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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

  1. Irreducible Ideals Let I = � x 2 − xy , xy − y 2 , x 4 , y 4 � ⊂ k [ x , y ], and let p = � x , y � . Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 9 / 36

  2. Irreducible Ideals Let I = � x 2 − xy , xy − y 2 , x 4 , y 4 � ⊂ k [ x , y ], and let p = � x , y � . x − y ∈ soc p ( I ) Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 9 / 36

  3. Irreducible Ideals Let I = � x 2 − xy , xy − y 2 , x 4 , y 4 � ⊂ k [ x , y ], and let p = � x , y � . x − y ∈ soc p ( I ) x 4 , x 3 y , x 2 y 2 , xy 3 , y 4 ∈ I Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 9 / 36

  4. Irreducible Ideals Let I = � x 2 − xy , xy − y 2 , x 4 , y 4 � ⊂ k [ x , y ], and let p = � x , y � . x − y ∈ soc p ( I ) x 4 , x 3 y , x 2 y 2 , xy 3 , y 4 ∈ I ⇒ x 3 , x 2 y , xy 2 , y 3 ∈ soc p ( I ) Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 9 / 36

  5. Irreducible Ideals Let I = � x 2 − xy , xy − y 2 , x 4 , y 4 � ⊂ k [ x , y ], and let p = � x , y � . x − y ∈ soc p ( I ) x 4 , x 3 y , x 2 y 2 , xy 3 , y 4 ∈ I ⇒ x 3 , x 2 y , xy 2 , y 3 ∈ soc p ( I ) soc p ( I ) / I = { f : p f ⊂ I } / I Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 9 / 36

  6. Irreducible Ideals Let I = � x 2 − xy , xy − y 2 , x 4 , y 4 � ⊂ k [ x , y ], and let p = � x , y � . x − y ∈ soc p ( I ) x 4 , x 3 y , x 2 y 2 , xy 3 , y 4 ∈ I ⇒ x 3 , x 2 y , xy 2 , y 3 ∈ soc p ( I ) soc p ( I ) / I = { f : p f ⊂ I } / I = { f : xf , yf ∈ I } / I Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 9 / 36

  7. Irreducible Ideals Let I = � x 2 − xy , xy − y 2 , x 4 , y 4 � ⊂ k [ x , y ], and let p = � x , y � . x − y ∈ soc p ( I ) x 4 , x 3 y , x 2 y 2 , xy 3 , y 4 ∈ I ⇒ x 3 , x 2 y , xy 2 , y 3 ∈ soc p ( I ) soc p ( I ) / I = { f : p f ⊂ I } / I = { f : xf , yf ∈ I } / I k { x − y , x 3 } = Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 9 / 36

  8. Irreducible Ideals Let I = � x 2 − xy , xy − y 2 , x 4 , y 4 � ⊂ k [ x , y ], and let p = � x , y � . x − y ∈ soc p ( I ) x 4 , x 3 y , x 2 y 2 , xy 3 , y 4 ∈ I ⇒ x 3 , x 2 y , xy 2 , y 3 ∈ soc p ( I ) soc p ( I ) / I = { f : p f ⊂ I } / I = { f : xf , yf ∈ I } / I k { x − y , x 3 } = so dim k (soc p ( I ) / I ) = 2. Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 9 / 36

  9. 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

  10. 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

  11. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 11 / 36

  12. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = x a = x a 1 1 · · · x a n n ∈ k [ x 1 , . . . , x n ] Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 11 / 36

  13. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = x a = x a 1 1 · · · x a n n ∈ k [ x 1 , . . . , x n ] → a = ( a 1 , . . . , a n ) ∈ N n ← Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 11 / 36

  14. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = x a = x a 1 1 · · · x a n n ∈ k [ x 1 , . . . , x n ] → a = ( a 1 , . . . , a n ) ∈ N n ← Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 11 / 36

  15. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = x a = x a 1 1 · · · x a n n ∈ k [ x 1 , . . . , x n ] → a = ( a 1 , . . . , a n ) ∈ N n ← Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 12 / 36

  16. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = x a = x a 1 1 · · · x a n n ∈ k [ x 1 , . . . , x n ] → a = ( a 1 , . . . , a n ) ∈ N n ← Connect all monomials x a ∈ I Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 12 / 36

