Irreducible decomposition of binomial ideals Christopher ONeill - - PowerPoint PPT Presentation

▶
irreducible decomposition of binomial ideals
SMART_READER_LITE
LIVE PREVIEW

Irreducible decomposition of binomial ideals Christopher ONeill - - PowerPoint PPT Presentation

Aug 12, 2023 260 likes •2.2k views

Irreducible decomposition of binomial ideals Christopher ONeill Duke University musicman@math.duke.edu Joint with Thomas Kahle and Ezra Miller January 18, 2014 Christopher ONeill (Duke University) Irreducible decomposition of binomial

slide-1
SLIDE 1

Irreducible decomposition of binomial ideals

Christopher O’Neill

Duke University musicman@math.duke.edu Joint with Thomas Kahle and Ezra Miller

January 18, 2014

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 1 / 36

slide-2
SLIDE 2

The Question

Definition

An ideal I ⊂ k[x1, . . . , xn] is a binomial ideal if it is generated by polynomials with at most two terms.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 2 / 36

slide-3
SLIDE 3

The Question

Definition

An ideal I ⊂ k[x1, . . . , xn] is a binomial ideal if it is generated by polynomials with at most two terms.

Example

x − y ⊂ k[x, y], x2 − xy, xy − y2 ⊂ k[x, y].

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 2 / 36

slide-4
SLIDE 4

The Question

Definition

An ideal I ⊂ k[x1, . . . , xn] is a binomial ideal if it is generated by polynomials with at most two terms.

Example

x − y ⊂ k[x, y], x2 − xy, xy − y2 ⊂ k[x, y].

Example

x2 − y, x2 + y = x2, y ⊂ k[x, y], x2y − xy2, x3, y3 ⊂ k[x, y].

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 2 / 36

slide-5
SLIDE 5

The Question

Definition

An ideal I ⊂ k[x1, . . . , xn] is a binomial ideal if it is generated by polynomials with at most two terms.

Example

x − y ⊂ k[x, y], x2 − xy, xy − y2 ⊂ k[x, y].

Example

x2 − y, x2 + y = x2, y ⊂ k[x, y], x2y − xy2, x3, y3 ⊂ k[x, y].

Example

x2 − xy, x3 − x2, x4y2 + xy2 ∈ x2, y2, xy ⊂ k[x, y].

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 2 / 36

slide-6
SLIDE 6

The Question

Definition

An ideal I ⊂ S is irreducible if whenever I = J1 ∩ J2 for ideals J1, J2 ⊂ S, either I = J1 or I = J2.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 3 / 36

slide-7
SLIDE 7

The Question

Definition

An ideal I ⊂ S is irreducible if whenever I = J1 ∩ J2 for ideals J1, J2 ⊂ S, either I = J1 or I = J2.

Fact

Every ideal I ⊂ k[x1, . . . , xn] can be written as a finite intersection I =

r

  • i=1

Ji

  • f irreducible ideals J1, . . . , Jr (an irreducible decomposition).

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 3 / 36

slide-8
SLIDE 8

The Question

Question (Eisenbud-Sturmfels, 1996)

Assume k is algebraically closed. Does every binomial ideal I have a binomial irreducible decomposition, that is, an expression I =

i Ji where

each Ji is irreducible and binomial?

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 4 / 36

slide-9
SLIDE 9

The Question

Question (Eisenbud-Sturmfels, 1996)

Assume k is algebraically closed. Does every binomial ideal I have a binomial irreducible decomposition, that is, an expression I =

i Ji where

each Ji is irreducible and binomial?

Example

If k = Q, then x4 + 4 = x2 − 2x + 2 ∩ x2 + 2x + 2.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 4 / 36

slide-10
SLIDE 10

The Question

Question (Eisenbud-Sturmfels, 1996)

Assume k is algebraically closed. Does every binomial ideal I have a binomial irreducible decomposition, that is, an expression I =

i Ji where

each Ji is irreducible and binomial?

Example

If k = Q, then x4 + 4 = x2 − 2x + 2 ∩ x2 + 2x + 2.

Answer (Kahle-Miller-O., 2014)

No.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 4 / 36

slide-11
SLIDE 11

The Question

Question (Eisenbud-Sturmfels, 1996)

Assume k is algebraically closed. Does every binomial ideal I have a binomial irreducible decomposition, that is, an expression I =

i Ji where

each Ji is irreducible and binomial?

Example

If k = Q, then x4 + 4 = x2 − 2x + 2 ∩ x2 + 2x + 2.

Answer (Kahle-Miller-O., 2014)

No.

Example

I = x2y − xy2, x3, y3 ⊂ k[x, y].

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 4 / 36

slide-12
SLIDE 12

The Question

State of affairs:

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 5 / 36

slide-13
SLIDE 13

The Question

State of affairs: Question: easy to state

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 5 / 36

slide-14
SLIDE 14

The Question

State of affairs: Question: easy to state Counter example: small

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 5 / 36

slide-15
SLIDE 15

The Question

State of affairs: Question: easy to state Counter example: small Proof: short and elementary

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 5 / 36

slide-16
SLIDE 16

The Question

State of affairs: Question: easy to state Counter example: small Proof: short and elementary So, why was this problem was open for almost 20 years?

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 5 / 36

slide-17
SLIDE 17

The Question

State of affairs: Question: easy to state Counter example: small Proof: short and elementary So, why was this problem was open for almost 20 years? Answer: Needed to know where to look.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 5 / 36

slide-18
SLIDE 18

Storytime!

Today:

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 6 / 36

slide-19
SLIDE 19

Storytime!

Today: Review primary decomposition

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 6 / 36

slide-20
SLIDE 20

Storytime!

Today: Review primary decomposition Irreducible decomposition of monomial ideals

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 6 / 36

slide-21
SLIDE 21

Storytime!

Today: Review primary decomposition Irreducible decomposition of monomial ideals Irreducible decomposition of binomial ideals

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 6 / 36

slide-22
SLIDE 22

Storytime!

Today: Review primary decomposition Irreducible decomposition of monomial ideals Irreducible decomposition of binomial ideals Examine the counterexample, with proof (time permitting).

