ n

Documentation

Mathlib.Algebra.MvPolynomial.Degrees

Degrees of polynomials #

Main declarations #

Notation #

`degrees` #

`degreeOf` #

`totalDegree` #

Degrees of polynomials #

Main declarations #

Notation #

degrees #

degreeOf #

totalDegree #

`degrees` #

`degreeOf` #

`totalDegree` #