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

▶
irreducible decompositions of binomial ideals
SMART_READER_LITE
LIVE PREVIEW

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

Jun 11, 2023 143 likes •751 views

Irreducible decompositions 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 decompositions of binomial

slide-1
SLIDE 1

Irreducible decompositions 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 decompositions of binomial ideals January 18, 2014 1 / 13

slide-2
SLIDE 2

Question

Fact

Every ideal I ⊂ k[x1, . . . , xn] can be written as a finite intersection of irreducible ideals (an irreducible decomposition).

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 2 / 13

slide-3
SLIDE 3

Question

Fact

Every ideal I ⊂ k[x1, . . . , xn] can be written as a finite intersection of irreducible ideals (an irreducible decomposition).

Definition

An ideal I ⊂ k[x1, . . . , xn] is a binomial ideal if it is generated by polynomials with at most two terms. Example: I = x − 2y, x2.

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 2 / 13

slide-4
SLIDE 4

Question

Fact

Every ideal I ⊂ k[x1, . . . , xn] can be written as a finite intersection of irreducible ideals (an irreducible decomposition).

Definition

An ideal I ⊂ k[x1, . . . , xn] is a binomial ideal if it is generated by polynomials with at most two terms. Example: I = x − 2y, x2.

Question (Eisenbud-Sturmfels, 1996)

Do binomial ideals have binomial irreducible decompositions?

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 2 / 13

slide-5
SLIDE 5

Question

Fact

Every ideal I ⊂ k[x1, . . . , xn] can be written as a finite intersection of irreducible ideals (an irreducible decomposition).

Definition

An ideal I ⊂ k[x1, . . . , xn] is a binomial ideal if it is generated by polynomials with at most two terms. Example: I = x − 2y, x2.

Question (Eisenbud-Sturmfels, 1996)

Do binomial ideals have binomial irreducible decompositions?

Answer (Kahle-Miller-O., 2014)

No.

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 2 / 13

slide-6
SLIDE 6

Monomial ideals

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

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 3 / 13

slide-7
SLIDE 7

Monomial ideals

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

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 3 / 13

slide-8
SLIDE 8

Monomial ideals

I = x4, x3y, x2y2, y4

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 4 / 13

slide-9
SLIDE 9

Monomial ideals

I = x4, x3y, x2y2, y4

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 4 / 13

slide-10
SLIDE 10

Monomial ideals

I = x4, x3y, x2y2, y4

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 4 / 13

slide-11
SLIDE 11

Monomial ideals

I = x4, x3y, x2y2, y4 “Staircase diagram”

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 5 / 13

slide-12
SLIDE 12

Monomial ideals

I = x4, x3y, x2y2, y4 “Staircase diagram”

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 5 / 13

slide-13
SLIDE 13

Monomial ideals

I = x4, x3y, x2y2, y4 “Staircase diagram”

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 5 / 13

slide-14
SLIDE 14

Monomial ideals

I = x4, x3y, x2y2, y4

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

slide-15
SLIDE 15

Monomial ideals

I = x4, x3y, x2y2, y4

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

slide-16
SLIDE 16

Monomial ideals

I = x4, x3y, x2y2, y4

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

slide-17
SLIDE 17

Monomial ideals

I = x4, x3y, x2y2, y4 =

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

slide-18
SLIDE 18

Monomial ideals

I = x4, x3y, x2y2, y4 = x4, y

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

slide-19
SLIDE 19

Monomial ideals

I = x4, x3y, x2y2, y4 = x4, y x3, y2

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

slide-20
SLIDE 20

Monomial ideals

I = x4, x3y, x2y2, y4 = x4, y x3, y2 x2, y4

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

slide-21
SLIDE 21

Monomial ideals

I = x4, x3y, x2y2, y4 = x4, y ∩ x3, y2 ∩ x2, y4

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

slide-22
SLIDE 22

Monomial ideals

I = x4, x3y, x2y2, y4 = x4, y ∩ x3, y2 ∩ x2, y4

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

slide-23
SLIDE 23

Monomial ideals

I = x4, x3y, x2y2, y4 = x4, y ∩ x3, y2 ∩ x2, y4 irreducible ⇔ “simple socle”

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

slide-24
SLIDE 24

Binomial ideals

And now, back to our original programming...

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 7 / 13

slide-25
SLIDE 25

Binomial ideals

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

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 7 / 13

slide-26
SLIDE 26

Binomial ideals

I = x2 − xy, xy − y2, x4, y4

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 8 / 13

slide-27
SLIDE 27

Binomial ideals

I = x2 − xy, xy − y2, x4, y4

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 8 / 13

slide-28
SLIDE 28

Binomial ideals

I = x2 − xy, xy − y2, x4, y4 x2 = xy in k[x, y]/I

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 8 / 13

slide-29
SLIDE 29

Binomial ideals

I = x2 − xy, xy − y2, x4, y4

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 8 / 13

slide-30
SLIDE 30

Binomial ideals

I = x2 − xy, xy − y2, x4, y4 “witnesses” = monomials that merge in all directions

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 9 / 13

slide-31
SLIDE 31

Binomial ideals

I = x2 − xy, xy − y2, x4, y4 “witnesses” = monomials that merge in all directions