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 6 / 36

slide-23
SLIDE 23

Primary Decomposition

Definition

An ideal I is primary if ab ∈ I implies aℓ ∈ I or bℓ ∈ I for some ℓ ≥ 1.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 7 / 36

slide-24
SLIDE 24

Primary Decomposition

Definition

An ideal I is primary if ab ∈ I implies aℓ ∈ I or bℓ ∈ I for some ℓ ≥ 1. If I is primary, then p = √ I is prime, and we say I is p-primary.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 7 / 36

slide-25
SLIDE 25

Primary Decomposition

Definition

An ideal I is primary if ab ∈ I implies aℓ ∈ I or bℓ ∈ I for some ℓ ≥ 1. If I is primary, then p = √ I is prime, and we say I is p-primary.

Fact

Any ideal in a Noetherean ring is a finite intersection of primary ideals (that is, admits a primary decomposition).

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 7 / 36

slide-26
SLIDE 26

Primary Decomposition

Definition

An ideal I is primary if ab ∈ I implies aℓ ∈ I or bℓ ∈ I for some ℓ ≥ 1. If I is primary, then p = √ I is prime, and we say I is p-primary.

Fact

Any ideal in a Noetherean ring is a finite intersection of primary ideals (that is, admits a primary decomposition).

Example

Primary ideals in Z are of the form pr for p prime, and

  • pr = p.

For a = pr1

1 · · · prℓ ℓ ∈ Z, a = ipri i .

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 7 / 36

slide-27
SLIDE 27

Irreducible Ideals

Fact

Irreducible ideals are primary.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 8 / 36

slide-28
SLIDE 28

Irreducible Ideals

Fact

Irreducible ideals are primary.

Definition

Given a p-primary ideal I ⊂ k[x1, . . . , xn], the socle of I is the ideal socp(I) = {f : pf ⊂ I} ⊂ I We say I has simple socle if dimk socp(I)/I = 1.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 8 / 36

slide-29
SLIDE 29

Irreducible Ideals

Fact

Irreducible ideals are primary.

Definition

Given a p-primary ideal I ⊂ k[x1, . . . , xn], the socle of I is the ideal socp(I) = {f : pf ⊂ I} ⊂ I We say I has simple socle if dimk socp(I)/I = 1.

Fact

A p-primary ideal I is irreducible if and only if it has simple socle.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 8 / 36

slide-30
SLIDE 30

Irreducible Ideals

Let I = x2 − xy, xy − y2, x4, y4 ⊂ k[x, y], and let p = x, y.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 9 / 36

slide-31
SLIDE 31

Irreducible Ideals

Let I = x2 − xy, xy − y2, x4, y4 ⊂ k[x, y], and let p = x, y. x − y ∈ socp(I)

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 9 / 36

slide-32
SLIDE 32

Irreducible Ideals

Let I = x2 − xy, xy − y2, x4, y4 ⊂ k[x, y], and let p = x, y. x − y ∈ socp(I) x4, x3y, x2y2, xy3, y4 ∈ I

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 9 / 36

slide-33
SLIDE 33

Irreducible Ideals

Let I = x2 − xy, xy − y2, x4, y4 ⊂ k[x, y], and let p = x, y. x − y ∈ socp(I) x4, x3y, x2y2, xy3, y4 ∈ I ⇒ x3, x2y, xy2, y3 ∈ socp(I)

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 9 / 36

slide-34
SLIDE 34

Irreducible Ideals

Let I = x2 − xy, xy − y2, x4, y4 ⊂ k[x, y], and let p = x, y. x − y ∈ socp(I) x4, x3y, x2y2, xy3, y4 ∈ I ⇒ x3, x2y, xy2, y3 ∈ socp(I) socp(I)/I = {f : pf ⊂ I}/I

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 9 / 36

slide-35
SLIDE 35

Irreducible Ideals

Let I = x2 − xy, xy − y2, x4, y4 ⊂ k[x, y], and let p = x, y. x − y ∈ socp(I) x4, x3y, x2y2, xy3, y4 ∈ I ⇒ x3, x2y, xy2, y3 ∈ socp(I) socp(I)/I = {f : pf ⊂ I}/I = {f : xf , yf ∈ I}/I

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 9 / 36

slide-36
SLIDE 36

Irreducible Ideals

Let I = x2 − xy, xy − y2, x4, y4 ⊂ k[x, y], and let p = x, y. x − y ∈ socp(I) x4, x3y, x2y2, xy3, y4 ∈ I ⇒ x3, x2y, xy2, y3 ∈ socp(I) socp(I)/I = {f : pf ⊂ I}/I = {f : xf , yf ∈ I}/I = k{x − y, x3}

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 9 / 36

slide-37
SLIDE 37

Irreducible Ideals

Let I = x2 − xy, xy − y2, x4, y4 ⊂ k[x, y], and let p = x, y. x − y ∈ socp(I) x4, x3y, x2y2, xy3, y4 ∈ I ⇒ x3, x2y, xy2, y3 ∈ socp(I) socp(I)/I = {f : pf ⊂ I}/I = {f : xf , yf ∈ I}/I = k{x − y, x3} so dimk(socp(I)/I) = 2.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 9 / 36

slide-38
SLIDE 38

Monomial Ideals

Long long ago, in an algebraic setting not far away...

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 10 / 36

slide-39
SLIDE 39

Monomial Ideals

Long long ago, in an algebraic setting not far away... Monomial Ideals

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 10 / 36

slide-40
SLIDE 40

Monomial Ideals

I = x4, x3y, x2y2, y4

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 11 / 36

slide-41
SLIDE 41

Monomial Ideals

I = x4, x3y, x2y2, y4 xa = xa1

1 · · · xan n ∈ k[x1, . . . , xn]

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 11 / 36

slide-42
SLIDE 42

Monomial Ideals

I = x4, x3y, x2y2, y4 xa = xa1

1 · · · xan n ∈ k[x1, . . . , xn]

← → a = (a1, . . . , an) ∈ Nn

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 11 / 36

slide-43
SLIDE 43

Monomial Ideals

I = x4, x3y, x2y2, y4 xa = xa1

1 · · · xan n ∈ k[x1, . . . , xn]

← → a = (a1, . . . , an) ∈ Nn

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 11 / 36

slide-44
SLIDE 44

Monomial Ideals

I = x4, x3y, x2y2, y4 xa = xa1

1 · · · xan n ∈ k[x1, . . . , xn]

← → a = (a1, . . . , an) ∈ Nn

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 12 / 36

slide-45
SLIDE 45

Monomial Ideals

I = x4, x3y, x2y2, y4 xa = xa1

1 · · · xan n ∈ k[x1, . . . , xn]

