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From Factorial Designs to Hilbert Schemes Lorenzo Robbiano Universit di Genova Dipartimento di Matematica Lorenzo Robbiano (Universit di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 1 / 31 Abstract This talk is meant

  1. Points and Statistics II The main task is to identify an unknown function ¯ f : D − → K called the model. In general it is not possible to perform all experiments corresponding to the points in D and measuring the value of ¯ f each time. A subset F of a full design D is called a fraction. We want to choose a fraction F ⊆ D that allows us to identify the model if we have some extra knowledge about the shape of ¯ f . In particular, we need to describe the sets of power products whose residue classes form a K -basis of P / I ( F ) . Statisticians express this property by saying that such sets of power products are identified by F . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 6 / 31

  2. Points and Statistics II The main task is to identify an unknown function ¯ f : D − → K called the model. In general it is not possible to perform all experiments corresponding to the points in D and measuring the value of ¯ f each time. A subset F of a full design D is called a fraction. We want to choose a fraction F ⊆ D that allows us to identify the model if we have some extra knowledge about the shape of ¯ f . In particular, we need to describe the sets of power products whose residue classes form a K -basis of P / I ( F ) . Statisticians express this property by saying that such sets of power products are identified by F . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 6 / 31

  3. Points and Statistics II The main task is to identify an unknown function ¯ f : D − → K called the model. In general it is not possible to perform all experiments corresponding to the points in D and measuring the value of ¯ f each time. A subset F of a full design D is called a fraction. We want to choose a fraction F ⊆ D that allows us to identify the model if we have some extra knowledge about the shape of ¯ f . In particular, we need to describe the sets of power products whose residue classes form a K -basis of P / I ( F ) . Statisticians express this property by saying that such sets of power products are identified by F . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 6 / 31

  4. Points and Statistics II The main task is to identify an unknown function ¯ f : D − → K called the model. In general it is not possible to perform all experiments corresponding to the points in D and measuring the value of ¯ f each time. A subset F of a full design D is called a fraction. We want to choose a fraction F ⊆ D that allows us to identify the model if we have some extra knowledge about the shape of ¯ f . In particular, we need to describe the sets of power products whose residue classes form a K -basis of P / I ( F ) . Statisticians express this property by saying that such sets of power products are identified by F . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 6 / 31

  5. A Proposition Proposition The following conditions are equivalent. • The order ideal O is identified by the fraction F . • The vanishing ideal I ( F ) has an O -border basis. • The evaluation matrix ( t i ( p j )) is invertible. The Inverse Problem Conversely, given O , how can we choose the fractions F such that the matrix of coefficients is invertible? In other words, given a full design D and an order ideal O ⊆ O D , which fractions F ⊆ D have the property that the residue classes of the elements of O are a K -basis of P / I ( F ) ? This is called the inverse problem of DoE. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 7 / 31

  6. A Proposition Proposition The following conditions are equivalent. • The order ideal O is identified by the fraction F . • The vanishing ideal I ( F ) has an O -border basis. • The evaluation matrix ( t i ( p j )) is invertible. The Inverse Problem Conversely, given O , how can we choose the fractions F such that the matrix of coefficients is invertible? In other words, given a full design D and an order ideal O ⊆ O D , which fractions F ⊆ D have the property that the residue classes of the elements of O are a K -basis of P / I ( F ) ? This is called the inverse problem of DoE. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 7 / 31

  7. A Proposition Proposition The following conditions are equivalent. • The order ideal O is identified by the fraction F . • The vanishing ideal I ( F ) has an O -border basis. • The evaluation matrix ( t i ( p j )) is invertible. The Inverse Problem Conversely, given O , how can we choose the fractions F such that the matrix of coefficients is invertible? In other words, given a full design D and an order ideal O ⊆ O D , which fractions F ⊆ D have the property that the residue classes of the elements of O are a K -basis of P / I ( F ) ? This is called the inverse problem of DoE. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 7 / 31