  17. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = x a = x a 1 1 · · · x a n n ∈ k [ x 1 , . . . , x n ] → a = ( a 1 , . . . , a n ) ∈ N n ← Connect all monomials x a ∈ I Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 12 / 36

  18. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram x a = x a 1 1 · · · x a n n ∈ k [ x 1 , . . . , x n ] → a = ( a 1 , . . . , a n ) ∈ N n ← Connect all monomials x a ∈ I Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 12 / 36

  19. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram x a = x a 1 1 · · · x a n n ∈ k [ x 1 , . . . , x n ] → a = ( a 1 , . . . , a n ) ∈ N n ← Connect all monomials x a ∈ I Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 12 / 36

  20. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram x a = x a 1 1 · · · x a n n ∈ k [ x 1 , . . . , x n ] → a = ( a 1 , . . . , a n ) ∈ N n ← Connect all monomials x a ∈ I Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 12 / 36

  21. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram x a = x a 1 1 · · · x a n n ∈ k [ x 1 , . . . , x n ] → a = ( a 1 , . . . , a n ) ∈ N n ← Connect all monomials x a ∈ I Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 12 / 36

  22. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram x a = x a 1 1 · · · x a n n ∈ k [ x 1 , . . . , x n ] → a = ( a 1 , . . . , a n ) ∈ N n ← Connect all monomials x a ∈ I Generators of I are “Inward-pointing corners” Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 12 / 36

  23. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram x a = x a 1 1 · · · x a n n ∈ k [ x 1 , . . . , x n ] → a = ( a 1 , . . . , a n ) ∈ N n ← Connect all monomials x a ∈ I Generators of I are “Inward-pointing corners” Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 12 / 36

  24. 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

  25. 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 r � I = J i i =1 for irreducible monomial ideals J 1 , . . . , J r . Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 13 / 36

  26. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 14 / 36

  27. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram I is p -primary, p = � x , y � Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 14 / 36

  28. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram I is p -primary, p = � x , y � soc p ( I ) / I = k { x 3 , x 2 y , xy 3 } Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 14 / 36

  29. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram I is p -primary, p = � x , y � soc p ( I ) / I = k { x 3 , x 2 y , xy 3 } Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 14 / 36

  30. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram I is p -primary, p = � x , y � soc p ( I ) / I = k { x 3 , x 2 y , xy 3 } Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 14 / 36

  31. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram I is p -primary, p = � x , y � soc p ( I ) / I = k { x 3 , x 2 y , xy 3 } “Outward-pointing corners” Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 14 / 36

  32. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram I is p -primary, p = � x , y � soc p ( I ) / I = k { x 3 , x 2 y , xy 3 } “Outward-pointing corners” Irreducible decomposition: I = J 1 ∩ J 2 ∩ J 3 Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 15 / 36

  33. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram I is p -primary, p = � x , y � soc p ( I ) / I = k { x 3 , x 2 y , xy 3 } “Outward-pointing corners” Irreducible decomposition: I = J 1 ∩ J 2 ∩ J 3 Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 15 / 36

  34. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram I is p -primary, p = � x , y � soc p ( I ) / I = k { x 3 , x 2 y , xy 3 } “Outward-pointing corners” Irreducible decomposition: I = J 1 ∩ J 2 ∩ J 3 J 1 = � x 4 , y � Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 15 / 36

  35. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram I is p -primary, p = � x , y � soc p ( I ) / I = k { x 3 , x 2 y , xy 3 } “Outward-pointing corners” Irreducible decomposition: I = J 1 ∩ J 2 ∩ J 3 J 1 = � x 4 , y � J 2 = � x 3 , y 2 � Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 15 / 36

  36. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram I is p -primary, p = � x , y � soc p ( I ) / I = k { x 3 , x 2 y , xy 3 } “Outward-pointing corners” Irreducible decomposition: I = J 1 ∩ J 2 ∩ J 3 J 1 = � x 4 , y � J 2 = � x 3 , y 2 � J 3 = � x 2 , y 4 � Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 15 / 36

  37. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram I is p -primary, p = � x , y � soc p ( I ) / I = k { x 3 , x 2 y , xy 3 } “Outward-pointing corners” Irreducible decomposition: I = J 1 ∩ J 2 ∩ J 3 J 1 = � x 4 , y � J 2 = � x 3 , y 2 � J 3 = � x 2 , y 4 � Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 15 / 36