← → a = (a1, . . . , an) ∈ Nn Connect all monomials xa ∈ I

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 12 / 36

slide-46
SLIDE 46

Monomial Ideals

I = x4, x3y, x2y2, y4 xa = xa1

1 · · · xan n ∈ k[x1, . . . , xn]

← → a = (a1, . . . , an) ∈ Nn Connect all monomials xa ∈ I

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 12 / 36

slide-47
SLIDE 47

Monomial Ideals

I = x4, x3y, x2y2, y4 xa = xa1

1 · · · xan n ∈ k[x1, . . . , xn]

← → a = (a1, . . . , an) ∈ Nn Connect all monomials xa ∈ I Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 12 / 36

slide-48
SLIDE 48

Monomial Ideals

I = x4, x3y, x2y2, y4 xa = xa1

1 · · · xan n ∈ k[x1, . . . , xn]

← → a = (a1, . . . , an) ∈ Nn Connect all monomials xa ∈ I Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 12 / 36

slide-49
SLIDE 49

Monomial Ideals

I = x4, x3y, x2y2, y4 xa = xa1

1 · · · xan n ∈ k[x1, . . . , xn]

← → a = (a1, . . . , an) ∈ Nn Connect all monomials xa ∈ I Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 12 / 36

slide-50
SLIDE 50

Monomial Ideals

I = x4, x3y, x2y2, y4 xa = xa1

1 · · · xan n ∈ k[x1, . . . , xn]

← → a = (a1, . . . , an) ∈ Nn Connect all monomials xa ∈ I Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 12 / 36

slide-51
SLIDE 51

Monomial Ideals

I = x4, x3y, x2y2, y4 xa = xa1

1 · · · xan n ∈ k[x1, . . . , xn]

← → a = (a1, . . . , an) ∈ Nn Connect all monomials xa ∈ I Generators of I are “Inward-pointing corners” Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 12 / 36

slide-52
SLIDE 52

Monomial Ideals

I = x4, x3y, x2y2, y4 xa = xa1

1 · · · xan n ∈ k[x1, . . . , xn]

← → a = (a1, . . . , an) ∈ Nn Connect all monomials xa ∈ I Generators of I are “Inward-pointing corners” Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 12 / 36

slide-53
SLIDE 53

Monomial Ideals

Fact

If a monomial ideal I is p-primary, then p is a monomial ideal.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 13 / 36

slide-54
SLIDE 54

Monomial Ideals

Fact

If a monomial ideal I is p-primary, then p is a monomial ideal.

Fact

Any monomial ideal I admits a monomial irreducible decomposition, that is, an expression of the form I =

r

  • i=1

Ji for irreducible monomial ideals J1, . . . , Jr.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 13 / 36

slide-55
SLIDE 55

Monomial Ideals

I = x4, x3y, x2y2, y4 Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 14 / 36

slide-56
SLIDE 56

Monomial Ideals

I = x4, x3y, x2y2, y4 I is p-primary, p = x, y Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 14 / 36

slide-57
SLIDE 57

Monomial Ideals

I = x4, x3y, x2y2, y4 I is p-primary, p = x, y socp(I)/I = k{x3, x2y, xy3} Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 14 / 36

slide-58
SLIDE 58

Monomial Ideals

I = x4, x3y, x2y2, y4 I is p-primary, p = x, y socp(I)/I = k{x3, x2y, xy3} Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 14 / 36

slide-59
SLIDE 59

Monomial Ideals

I = x4, x3y, x2y2, y4 I is p-primary, p = x, y socp(I)/I = k{x3, x2y, xy3} Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 14 / 36

slide-60
SLIDE 60

Monomial Ideals

I = x4, x3y, x2y2, y4 I is p-primary, p = x, y socp(I)/I = k{x3, x2y, xy3} “Outward-pointing corners” Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 14 / 36

slide-61
SLIDE 61

Monomial Ideals

I = x4, x3y, x2y2, y4 I is p-primary, p = x, y socp(I)/I = k{x3, x2y, xy3} “Outward-pointing corners” Irreducible decomposition: I = J1 ∩ J2 ∩ J3 Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 15 / 36

slide-62
SLIDE 62

Monomial Ideals

I = x4, x3y, x2y2, y4 I is p-primary, p = x, y socp(I)/I = k{x3, x2y, xy3} “Outward-pointing corners” Irreducible decomposition: I = J1 ∩ J2 ∩ J3 Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 15 / 36

slide-63
SLIDE 63

Monomial Ideals

I = x4, x3y, x2y2, y4 I is p-primary, p = x, y socp(I)/I = k{x3, x2y, xy3} “Outward-pointing corners” Irreducible decomposition: I = J1 ∩ J2 ∩ J3 J1 = x4, y Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 15 / 36

slide-64
SLIDE 64

Monomial Ideals

I = x4, x3y, x2y2, y4 I is p-primary, p = x, y socp(I)/I = k{x3, x2y, xy3} “Outward-pointing corners” Irreducible decomposition: I = J1 ∩ J2 ∩ J3 J1 = x4, y J2 = x3, y2 Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 15 / 36

slide-65
SLIDE 65

Monomial Ideals

I = x4, x3y, x2y2, y4 I is p-primary, p = x, y socp(I)/I = k{x3, x2y, xy3} “Outward-pointing corners” Irreducible decomposition: I = J1 ∩ J2 ∩ J3 J1 = x4, y J2 = x3, y2 J3 = x2, y4 Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 15 / 36

slide-66
SLIDE 66

Monomial Ideals

I = x4, x3y, x2y2, y4 I is p-primary, p = x, y socp(I)/I = k{x3, x2y, xy3} “Outward-pointing corners” Irreducible decomposition: I = J1 ∩ J2 ∩ J3 J1 = x4, y J2 = x3, y2 J3 = x2, y4 Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 15 / 36

slide-67
SLIDE 67

Monomial Ideals

I = x4, x3y, x2y2, y4 I is p-primary, p = x, y socp(I)/I = k{x3, x2y, xy3} “Outward-pointing corners” Irreducible decomposition: I = J1 ∩ J2 ∩ J3 J1 = x4, y J2 = x3, y2 J3 = x2, y4 Staircase Diagram