  8. A Proposition Proposition The following conditions are equivalent. • The order ideal O is identified by the fraction F . • The vanishing ideal I ( F ) has an O -border basis. • The evaluation matrix ( t i ( p j )) is invertible. The Inverse Problem Conversely, given O , how can we choose the fractions F such that the matrix of coefficients is invertible? In other words, given a full design D and an order ideal O ⊆ O D , which fractions F ⊆ D have the property that the residue classes of the elements of O are a K -basis of P / I ( F ) ? This is called the inverse problem of DoE. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 7 / 31

  9. A Proposition Proposition The following conditions are equivalent. • The order ideal O is identified by the fraction F . • The vanishing ideal I ( F ) has an O -border basis. • The evaluation matrix ( t i ( p j )) is invertible. The Inverse Problem Conversely, given O , how can we choose the fractions F such that the matrix of coefficients is invertible? In other words, given a full design D and an order ideal O ⊆ O D , which fractions F ⊆ D have the property that the residue classes of the elements of O are a K -basis of P / I ( F ) ? This is called the inverse problem of DoE. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 7 / 31

  10. A Proposition Proposition The following conditions are equivalent. • The order ideal O is identified by the fraction F . • The vanishing ideal I ( F ) has an O -border basis. • The evaluation matrix ( t i ( p j )) is invertible. The Inverse Problem Conversely, given O , how can we choose the fractions F such that the matrix of coefficients is invertible? In other words, given a full design D and an order ideal O ⊆ O D , which fractions F ⊆ D have the property that the residue classes of the elements of O are a K -basis of P / I ( F ) ? This is called the inverse problem of DoE. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 7 / 31

  11. A Proposition Proposition The following conditions are equivalent. • The order ideal O is identified by the fraction F . • The vanishing ideal I ( F ) has an O -border basis. • The evaluation matrix ( t i ( p j )) is invertible. The Inverse Problem Conversely, given O , how can we choose the fractions F such that the matrix of coefficients is invertible? In other words, given a full design D and an order ideal O ⊆ O D , which fractions F ⊆ D have the property that the residue classes of the elements of O are a K -basis of P / I ( F ) ? This is called the inverse problem of DoE. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 7 / 31

  12. Solution This problem was partially solved in M. Caboara and L. Robbiano: Families of Ideals in Statistics , Proc. of ISSAC-1997 (Maui, Hawaii, July 1997) (New York, N.Y.), W.W. Küchlin, Ed. (1997), 404–409. with the use of Gröbner bases, and totally solved in M. Caboara and L. Robbiano: Families of Estimable Terms Proc. of ISSAC 2001, (London, Ontario, Canada, July 2001) (New York, N.Y.), B. Mourrain, Ed. ed (2001) 56–63. with the use of Border bases. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 8 / 31

  13. Solution This problem was partially solved in M. Caboara and L. Robbiano: Families of Ideals in Statistics , Proc. of ISSAC-1997 (Maui, Hawaii, July 1997) (New York, N.Y.), W.W. Küchlin, Ed. (1997), 404–409. with the use of Gröbner bases, and totally solved in M. Caboara and L. Robbiano: Families of Estimable Terms Proc. of ISSAC 2001, (London, Ontario, Canada, July 2001) (New York, N.Y.), B. Mourrain, Ed. ed (2001) 56–63. with the use of Border bases. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 8 / 31

  14. An Example Let D be the full design D = {− 1 , 0 , 1 } × {− 1 , 0 , 1 } . The task it to solve the inverse problem for the order ideal O = { 1 , x , y , x 2 , y 2 } It turns out that we have to solve a system defined by 20 quadratic polynomials. � 9 � Using CoCoA, we check that among the 126 = five-tuples of points in D 5 there are exactly 81 five-tuples which solve the inverse problem. It is natural to ask how many of these 81 fractions have the property that O is of the form T n \ LT σ {I ( F ) } with σ varying among the term orderings. One can prove that 36 of those 81 fractions are not of that type. This is a surprisingly high number which shows that border bases provide a much more flexible environment for working with zero-dimensional ideals than Gröbner bases do. The details are explained in M. Kreuzer – L. Robbiano: Computational Commutative Algebra 2, Springer (2005), Tutorial 92. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 9 / 31