  38. Monomial Ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Staircase Diagram I is p -primary, p = � x , y � soc p ( I ) / I = k { x 3 , x 2 y , xy 3 } “Outward-pointing corners” Irreducible decomposition: I = J 1 ∩ J 2 ∩ J 3 J 1 = � x 4 , y � J 2 = � x 3 , y 2 � J 3 = � x 2 , y 4 � Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 15 / 36

  39. Irreducible Decomposition Facts Fix an irredundant irreducible decomposition r � I = J i i =1 for a p -primary ideal I . Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 16 / 36

  40. Irreducible Decomposition Facts Fix an irredundant irreducible decomposition r � I = J i i =1 for a p -primary ideal I . r = dim k soc p ( I ) / I . Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 16 / 36

  41. Irreducible Decomposition Facts Fix an irredundant irreducible decomposition r � I = J i i =1 for a p -primary ideal I . r = dim k soc p ( I ) / I . For each i , the map R / I ։ R / J i induces a nonzero map on socles. Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 16 / 36

  42. Irreducible Decomposition Facts Fix an irredundant irreducible decomposition r � I = J i i =1 for a p -primary ideal I . r = dim k soc p ( I ) / I . For each i , the map R / I ։ R / J i induces a nonzero map on socles. More generally, soc p ( I ) / I ∼ = � r i =1 soc p ( J i ) / J i . Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 16 / 36

  43. Irreducible Decomposition Facts Fix an irredundant irreducible decomposition r � I = J i i =1 for a p -primary ideal I . r = dim k soc p ( I ) / I . For each i , the map R / I ։ R / J i induces a nonzero map on socles. More generally, soc p ( I ) / I ∼ = � r i =1 soc p ( J i ) / J i . If I is monomial ideal, then soc p ( I ) is monomial. Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 16 / 36

  44. Binomial Ideals And now, back to our original programming... Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 17 / 36

  45. 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

  46. 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

  47. 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

  48. 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

  49. 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

  50. Binomial Ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � I = Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

  51. Binomial Ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � I = x a ∈ k [ x 1 , . . . , x n ] ← → a ∈ N n Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

  52. Binomial Ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � I = x a ∈ k [ x 1 , . . . , x n ] ← → a ∈ N n Define relation ∼ I on N n : Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

  53. Binomial Ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � I = x a ∈ k [ x 1 , . . . , x n ] ← → a ∈ N n Define relation ∼ I on N n : a ∼ I b ∈ N n ← → x a − λ x b ∈ I for some nonzero λ ∈ k Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

  54. Binomial Ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � I = x a ∈ k [ x 1 , . . . , x n ] ← → a ∈ N n Define relation ∼ I on N n : a ∼ I b ∈ N n ← → x a − λ x b ∈ I for some nonzero λ ∈ k x 2 − xy ∈ I , Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

  55. Binomial Ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � I = x a ∈ k [ x 1 , . . . , x n ] ← → a ∈ N n Define relation ∼ I on N n : a ∼ I b ∈ N n ← → x a − λ x b ∈ I for some nonzero λ ∈ k x 2 − xy ∈ I , xy − y 2 ∈ I , Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

  56. Binomial Ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � I = x a ∈ k [ x 1 , . . . , x n ] ← → a ∈ N n Define relation ∼ I on N n : a ∼ I b ∈ N n ← → x a − λ x b ∈ I for some nonzero λ ∈ k x 2 − xy ∈ I , xy − y 2 ∈ I , x ( x 2 − xy ) = x 3 − x 2 y ∈ I , Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

  57. Binomial Ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � I = x a ∈ k [ x 1 , . . . , x n ] ← → a ∈ N n Define relation ∼ I on N n : a ∼ I b ∈ N n ← → x a − λ x b ∈ I for some nonzero λ ∈ k x 2 − xy ∈ I , xy − y 2 ∈ I , x ( x 2 − xy ) = x 3 − x 2 y ∈ I , . . . Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