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 15 / 36

slide-68
SLIDE 68

Irreducible Decomposition

Facts

Fix an irredundant irreducible decomposition I =

r

  • i=1

Ji for a p-primary ideal I.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 16 / 36

slide-69
SLIDE 69

Irreducible Decomposition

Facts

Fix an irredundant irreducible decomposition I =

r

  • i=1

Ji for a p-primary ideal I. r = dimk socp(I)/I.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 16 / 36

slide-70
SLIDE 70

Irreducible Decomposition

Facts

Fix an irredundant irreducible decomposition I =

r

  • i=1

Ji for a p-primary ideal I. r = dimk socp(I)/I. For each i, the map R/I ։ R/Ji induces a nonzero map on socles.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 16 / 36

slide-71
SLIDE 71

Irreducible Decomposition

Facts

Fix an irredundant irreducible decomposition I =

r

  • i=1

Ji for a p-primary ideal I. r = dimk socp(I)/I. For each i, the map R/I ։ R/Ji induces a nonzero map on socles. More generally, socp(I)/I ∼ = r

i=1 socp(Ji)/Ji.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 16 / 36

slide-72
SLIDE 72

Irreducible Decomposition

Facts

Fix an irredundant irreducible decomposition I =

r

  • i=1

Ji for a p-primary ideal I. r = dimk socp(I)/I. For each i, the map R/I ։ R/Ji induces a nonzero map on socles. More generally, socp(I)/I ∼ = r

i=1 socp(Ji)/Ji.

If I is monomial ideal, then socp(I) is monomial.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 16 / 36

slide-73
SLIDE 73

Binomial Ideals

And now, back to our original programming...

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 17 / 36

slide-74
SLIDE 74

Binomial Ideals

And now, back to our original programming... Binomial ideals

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 17 / 36

slide-75
SLIDE 75

Binomial Ideals

Theorem (Eisenbud-Sturmfels, 1996)

If k = k, every binomial ideal admits a binomial primary decomposition.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 18 / 36

slide-76
SLIDE 76

Binomial Ideals

Theorem (Eisenbud-Sturmfels, 1996)

If k = k, every binomial ideal admits a binomial primary decomposition.

Question (Eisenbud-Sturmfels, 1996)

Does the same hold for irreducible decomposition?

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 18 / 36

slide-77
SLIDE 77

Binomial Ideals

Theorem (Eisenbud-Sturmfels, 1996)

If k = k, every binomial ideal admits a binomial primary decomposition.

Question (Eisenbud-Sturmfels, 1996)

Does the same hold for irreducible decomposition? In 2002, Dickenstein, Matusevich and Miller investigate the combinatorics of binomial primary decomposition.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 18 / 36

slide-78
SLIDE 78

Binomial Ideals

Theorem (Eisenbud-Sturmfels, 1996)

If k = k, every binomial ideal admits a binomial primary decomposition.

Question (Eisenbud-Sturmfels, 1996)

Does the same hold for irreducible decomposition? In 2002, Dickenstein, Matusevich and Miller investigate the combinatorics of binomial primary decomposition. In 2013, Kahle and Miller give a combinatorial method of explicitly constructing binomial primary decomposition.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 18 / 36

slide-79
SLIDE 79

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

slide-80
SLIDE 80

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 xa ∈ k[x1, . . . , xn] ← → a ∈ Nn

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

slide-81
SLIDE 81

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 xa ∈ k[x1, . . . , xn] ← → a ∈ Nn Define relation ∼I on Nn:

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

slide-82
SLIDE 82

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 xa ∈ k[x1, . . . , xn] ← → a ∈ Nn Define relation ∼I on Nn: a ∼I b ∈ Nn ← → xa − λxb ∈ I for some nonzero λ ∈ k

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

slide-83
SLIDE 83

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 xa ∈ k[x1, . . . , xn] ← → a ∈ Nn Define relation ∼I on Nn: a ∼I b ∈ Nn ← → xa − λxb ∈ I for some nonzero λ ∈ k x2 − xy ∈ I,

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

slide-84
SLIDE 84

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 xa ∈ k[x1, . . . , xn] ← → a ∈ Nn Define relation ∼I on Nn: a ∼I b ∈ Nn ← → xa − λxb ∈ I for some nonzero λ ∈ k x2 − xy ∈ I, xy − y2 ∈ I,

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

slide-85
SLIDE 85

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 xa ∈ k[x1, . . . , xn] ← → a ∈ Nn Define relation ∼I on Nn: a ∼I b ∈ Nn ← → xa − λxb ∈ I for some nonzero λ ∈ k x2 − xy ∈ I, xy − y2 ∈ I, x(x2 − xy) = x3 − x2y ∈ I,

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

slide-86
SLIDE 86

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 xa ∈ k[x1, . . . , xn] ← → a ∈ Nn Define relation ∼I on Nn: a ∼I b ∈ Nn ← → xa − λxb ∈ I for some nonzero λ ∈ k x2 − xy ∈ I, xy − y2 ∈ I, x(x2 − xy) = x3 − x2y ∈ I, . . .

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

slide-87
SLIDE 87

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 xa ∈ k[x1, . . . , xn] ← → a ∈ Nn Define relation ∼I on Nn: a ∼I b ∈ Nn ← → xa − λxb ∈ I for some nonzero λ ∈ k x2 − xy ∈ I, xy − y2 ∈ I, x(x2 − xy) = x3 − x2y ∈ I, . . . xa, xb ∈ I ⇒ xa − xb ∈ I

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

slide-88
SLIDE 88

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 xa ∈ k[x1, . . . , xn] ← → a ∈ Nn Define relation ∼I on Nn: a ∼I b ∈ Nn ← → xa − λxb ∈ I for some nonzero λ ∈ k x2 − xy ∈ I, xy − y2 ∈ I, x(x2 − xy) = x3 − x2y ∈ I, . . . xa, xb ∈ I ⇒ xa − xb ∈ I

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

slide-89
SLIDE 89

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 xa ∈ k[x1, . . . , xn] ← → a ∈ Nn Define relation ∼I on Nn: a ∼I b ∈ Nn ← → xa − λxb ∈ I for some nonzero λ ∈ k x2 − xy ∈ I, xy − y2 ∈ I, x(x2 − xy) = x3 − x2y ∈ I, . . . xa, xb ∈ I ⇒ xa − xb ∈ I (x2 = xy in k[x, y]/I)

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

slide-90
SLIDE 90

Binomial Ideals

Fix a binomial ideal I ⊂ k[x1, . . . , xn].