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 9 / 13

slide-32
SLIDE 32

Binomial ideals

I = x2 − xy, xy − y2, x4, y4 “witnesses” = monomials that merge in all directions

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 9 / 13

slide-33
SLIDE 33

Binomial ideals

I = x2 − xy, xy − y2, x4, y4 “witnesses” = monomials that merge in all directions

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 9 / 13

slide-34
SLIDE 34

Binomial ideals

I = x2 − xy, xy − y2, x4, y4 “witnesses” = monomials that merge in all directions To decompose I: force each witness to be a simple socle

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 9 / 13

slide-35
SLIDE 35

Binomial ideals

I = x2 − xy, xy − y2, x4, y4

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 10 / 13

slide-36
SLIDE 36

Binomial ideals

I = x2 − xy, xy − y2, x4, y4 =

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 10 / 13

slide-37
SLIDE 37

Binomial ideals

I = x2 − xy, xy − y2, x4, y4 = x − y ∈ socle

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 10 / 13

slide-38
SLIDE 38

Binomial ideals

I = x2 − xy, xy − y2, x4, y4 = x − y, x4, y4

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 10 / 13

slide-39
SLIDE 39

Binomial ideals

I = x2 − xy, xy − y2, x4, y4 = x − y, x4, y4

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 10 / 13

slide-40
SLIDE 40

Binomial ideals

I = x2 − xy, xy − y2, x4, y4 = x − y, x4, y4 x2, y

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 10 / 13

slide-41
SLIDE 41

Binomial ideals

I = x2 − xy, xy − y2, x4, y4 = x − y, x4, y4 x2, y x, y2

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 10 / 13

slide-42
SLIDE 42

Binomial ideals

I = x2 − xy, xy − y2, x4, y4 = x − y, x4, y4 ∩ x2, y ∩ x, y2

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 10 / 13

slide-43
SLIDE 43

Binomial ideals

I = x2 − xy, xy − y2, x4, y4 = x − y, x4, y4 ∩ x2, y ∩ x, y2

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 10 / 13

slide-44
SLIDE 44

Producing a binomial irreducible decomposition

(Kahle, Miller, O.) To construct binomial irreducible decompositions: One component per witness

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 11 / 13

slide-45
SLIDE 45

Producing a binomial irreducible decomposition

(Kahle, Miller, O.) To construct binomial irreducible decompositions: One component per witness For each component, “soccularize” by merging paired witnesses

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 11 / 13

slide-46
SLIDE 46

Producing a binomial irreducible decomposition

(Kahle, Miller, O.) To construct binomial irreducible decompositions: One component per witness For each component, “soccularize” by merging paired witnesses Merge new witness pairs

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 11 / 13

slide-47
SLIDE 47

Producing a binomial irreducible decomposition

(Kahle, Miller, O.) To construct binomial irreducible decompositions: One component per witness For each component, “soccularize” by merging paired witnesses Merge new witness pairs (“protected witnesses”)

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 11 / 13

slide-48
SLIDE 48

Producing a binomial irreducible decomposition

(Kahle, Miller, O.) To construct binomial irreducible decompositions: One component per witness For each component, “soccularize” by merging paired witnesses Merge new witness pairs (“protected witnesses”) Repeat until no witness pairs remain

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 11 / 13

slide-49
SLIDE 49

Producing a binomial irreducible decomposition

(Kahle, Miller, O.) To construct binomial irreducible decompositions: One component per witness For each component, “soccularize” by merging paired witnesses Merge new witness pairs (“protected witnesses”) Repeat until no witness pairs remain Result should have simple socle

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 11 / 13

slide-50
SLIDE 50

Producing a binomial irreducible decomposition

(Kahle, Miller, O.) To construct binomial irreducible decompositions: One component per witness For each component, “soccularize” by merging paired witnesses Merge new witness pairs (“protected witnesses”) Repeat until no witness pairs remain Result should have simple socle This means binomial irreducible decompositions exist!

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 11 / 13

slide-51
SLIDE 51

Producing a binomial irreducible decomposition

(Kahle, Miller, O.) To construct binomial irreducible decompositions: One component per witness For each component, “soccularize” by merging paired witnesses Merge new witness pairs (“protected witnesses”) Repeat until no witness pairs remain Result should have simple socle This means binomial irreducible decompositions exist! ...almost

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 11 / 13

slide-52
SLIDE 52

Counterexample revisited

I = x2y − xy2, x3, y3

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 12 / 13

slide-53
SLIDE 53

Counterexample revisited

I = x2y − xy2, x3, y3

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 12 / 13

slide-54
SLIDE 54

Counterexample revisited

I = x2y − xy2, x3, y3

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 12 / 13

slide-55
SLIDE 55

Counterexample revisited

I = x2y − xy2, x3, y3

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 12 / 13

slide-56
SLIDE 56

Counterexample revisited

I = x2y − xy2, x3, y3 x2 + y2 − xy ∈ soc(I)

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 12 / 13

slide-57
SLIDE 57

Counterexample revisited

I = x2y − xy2, x3, y3 x2 + y2 − xy ∈ soc(I) I admits no binomial irreducible decompositions

Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 12 / 13

slide-58
SLIDE 58

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 decompositions of binomial ideals January 18, 2014 13 / 13

slide-59
SLIDE 59

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 decompositions of binomial ideals January 18, 2014 13 / 13