  58. Binomial Ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � I = x a ∈ k [ x 1 , . . . , x n ] ← → a ∈ N n Define relation ∼ I on N n : a ∼ I b ∈ N n ← → x a − λ x b ∈ I for some nonzero λ ∈ k x 2 − xy ∈ I , xy − y 2 ∈ I , x ( x 2 − xy ) = x 3 − x 2 y ∈ I , . . . x a , x b ∈ I ⇒ x a − x b ∈ I Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

  59. Binomial Ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � I = x a ∈ k [ x 1 , . . . , x n ] ← → a ∈ N n Define relation ∼ I on N n : a ∼ I b ∈ N n ← → x a − λ x b ∈ I for some nonzero λ ∈ k x 2 − xy ∈ I , xy − y 2 ∈ I , x ( x 2 − xy ) = x 3 − x 2 y ∈ I , . . . x a , x b ∈ I ⇒ x a − x b ∈ I Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

  60. Binomial Ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � I = x a ∈ k [ x 1 , . . . , x n ] ← → a ∈ N n Define relation ∼ I on N n : a ∼ I b ∈ N n ← → x a − λ x b ∈ I for some nonzero λ ∈ k x 2 − xy ∈ I , xy − y 2 ∈ I , x ( x 2 − xy ) = x 3 − x 2 y ∈ I , . . . x a , x b ∈ I ⇒ x a − x b ∈ I ( x 2 = xy in k [ x , y ] / I ) Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 19 / 36

  61. Binomial Ideals Fix a binomial ideal I ⊂ k [ x 1 , . . . , x n ]. Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 20 / 36

  62. Binomial Ideals Fix a binomial ideal I ⊂ k [ x 1 , . . . , x n ]. The equivalence relation ∼ I induced by I on N n is a congruence : a ∼ I b implies a + c ∼ I b + c for a , b , c ∈ N n . In particular, ( N n / ∼ I , +) is well defined. Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 20 / 36

  63. Binomial Ideals Fix a binomial ideal I ⊂ k [ x 1 , . . . , x n ]. The equivalence relation ∼ I induced by I on N n is a congruence : a ∼ I b implies a + c ∼ I b + c for a , b , c ∈ N n . In particular, ( N n / ∼ I , +) is well defined. The monomials in I form a single class ∞ ∈ N n / ∼ I , called the nil . Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 20 / 36

  64. Binomial Ideals Fix a binomial ideal I ⊂ k [ x 1 , . . . , x n ]. The equivalence relation ∼ I induced by I on N n is a congruence : a ∼ I b implies a + c ∼ I b + c for a , b , c ∈ N n . In particular, ( N n / ∼ I , +) is well defined. The monomials in I form a single class ∞ ∈ N n / ∼ I , called the nil . The nil ∞ corresponds to 0 in the quotient k [ x 1 , . . . , x n ] / I . Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 20 / 36

  65. Binomial Ideals Fix a binomial ideal I ⊂ k [ x 1 , . . . , x n ]. The equivalence relation ∼ I induced by I on N n is a congruence : a ∼ I b implies a + c ∼ I b + c for a , b , c ∈ N n . In particular, ( N n / ∼ I , +) is well defined. The monomials in I form a single class ∞ ∈ N n / ∼ I , called the nil . The nil ∞ corresponds to 0 in the quotient k [ x 1 , . . . , x n ] / I . Each non-nil a ∈ N n / ∼ I represents a distinct monomial modulo I . Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 20 / 36

  66. Binomial Ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � I = Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 21 / 36

  67. Binomial Ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � Monoid N 2 / ∼ I I = Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 21 / 36

  68. Binomial Ideals Theorem (Kahle-Miller, 2013) For k = k , every binomial ideal has an expression of the form r � I = J i i =1 where each J i 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

  69. Binomial Ideals Theorem (Kahle-Miller, 2013) For k = k , every binomial ideal has an expression of the form r � I = J i i =1 where each J i 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

  70. Binomial Ideals Theorem (Kahle-Miller, 2013) For k = k , every binomial ideal has an expression of the form r � I = J i i =1 where each J i 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

  71. Binomial Ideals Theorem (Kahle-Miller, 2013) For k = k , every binomial ideal has an expression of the form r � I = J i i =1 where each J i 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 , soc m ( I ) / I has a unique monomial. Christopher O’Neill (Duke University) Irreducible decomposition of binomial ideals January 18, 2014 22 / 36

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