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 20 / 36

slide-91
SLIDE 91

Binomial Ideals

Fix a binomial ideal I ⊂ k[x1, . . . , xn]. The equivalence relation ∼I induced by I on Nn is a congruence: a ∼I b implies a + c ∼I b + c for a, b, c ∈ Nn. In particular, (Nn/∼I, +) is well defined.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 20 / 36

slide-92
SLIDE 92

Binomial Ideals

Fix a binomial ideal I ⊂ k[x1, . . . , xn]. The equivalence relation ∼I induced by I on Nn is a congruence: a ∼I b implies a + c ∼I b + c for a, b, c ∈ Nn. In particular, (Nn/∼I, +) is well defined. The monomials in I form a single class ∞ ∈ Nn/∼I, called the nil.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 20 / 36

slide-93
SLIDE 93

Binomial Ideals

Fix a binomial ideal I ⊂ k[x1, . . . , xn]. The equivalence relation ∼I induced by I on Nn is a congruence: a ∼I b implies a + c ∼I b + c for a, b, c ∈ Nn. In particular, (Nn/∼I, +) is well defined. The monomials in I form a single class ∞ ∈ Nn/∼I, called the nil. The nil ∞ corresponds to 0 in the quotient k[x1, . . . , xn]/I.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 20 / 36

slide-94
SLIDE 94

Binomial Ideals

Fix a binomial ideal I ⊂ k[x1, . . . , xn]. The equivalence relation ∼I induced by I on Nn is a congruence: a ∼I b implies a + c ∼I b + c for a, b, c ∈ Nn. In particular, (Nn/∼I, +) is well defined. The monomials in I form a single class ∞ ∈ Nn/∼I, called the nil. The nil ∞ corresponds to 0 in the quotient k[x1, . . . , xn]/I. Each non-nil a ∈ Nn/∼I represents a distinct monomial modulo I.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 20 / 36

slide-95
SLIDE 95

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 21 / 36

slide-96
SLIDE 96

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 Monoid N2/∼I

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 21 / 36

slide-97
SLIDE 97

Binomial Ideals

Theorem (Kahle-Miller, 2013)

For k = k, every binomial ideal has an expression of the form I =

r

  • i=1

Ji where each Ji is binomial, primary, and has a unique monomial in its socle.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 22 / 36

slide-98
SLIDE 98

Binomial Ideals

Theorem (Kahle-Miller, 2013)

For k = k, every binomial ideal has an expression of the form I =

r

  • i=1

Ji where each Ji is binomial, primary, and has a unique monomial in its socle. To construct a binomial irreducible decomposition for I, we can assume

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 22 / 36

slide-99
SLIDE 99

Binomial Ideals

Theorem (Kahle-Miller, 2013)

For k = k, every binomial ideal has an expression of the form I =

r

  • i=1

Ji where each Ji is binomial, primary, and has a unique monomial in its socle. To construct a binomial irreducible decomposition for I, we can assume I is primary to the maximal ideal m,

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 22 / 36

slide-100
SLIDE 100

Binomial Ideals

Theorem (Kahle-Miller, 2013)

For k = k, every binomial ideal has an expression of the form I =

r

  • i=1

Ji where each Ji is binomial, primary, and has a unique monomial in its socle. To construct a binomial irreducible decomposition for I, we can assume I is primary to the maximal ideal m, socm(I)/I has a unique monomial.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 22 / 36

slide-101
SLIDE 101

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 23 / 36

slide-102
SLIDE 102

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 Nn/∼I ← → monomials mod I

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 23 / 36

slide-103
SLIDE 103

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 Nn/∼I ← → monomials mod I socm(I)/I = k{x3, x − y}

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 23 / 36

slide-104
SLIDE 104

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 Nn/∼I ← → monomials mod I socm(I)/I = k{x3, x − y}

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 23 / 36

slide-105
SLIDE 105

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 Nn/∼I ← → monomials mod I socm(I)/I = k{x3, x − y}

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 23 / 36

slide-106
SLIDE 106

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 Nn/∼I ← → monomials mod I socm(I)/I = k{x3, x − y}

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 23 / 36

slide-107
SLIDE 107

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 Nn/∼I ← → monomials mod I socm(I)/I = k{x3, x − y}

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 23 / 36

slide-108
SLIDE 108

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 Nn/∼I ← → monomials mod I socm(I)/I = k{x3, x − y} witnesses: monomials that merge with something in each direction

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 23 / 36

slide-109
SLIDE 109

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 Nn/∼I ← → monomials mod I socm(I)/I = k{x3, x − y} witnesses: monomials that merge with something in each direction I-witnesses: x3, x, y

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 23 / 36

slide-110
SLIDE 110

Binomial Ideals

Definition

A monomial xa is a witness for I if for each xp ∈ p, p + a ∼I p + a′ for some a′ ∼I a, that is, xa merges with another monomial modulo I when multiplied by any monomial in p.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 24 / 36

slide-111
SLIDE 111

Binomial Ideals

Definition

A monomial xa is a witness for I if for each xp ∈ p, p + a ∼I p + a′ for some a′ ∼I a, that is, xa merges with another monomial modulo I when multiplied by any monomial in p.

Theorem (Kahle-Miller, 2013)

For any p-primary binomial ideal I, any f ∈ socp(I)/I is a sum of witnesses.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 24 / 36

slide-112
SLIDE 112

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 25 / 36

slide-113
SLIDE 113

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 socm(I)/I = k{x3, x − y}

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 25 / 36

slide-114
SLIDE 114

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 socm(I)/I = k{x3, x − y} soccularize I: “Force simple socle”

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 25 / 36

slide-115
SLIDE 115

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 socm(I)/I = k{x3, x − y} soccularize I: “Force simple socle”

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 25 / 36

slide-116
SLIDE 116

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 socm(I)/I = k{x3, x − y} soccularize I: “Force simple socle” J = x − y, x4, y4

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 25 / 36

slide-117
SLIDE 117

Binomial Ideals

I = x2 − xy, xy − y2, x4, y4 socm(I)/I = k{x3, x − y} soccularize I: “Force simple socle” J = x − y, x4, y4 socm(J)/J = k{x3}

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 25 / 36

slide-118
SLIDE 118

Soccular Decomposition

Plan of attack:

