LexPlusPowers.lean`

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7 declarations, 0 contain sorry, 0 declared as axioms

Hilbert functions, lex ideals and ideals of pure powers

Let k be a field and let S = k[x₁, …, xₙ], graded by total degree, with the variables ordered by x₁ > x₂ > ⋯ > xₙ. This file defines the Hilbert function d ↦ dim_k I_d of an ideal I of S, monomial ideals, lex ideals, and the ideal generated by pure powers of the first c variables. The sum of a lex ideal and an ideal of pure powers is a lex-plus-powers ideal.

Main definitions

MvPolynomial.hilbertFunction: the Hilbert function d ↦ dim_k I_d of an ideal I.
MvPolynomial.IsMonomialIdeal: the ideal is generated by monomials.
MvPolynomial.IsLexIdeal: a monomial ideal that, in each degree, contains every monomial that is lexicographically larger than one of its monomials.
MvPolynomial.purePowersIdeal: the ideal (x₁ ^ d₁, …, x_c ^ d_c).

References

C. A. Francisco, B. P. Richert, Lex-plus-powers ideals, in Syzygies and Hilbert functions, Lect. Notes Pure Appl. Math. 254, 113-144.

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

def MvPolynomial.hilbertFunction The Hilbert function of a homogeneous ideal I of $k[x_1, \dots, x_n]$, sending d to $\dim_k I_d$ line 55 source
Source documentation
The Hilbert function of a homogeneous ideal I of $k[x_1, \dots, x_n]$, sending d to $\dim_k I_d$. This is finite, since $I_d$ is a subspace of the finite dimensional space $S_d$. Two homogeneous ideals have the same Hilbert function in this sense if and only if the graded quotients by them do, since $\dim_k (S/I)_d = \dim_k S_d - \dim_k I_d$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Field, Ideal, MvPolynomial, Fin, Finsupp
Proof uses: Module.finrank, Subtype, Membership.mem, Submodule, Min.min, Submodule.restrictScalars, MvPolynomial.homogeneousSubmodule
30 instances, coercions and auxiliary constants left out.
```
noncomputable def hilbertFunction (I : Ideal (MvPolynomial (Fin n) k)) (d : ℕ) : ℕ :=
…
```
def MvPolynomial.IsMonomialIdeal An ideal of a polynomial ring is a monomial ideal if it is generated by monomials line 59 source
Source documentation
An ideal of a polynomial ring is a monomial ideal if it is generated by monomials.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Field, Ideal, MvPolynomial, Fin, Finsupp
Proof uses: Exists, Set, Eq, Ideal.span, Set.image, LinearMap, RingHom.id, MvPolynomial.monomial, OfNat.ofNat
29 instances, coercions and auxiliary constants left out.
```
def IsMonomialIdeal (I : Ideal (MvPolynomial (Fin n) k)) : Prop :=
…
```
def MvPolynomial.IsLexIdeal A lex ideal is a monomial ideal L such that for monomials u, v of the same degree with u larger than v in the lexicographic order, v ∈ L implies u ∈ L line 67 source
Source documentation
A lex ideal is a monomial ideal L such that for monomials u, v of the same degree with u larger than v in the lexicographic order, v ∈ L implies u ∈ L. Here the variables are ordered by $x_1 > x_2 > \dots > x_n$, which for exponent vectors is toLex.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Field, Ideal, MvPolynomial, Fin, Finsupp
Proof uses: And, MvPolynomial.IsMonomialIdeal, Eq, AddMonoidHom, Finsupp.degree, LT.lt, Lex, Equiv, toLex, Membership.mem, LinearMap, RingHom.id, MvPolynomial.monomial, OfNat.ofNat
43 instances, coercions and auxiliary constants left out.
```
def IsLexIdeal (L : Ideal (MvPolynomial (Fin n) k)) : Prop :=
…
```
def MvPolynomial.purePowersIdeal The ideal generated by the pure powers $x_1^{d_1}, \dots, x_c^{d_c}$ line 72 source
Source documentation
The ideal generated by the pure powers $x_1^{d_1}, \dots, x_c^{d_c}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Field, LE.le, Fin, Ideal, MvPolynomial, Finsupp
Proof uses: Ideal.span, Set.range, HPow.hPow, MvPolynomial.X, Fin.castLE
22 instances, coercions and auxiliary constants left out.
```
noncomputable def purePowersIdeal {c : ℕ} (hc : c ≤ n) (d : Fin c → ℕ) :
    Ideal (MvPolynomial (Fin n) k) :=
…
```
Mentions: c
theorem MvPolynomial.hilbertFunction_bot The zero ideal has Hilbert function zero line 77 source
Source documentation
The zero ideal has Hilbert function zero.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Field, Eq, MvPolynomial.hilbertFunction, Bot.bot, Ideal, MvPolynomial, Fin, Finsupp, OfNat.ofNat
Proof uses: Eq.mpr, Module.finrank, Subtype, Membership.mem, Submodule, Min.min, Submodule.restrictScalars, MvPolynomial.homogeneousSubmodule, id, congrArg, MvPolynomial.hilbertFunction.eq_1, Submodule.restrictScalars_bot, bot_inf_eq, finrank_bot, Eq.refl
48 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem hilbertFunction_bot (d : ℕ) :
    hilbertFunction (⊥ : Ideal (MvPolynomial (Fin n) k)) d = 0 := by
…
```
Mentions: MvPolynomial.hilbertFunction
theorem MvPolynomial.purePowersIdeal_zero For c = 0 there are no pure powers, so purePowersIdeal is the zero ideal line 82 source
Source documentation
For c = 0 there are no pure powers, so purePowersIdeal is the zero ideal.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Field, LE.le, OfNat.ofNat, Fin, Eq, Ideal, MvPolynomial, Finsupp, MvPolynomial.purePowersIdeal, Bot.bot
Proof uses: of_eq_true, Eq.trans, Submodule, True, congrFun', congrArg, Ideal.span, Set.range, HPow.hPow, MvPolynomial.X, Fin.castLE, EmptyCollection.emptyCollection, Set, Matrix.range_empty, Submodule.span_empty, eq_self
31 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem purePowersIdeal_zero (hc : 0 ≤ n) (d : Fin 0 → ℕ) :
    (purePowersIdeal hc d : Ideal (MvPolynomial (Fin n) k)) = ⊥ := by
…
```
Mentions: MvPolynomial.purePowersIdeal
theorem MvPolynomial.isLexIdeal_top The unit ideal is a lex ideal line 87 source
Source documentation
The unit ideal is a lex ideal.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Field, MvPolynomial.IsLexIdeal, Top.top, Ideal, MvPolynomial, Fin, Finsupp
Proof uses: And.intro, MvPolynomial.IsMonomialIdeal, Eq, AddMonoidHom, Finsupp.degree, LT.lt, Lex, Equiv, toLex, Membership.mem, LinearMap, RingHom.id, MvPolynomial.monomial, OfNat.ofNat, Exists.intro, Set, Ideal.span, Set.image, Set.univ, Eq.mpr and 11 more
53 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem isLexIdeal_top : IsLexIdeal (⊤ : Ideal (MvPolynomial (Fin n) k)) := by
…
```
Mentions: MvPolynomial.IsLexIdeal