Irreducible decompositions of binomial ideals Christopher ONeill - PowerPoint PPT Presentation

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

Question Fact Every ideal I ⊂ k [ x 1 , . . . , x n ] 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

Question Fact Every ideal I ⊂ k [ x 1 , . . . , x n ] can be written as a finite intersection of irreducible ideals (an irreducible decomposition). Definition An ideal I ⊂ k [ x 1 , . . . , x n ] is a binomial ideal if it is generated by polynomials with at most two terms. Example: I = � x − 2 y , x 2 � . Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 2 / 13

Question Fact Every ideal I ⊂ k [ x 1 , . . . , x n ] can be written as a finite intersection of irreducible ideals (an irreducible decomposition). Definition An ideal I ⊂ k [ x 1 , . . . , x n ] is a binomial ideal if it is generated by polynomials with at most two terms. Example: I = � x − 2 y , x 2 � . 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

Question Fact Every ideal I ⊂ k [ x 1 , . . . , x n ] can be written as a finite intersection of irreducible ideals (an irreducible decomposition). Definition An ideal I ⊂ k [ x 1 , . . . , x n ] is a binomial ideal if it is generated by polynomials with at most two terms. Example: I = � x − 2 y , x 2 � . 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

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

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

Monomial ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 4 / 13

Monomial ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 4 / 13

Monomial ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 4 / 13

Monomial ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = “Staircase diagram” Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 5 / 13

Monomial ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = “Staircase diagram” Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 5 / 13

Monomial ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = “Staircase diagram” Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 5 / 13

Monomial ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

Monomial ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

Monomial ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

Monomial ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = = Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

Monomial ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = � x 4 , y � = Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

Monomial ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = � x 4 , y � � x 3 , y 2 � = Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

Monomial ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = � x 4 , y � � x 3 , y 2 � � x 2 , y 4 � = Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

Monomial ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = � x 4 , y � ∩ � x 3 , y 2 � ∩ � x 2 , y 4 � = Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

Monomial ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = � x 4 , y � ∩ � x 3 , y 2 � ∩ � x 2 , y 4 � = Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

Monomial ideals � x 4 , x 3 y , x 2 y 2 , y 4 � I = � x 4 , y � ∩ � x 3 , y 2 � ∩ � x 2 , y 4 � = irreducible ⇔ “simple socle” Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 6 / 13

Binomial ideals And now, back to our original programming... Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 7 / 13

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

Binomial ideals I = � x 2 − xy , xy − y 2 , x 4 , y 4 � Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 8 / 13

Binomial ideals I = � x 2 − xy , xy − y 2 , x 4 , y 4 � Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 8 / 13

Binomial ideals I = � x 2 − xy , xy − y 2 , x 4 , y 4 � x 2 = xy in k [ x , y ] / I Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 8 / 13

Binomial ideals I = � x 2 − xy , xy − y 2 , x 4 , y 4 � Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 8 / 13

Binomial ideals I = � x 2 − xy , xy − y 2 , x 4 , y 4 � “witnesses” = monomials that merge in all directions Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 9 / 13

Binomial ideals I = � x 2 − xy , xy − y 2 , x 4 , y 4 � “witnesses” = monomials that merge in all directions Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 9 / 13

Binomial ideals I = � x 2 − xy , xy − y 2 , x 4 , y 4 � “witnesses” = monomials that merge in all directions Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 9 / 13

Binomial ideals I = � x 2 − xy , xy − y 2 , x 4 , y 4 � “witnesses” = monomials that merge in all directions Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 9 / 13

Binomial ideals I = � x 2 − xy , xy − y 2 , x 4 , y 4 � “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

Binomial ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � I = Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 10 / 13

Binomial ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � I = = Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 10 / 13

Binomial ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � I = = x − y ∈ socle Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 10 / 13

Binomial ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � I = � x − y , x 4 , y 4 � = Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 10 / 13

Binomial ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � I = � x − y , x 4 , y 4 � = Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 10 / 13

Binomial ideals � x 2 − xy , xy − y 2 , x 4 , y 4 � I = � x − y , x 4 , y 4 � � x 2 , y � = Christopher O’Neill (Duke University) Irreducible decompositions of binomial ideals January 18, 2014 10 / 13