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 26 / 36

slide-119
SLIDE 119

Soccular Decomposition

Plan of attack: One irreducible component per witness monomial.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 26 / 36

slide-120
SLIDE 120

Soccular Decomposition

Plan of attack: One irreducible component per witness monomial. For each component, force chosen witness to be maximal.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 26 / 36

slide-121
SLIDE 121

Soccular Decomposition

Plan of attack: One irreducible component per witness monomial. For each component, force chosen witness to be maximal. Soccularize to remove other socle elements.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 26 / 36

slide-122
SLIDE 122

Soccular Decomposition

I = x2 − xy, xy − y2, x4, y4

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 27 / 36

slide-123
SLIDE 123

Soccular Decomposition

I = x2 − xy, xy − y2, x4, y4 Witnesses: x3, x, y

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 27 / 36

slide-124
SLIDE 124

Soccular Decomposition

I = x2 − xy, xy − y2, x4, y4 Witnesses: x3, x, y

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 27 / 36

slide-125
SLIDE 125

Soccular Decomposition

I = x2 − xy, xy − y2, x4, y4 Witnesses: x3, x, y

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 27 / 36

slide-126
SLIDE 126

Soccular Decomposition

I = x2 − xy, xy − y2, x4, y4 Witnesses: x3, x, y J1 = x − y, x4, y4,

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 27 / 36

slide-127
SLIDE 127

Soccular Decomposition

I = x2 − xy, xy − y2, x4, y4 Witnesses: x3, x, y J1 = x − y, x4, y4,

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 27 / 36

slide-128
SLIDE 128

Soccular Decomposition

I = x2 − xy, xy − y2, x4, y4 Witnesses: x3, x, y J1 = x − y, x4, y4, J2 = x2, y,

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 27 / 36

slide-129
SLIDE 129

Soccular Decomposition

I = x2 − xy, xy − y2, x4, y4 Witnesses: x3, x, y J1 = x − y, x4, y4, J2 = x2, y,

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 27 / 36

slide-130
SLIDE 130

Soccular Decomposition

I = x2 − xy, xy − y2, x4, y4 Witnesses: x3, x, y J1 = x − y, x4, y4, J2 = x2, y, J3 = x, y2

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 27 / 36

slide-131
SLIDE 131

Soccular Decomposition

I = x2 − xy, xy − y2, x4, y4 Witnesses: x3, x, y J1 = x − y, x4, y4, J2 = x2, y, J3 = x, y2 I = J1 ∩ J2 ∩ J3

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 27 / 36

slide-132
SLIDE 132

Soccular Decomposition

I = x2 − xy, xy − y2, x4, y4 Witnesses: x3, x, y J1 = x − y, x4, y4, J2 = x2, y, J3 = x, y2 I = J1 ∩ J2 ∩ J3 = J1 ∩ J2

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 27 / 36

slide-133
SLIDE 133

Soccular Decomposition

I = x2 − xy, xy + y2, x4, y4

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 28 / 36

slide-134
SLIDE 134

Soccular Decomposition

I = x2 − xy, xy + y2, x4, y4

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 28 / 36

slide-135
SLIDE 135

Soccular Decomposition

I = x2 − xy, xy + y2, x4, y4 Witnesses: x3, x, y

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 28 / 36

slide-136
SLIDE 136

Soccular Decomposition

I = x2 − xy, xy + y2, x4, y4 Witnesses: x3, x, y socm(I)/I = k{x3}

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 28 / 36

slide-137
SLIDE 137

Soccular Decomposition

I = x2 − xy, xy + y2, x4, y4 Witnesses: x3, x, y socm(I)/I = k{x3} I = I ∩ x2, y ∩ x, y2

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 28 / 36

slide-138
SLIDE 138

Soccular Decomposition

I = xy − y2, x3 − xy2, x4, y4

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 29 / 36

slide-139
SLIDE 139

Soccular Decomposition

I = xy − y2, x3 − xy2, x4, y4

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 29 / 36

slide-140
SLIDE 140

Soccular Decomposition

I = xy − y2, x3 − xy2, x4, y4 Witnesses: x3, x2, xy

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 29 / 36

slide-141
SLIDE 141

Soccular Decomposition

I = xy − y2, x3 − xy2, x4, y4 Witnesses: x3, x2, xy socm(I)/I = k{x3, x2 − xy}

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 29 / 36

slide-142
SLIDE 142

Soccular Decomposition

I = xy − y2, x3 − xy2, x4, y4 Witnesses: x3, x2, xy socm(I)/I = k{x3, x2 − xy} Soccularize:

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 29 / 36

slide-143
SLIDE 143

Soccular Decomposition

I = xy − y2, x3 − xy2, x4, y4 Witnesses: x3, x2, xy socm(I)/I = k{x3, x2 − xy} Soccularize:

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 29 / 36

slide-144
SLIDE 144

Soccular Decomposition

I = xy − y2, x3 − xy2, x4, y4 Witnesses: x3, x2, xy socm(I)/I = k{x3, x2 − xy} Soccularize:

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 29 / 36

slide-145
SLIDE 145

Soccular Decomposition

I = xy − y2, x3 − xy2, x4, y4 Witnesses: x3, x2, xy socm(I)/I = k{x3, x2 − xy} Soccularize: New witnesses!

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 29 / 36

slide-146
SLIDE 146

Soccular Decomposition

I = xy − y2, x3 − xy2, x4, y4 Witnesses: x3, x2, xy socm(I)/I = k{x3, x2 − xy} Soccularize: New witnesses! Protected witnesses: x, y

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 29 / 36

slide-147
SLIDE 147

Soccular Decomposition

I = xy − y2, x3 − xy2, x4, y4 Witnesses: x3, x2, xy socm(I)/I = k{x3, x2 − xy} Soccularize: New witnesses! Protected witnesses: x, y

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 29 / 36

slide-148
SLIDE 148

Soccular Decomposition

I = xy − y2, x3 − xy2, x4, y4 Witnesses: x3, x2, xy socm(I)/I = k{x3, x2 − xy} Soccularize: New witnesses! Protected witnesses: x, y J1 = x − y, x4, y4

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 29 / 36

slide-149
SLIDE 149

Soccular Decomposition

I = xy − y2, x3 − xy2, x4, y4 Witnesses: x3, x2, xy socm(I)/I = k{x3, x2 − xy} Soccularize: New witnesses! Protected witnesses: x, y J1 = x − y, x4, y4

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 30 / 36

slide-150
SLIDE 150

Soccular Decomposition

I = xy − y2, x3 − xy2, x4, y4 Witnesses: x3, x2, xy socm(I)/I = k{x3, x2 − xy} Soccularize: New witnesses! Protected witnesses: x, y J1 = x − y, x4, y4 J2 = x3, y

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 30 / 36

slide-151
SLIDE 151

Soccular Decomposition

I = xy − y2, x3 − xy2, x4, y4 Witnesses: x3, x2, xy socm(I)/I = k{x3, x2 − xy} Soccularize: New witnesses! Protected witnesses: x, y J1 = x − y, x4, y4 J2 = x3, y