  15. An Example Let D be the full design D = {− 1 , 0 , 1 } × {− 1 , 0 , 1 } . The task it to solve the inverse problem for the order ideal O = { 1 , x , y , x 2 , y 2 } It turns out that we have to solve a system defined by 20 quadratic polynomials. � 9 � Using CoCoA, we check that among the 126 = five-tuples of points in D 5 there are exactly 81 five-tuples which solve the inverse problem. It is natural to ask how many of these 81 fractions have the property that O is of the form T n \ LT σ {I ( F ) } with σ varying among the term orderings. One can prove that 36 of those 81 fractions are not of that type. This is a surprisingly high number which shows that border bases provide a much more flexible environment for working with zero-dimensional ideals than Gröbner bases do. The details are explained in M. Kreuzer – L. Robbiano: Computational Commutative Algebra 2, Springer (2005), Tutorial 92. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 9 / 31

  16. An Example Let D be the full design D = {− 1 , 0 , 1 } × {− 1 , 0 , 1 } . The task it to solve the inverse problem for the order ideal O = { 1 , x , y , x 2 , y 2 } It turns out that we have to solve a system defined by 20 quadratic polynomials. � 9 � Using CoCoA, we check that among the 126 = five-tuples of points in D 5 there are exactly 81 five-tuples which solve the inverse problem. It is natural to ask how many of these 81 fractions have the property that O is of the form T n \ LT σ {I ( F ) } with σ varying among the term orderings. One can prove that 36 of those 81 fractions are not of that type. This is a surprisingly high number which shows that border bases provide a much more flexible environment for working with zero-dimensional ideals than Gröbner bases do. The details are explained in M. Kreuzer – L. Robbiano: Computational Commutative Algebra 2, Springer (2005), Tutorial 92. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 9 / 31

  17. An Example Let D be the full design D = {− 1 , 0 , 1 } × {− 1 , 0 , 1 } . The task it to solve the inverse problem for the order ideal O = { 1 , x , y , x 2 , y 2 } It turns out that we have to solve a system defined by 20 quadratic polynomials. � 9 � Using CoCoA, we check that among the 126 = five-tuples of points in D 5 there are exactly 81 five-tuples which solve the inverse problem. It is natural to ask how many of these 81 fractions have the property that O is of the form T n \ LT σ {I ( F ) } with σ varying among the term orderings. One can prove that 36 of those 81 fractions are not of that type. This is a surprisingly high number which shows that border bases provide a much more flexible environment for working with zero-dimensional ideals than Gröbner bases do. The details are explained in M. Kreuzer – L. Robbiano: Computational Commutative Algebra 2, Springer (2005), Tutorial 92. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 9 / 31

  18. An Example Let D be the full design D = {− 1 , 0 , 1 } × {− 1 , 0 , 1 } . The task it to solve the inverse problem for the order ideal O = { 1 , x , y , x 2 , y 2 } It turns out that we have to solve a system defined by 20 quadratic polynomials. � 9 � Using CoCoA, we check that among the 126 = five-tuples of points in D 5 there are exactly 81 five-tuples which solve the inverse problem. It is natural to ask how many of these 81 fractions have the property that O is of the form T n \ LT σ {I ( F ) } with σ varying among the term orderings. One can prove that 36 of those 81 fractions are not of that type. This is a surprisingly high number which shows that border bases provide a much more flexible environment for working with zero-dimensional ideals than Gröbner bases do. The details are explained in M. Kreuzer – L. Robbiano: Computational Commutative Algebra 2, Springer (2005), Tutorial 92. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 9 / 31

  19. An Example Let D be the full design D = {− 1 , 0 , 1 } × {− 1 , 0 , 1 } . The task it to solve the inverse problem for the order ideal O = { 1 , x , y , x 2 , y 2 } It turns out that we have to solve a system defined by 20 quadratic polynomials. � 9 � Using CoCoA, we check that among the 126 = five-tuples of points in D 5 there are exactly 81 five-tuples which solve the inverse problem. It is natural to ask how many of these 81 fractions have the property that O is of the form T n \ LT σ {I ( F ) } with σ varying among the term orderings. One can prove that 36 of those 81 fractions are not of that type. This is a surprisingly high number which shows that border bases provide a much more flexible environment for working with zero-dimensional ideals than Gröbner bases do. The details are explained in M. Kreuzer – L. Robbiano: Computational Commutative Algebra 2, Springer (2005), Tutorial 92. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 9 / 31