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 30 / 36

slide-152
SLIDE 152

Soccular Decomposition

I = xy − y2, x3 − xy2, x4, y4 Witnesses: x3, x2, xy socm(I)/I = k{x3, x2 − xy} Soccularize: New witnesses! Protected witnesses: x, y J1 = x − y, x4, y4 J2 = x3, y J3 = xy − y2, x2, y3

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 30 / 36

slide-153
SLIDE 153

Soccular Decomposition

I = xy − y2, x3 − xy2, x4, y4 Witnesses: x3, x2, xy socm(I)/I = k{x3, x2 − xy} Soccularize: New witnesses! Protected witnesses: x, y J1 = x − y, x4, y4 J2 = x3, y J3 = xy − y2, x2, y3 I = J1 ∩ J2 ∩ J3

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 30 / 36

slide-154
SLIDE 154

Soccular Decomposition

Algorithm for decompositing a binomial ideal I:

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 31 / 36

slide-155
SLIDE 155

Soccular Decomposition

Algorithm for decompositing a binomial ideal I: One component for each I-witness.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 31 / 36

slide-156
SLIDE 156

Soccular Decomposition

Algorithm for decompositing a binomial ideal I: One component for each I-witness. For the component at a witness w:

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 31 / 36

slide-157
SLIDE 157

Soccular Decomposition

Algorithm for decompositing a binomial ideal I: One component for each I-witness. For the component at a witness w:

Add monomials not below w, so w is a unique monomial socle element.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 31 / 36

slide-158
SLIDE 158

Soccular Decomposition

Algorithm for decompositing a binomial ideal I: One component for each I-witness. For the component at a witness w:

Add monomials not below w, so w is a unique monomial socle element. “Soccularize” by merging witness pairs below w.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 31 / 36

slide-159
SLIDE 159

Soccular Decomposition

Algorithm for decompositing a binomial ideal I: One component for each I-witness. For the component at a witness w:

Add monomials not below w, so w is a unique monomial socle element. “Soccularize” by merging witness pairs below w. Repeat with protected witnesses until no new witness pairs are created

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 31 / 36

slide-160
SLIDE 160

Soccular Decomposition

Algorithm for decompositing a binomial ideal I: One component for each I-witness. For the component at a witness w:

Add monomials not below w, so w is a unique monomial socle element. “Soccularize” by merging witness pairs below w. Repeat with protected witnesses until no new witness pairs are created

Theorem (Kahle-Miller-O., 2014)

For k = k, any binomial ideal I can be written as I = r

i=1 Ji, where each

Ji is binomial and pi-primary, and the socle socpi(Ji)/Ji contains a unique monomial and no other binomials.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 31 / 36

slide-161
SLIDE 161

The Counterexample

I = x2y − xy2, x3, y3

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 32 / 36

slide-162
SLIDE 162

The Counterexample

I = x2y − xy2, x3, y3

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 32 / 36

slide-163
SLIDE 163

The Counterexample

I = x2y − xy2, x3, y3 Witnesses: x2y, x2, xy, y2

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 32 / 36

slide-164
SLIDE 164

The Counterexample

I = x2y − xy2, x3, y3 Witnesses: x2y, x2, xy, y2 socm(I)/I = k{x2y, x2 + y2 − xy}

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 32 / 36

slide-165
SLIDE 165

The Counterexample

I = x2y − xy2, x3, y3 Witnesses: x2y, x2, xy, y2 socm(I)/I = k{x2y, x2 + y2 − xy}

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 32 / 36

slide-166
SLIDE 166

The Counterexample

I = x2y − xy2, x3, y3 Witnesses: x2y, x2, xy, y2 socm(I)/I = k{x2y, x2 + y2 − xy}

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 32 / 36

slide-167
SLIDE 167

The Counterexample

I = x2y − xy2, x3, y3 Witnesses: x2y, x2, xy, y2 socm(I)/I = k{x2y, x2 + y2 − xy}

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 32 / 36

slide-168
SLIDE 168

The Counterexample

I = x2y − xy2, x3, y3 Witnesses: x2y, x2, xy, y2 socm(I)/I = k{x2y, x2 + y2 − xy}

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 32 / 36

slide-169
SLIDE 169

The Counterexample

I = x2y − xy2, x3, y3 Witnesses: x2y, x2, xy, y2 socm(I)/I = k{x2y, x2 + y2 − xy} I = x2 + y2 − xy, x3, y3 ∩ x3, y

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 32 / 36

slide-170
SLIDE 170

The Counterexample

Theorem (Kahle-Miller-O., 2014)

I = x2y − xy2, x3, y3 admits no binomial irreducible decomposition.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 33 / 36

slide-171
SLIDE 171

The Counterexample

Theorem (Kahle-Miller-O., 2014)

I = x2y − xy2, x3, y3 admits no binomial irreducible decomposition.

Proof.

Fix an irredundant irreducible decomposition I = r

i=1 Ji.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 33 / 36

slide-172
SLIDE 172

The Counterexample

Theorem (Kahle-Miller-O., 2014)

I = x2y − xy2, x3, y3 admits no binomial irreducible decomposition.

Proof.

Fix an irredundant irreducible decomposition I = r

i=1 Ji.

We have r = dimk(socm(I)/I) = 2, so I = J1 ∩ J2.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 33 / 36

slide-173
SLIDE 173

The Counterexample

Theorem (Kahle-Miller-O., 2014)

I = x2y − xy2, x3, y3 admits no binomial irreducible decomposition.

Proof.

Fix an irredundant irreducible decomposition I = r

i=1 Ji.

We have r = dimk(socm(I)/I) = 2, so I = J1 ∩ J2. Write α = x2 + y2 − xy, β = x2y, so socm(I)/I = k{α, β}.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 33 / 36

slide-174
SLIDE 174

The Counterexample

Theorem (Kahle-Miller-O., 2014)

I = x2y − xy2, x3, y3 admits no binomial irreducible decomposition.

Proof.

Fix an irredundant irreducible decomposition I = r

i=1 Ji.

We have r = dimk(socm(I)/I) = 2, so I = J1 ∩ J2. Write α = x2 + y2 − xy, β = x2y, so socm(I)/I = k{α, β}. We know socm(I)/I ∼ = socm(J1)/J1 ⊕ socm(J2)/J2, so we have α + λβ ∈ socm(Ji)/Ji for some i, say i = 1.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 33 / 36

slide-175
SLIDE 175

The Counterexample

Theorem (Kahle-Miller-O., 2014)

I = x2y − xy2, x3, y3 admits no binomial irreducible decomposition.

Proof.

Fix an irredundant irreducible decomposition I = r

i=1 Ji.

We have r = dimk(socm(I)/I) = 2, so I = J1 ∩ J2. Write α = x2 + y2 − xy, β = x2y, so socm(I)/I = k{α, β}. We know socm(I)/I ∼ = socm(J1)/J1 ⊕ socm(J2)/J2, so we have α + λβ ∈ socm(Ji)/Ji for some i, say i = 1. This means I + α + λβ ⊂ J1.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 33 / 36

slide-176
SLIDE 176

The Counterexample

Theorem (Kahle-Miller-O., 2014)

I = x2y − xy2, x3, y3 admits no binomial irreducible decomposition.

Proof.

Fix an irredundant irreducible decomposition I = r

i=1 Ji.

We have r = dimk(socm(I)/I) = 2, so I = J1 ∩ J2. Write α = x2 + y2 − xy, β = x2y, so socm(I)/I = k{α, β}. We know socm(I)/I ∼ = socm(J1)/J1 ⊕ socm(J2)/J2, so we have α + λβ ∈ socm(Ji)/Ji for some i, say i = 1. This means I + α + λβ ⊂ J1. But I + α + λβ already has simple socle, so J1 = I + α + λβ.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 33 / 36

slide-177
SLIDE 177

References

David Eisenbud, Bernd Sturmfels (1996) Binomial ideals. Duke Math J. 84 (1996), no. 1, 145. Ezra Miller, Bernd Sturmfels (2005) Combinatorial commutative algebra. Graduate Texts in Mathematics 227. Springer-Verlag, New York, 2005. Thomas Kahle, Ezra Miller (2013) Decompositions of commutative monoid congruences and binomial ideals. arXiv:1107.4699 [math]. Thomas Kahle, Ezra Miller, Christopher O’Neill (2014) Irreducible decompositions of binomial ideals. To appear.

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 34 / 36

slide-178
SLIDE 178

References

David Eisenbud, Bernd Sturmfels (1996) Binomial ideals. Duke Math J. 84 (1996), no. 1, 145. Ezra Miller, Bernd Sturmfels (2005) Combinatorial commutative algebra. Graduate Texts in Mathematics 227. Springer-Verlag, New York, 2005. Thomas Kahle, Ezra Miller (2013) Decompositions of commutative monoid congruences and binomial ideals. arXiv:1107.4699 [math]. Thomas Kahle, Ezra Miller, Christopher O’Neill (2014) Irreducible decompositions of binomial ideals. To appear. Thanks!

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 34 / 36

slide-179
SLIDE 179

When do they exist?

I = x2y − xy2, x4 − x3y, xy3 − y4, x5, y5

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 35 / 36

slide-180
SLIDE 180

When do they exist?

I = x2y − xy2, x4 − x3y, xy3 − y4, x5, y5 Witnesses: x4, x3, x2y, y3

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 35 / 36

slide-181
SLIDE 181

When do they exist?

I = x2y − xy2, x4 − x3y, xy3 − y4, x5, y5 Witnesses: x4, x3, x2y, y3 I = J1 ∩ J2 ∩ J3 ∩ J4

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 35 / 36

slide-182
SLIDE 182

When do they exist?

I = x2y − xy2, x4 − x3y, xy3 − y4, x5, y5 Witnesses: x4, x3, x2y, y3 I = J1 ∩ J2 ∩ J3 ∩ J4 J3 not binomial

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 35 / 36

slide-183
SLIDE 183

When do they exist?

I = x2y − xy2, x4 − x3y, xy3 − y4, x5, y5 Witnesses: x4, x3, x2y, y3 I = J1 ∩ J2 ∩ J3 ∩ J4 J3 not binomial

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 35 / 36

slide-184
SLIDE 184

When do they exist?

I = x2y − xy2, x4 − x3y, xy3 − y4, x5, y5 Witnesses: x4, x3, x2y, y3 I = J1 ∩ J2 ∩ J3 ∩ J4 J3 not binomial

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 35 / 36

slide-185
SLIDE 185

When do they exist?

I = x2y − xy2, x4 − x3y, xy3 − y4, x5, y5 Witnesses: x4, x3, x2y, y3 I = J1 ∩ J2 ∩ J3 ∩ J4 J3 not binomial Can omit one of J2, J3, J4

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 35 / 36

slide-186
SLIDE 186

When do they exist?

I ′ = whatever is necessary

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 36 / 36

slide-187
SLIDE 187

When do they exist?

I ′ = whatever is necessary Witnesses: x6, x5, x4y, xy4, y5

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 36 / 36

slide-188
SLIDE 188

When do they exist?

I ′ = whatever is necessary Witnesses: x6, x5, x4y, xy4, y5 I = J1 ∩ J2 ∩ J3 ∩ J4 ∩ J5

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 36 / 36

slide-189
SLIDE 189

When do they exist?

I ′ = whatever is necessary Witnesses: x6, x5, x4y, xy4, y5 I = J1 ∩ J2 ∩ J3 ∩ J4 ∩ J5 J3, J4 not binomial

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 36 / 36

slide-190
SLIDE 190

When do they exist?

I ′ = whatever is necessary Witnesses: x6, x5, x4y, xy4, y5 I = J1 ∩ J2 ∩ J3 ∩ J4 ∩ J5 J3, J4 not binomial

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 36 / 36

slide-191
SLIDE 191

When do they exist?

I ′ = whatever is necessary Witnesses: x6, x5, x4y, xy4, y5 I = J1 ∩ J2 ∩ J3 ∩ J4 ∩ J5 J3, J4 not binomial

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 36 / 36

slide-192
SLIDE 192

When do they exist?

I ′ = whatever is necessary Witnesses: x6, x5, x4y, xy4, y5 I = J1 ∩ J2 ∩ J3 ∩ J4 ∩ J5 J3, J4 not binomial

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 36 / 36

slide-193
SLIDE 193

When do they exist?

I ′ = whatever is necessary Witnesses: x6, x5, x4y, xy4, y5 I = J1 ∩ J2 ∩ J3 ∩ J4 ∩ J5 J3, J4 not binomial Can omit one of J2, J3, J4, J5

Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 36 / 36