  20. PART 2 Border Bases: The Continuous Case Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 10 / 31

  21. Two conics I Example Consider the polynomial system 4 x 2 + y 2 − 1 1 f 1 = = 0 x 2 + 1 4 y 2 − 1 = = f 2 0 � � X = Z ( f 1 ) ∩ Z ( f 2 ) consists of the four points X = { ( ± 4 / 5 , ± 4 / 5 ) } . The set { x 2 − 4 5 , y 2 − 4 5 } is the universal reduced Gröbner basis of the ideal I = ( f 1 , f 2 ) ⊆ C [ x , y ] , in particular with respect to σ = DegRevLex . LT σ ( I ) = ( x 2 , y 2 ) , and the residue classes of the terms in T 2 \ LT σ { I } = { 1 , x , y , xy } form a C -vector space basis of C [ x , y ] / I . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 11 / 31

  22. Two conics I Example Consider the polynomial system 4 x 2 + y 2 − 1 1 f 1 = = 0 x 2 + 1 4 y 2 − 1 = = f 2 0 � � X = Z ( f 1 ) ∩ Z ( f 2 ) consists of the four points X = { ( ± 4 / 5 , ± 4 / 5 ) } . The set { x 2 − 4 5 , y 2 − 4 5 } is the universal reduced Gröbner basis of the ideal I = ( f 1 , f 2 ) ⊆ C [ x , y ] , in particular with respect to σ = DegRevLex . LT σ ( I ) = ( x 2 , y 2 ) , and the residue classes of the terms in T 2 \ LT σ { I } = { 1 , x , y , xy } form a C -vector space basis of C [ x , y ] / I . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 11 / 31

  23. Two conics I Example Consider the polynomial system 4 x 2 + y 2 − 1 1 f 1 = = 0 x 2 + 1 4 y 2 − 1 = = f 2 0 � � X = Z ( f 1 ) ∩ Z ( f 2 ) consists of the four points X = { ( ± 4 / 5 , ± 4 / 5 ) } . The set { x 2 − 4 5 , y 2 − 4 5 } is the universal reduced Gröbner basis of the ideal I = ( f 1 , f 2 ) ⊆ C [ x , y ] , in particular with respect to σ = DegRevLex . LT σ ( I ) = ( x 2 , y 2 ) , and the residue classes of the terms in T 2 \ LT σ { I } = { 1 , x , y , xy } form a C -vector space basis of C [ x , y ] / I . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 11 / 31

  24. Two conics I Example Consider the polynomial system 4 x 2 + y 2 − 1 1 f 1 = = 0 x 2 + 1 4 y 2 − 1 = = f 2 0 � � X = Z ( f 1 ) ∩ Z ( f 2 ) consists of the four points X = { ( ± 4 / 5 , ± 4 / 5 ) } . The set { x 2 − 4 5 , y 2 − 4 5 } is the universal reduced Gröbner basis of the ideal I = ( f 1 , f 2 ) ⊆ C [ x , y ] , in particular with respect to σ = DegRevLex . LT σ ( I ) = ( x 2 , y 2 ) , and the residue classes of the terms in T 2 \ LT σ { I } = { 1 , x , y , xy } form a C -vector space basis of C [ x , y ] / I . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 11 / 31

  25. Two conics I Example Consider the polynomial system 4 x 2 + y 2 − 1 1 f 1 = = 0 x 2 + 1 4 y 2 − 1 = = f 2 0 � � X = Z ( f 1 ) ∩ Z ( f 2 ) consists of the four points X = { ( ± 4 / 5 , ± 4 / 5 ) } . The set { x 2 − 4 5 , y 2 − 4 5 } is the universal reduced Gröbner basis of the ideal I = ( f 1 , f 2 ) ⊆ C [ x , y ] , in particular with respect to σ = DegRevLex . LT σ ( I ) = ( x 2 , y 2 ) , and the residue classes of the terms in T 2 \ LT σ { I } = { 1 , x , y , xy } form a C -vector space basis of C [ x , y ] / I . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 11 / 31

  26. Two conics II Now consider the slightly perturbed polynomial system 4 x 2 + y 2 + ε xy − 1 ˜ 1 f 1 = = 0 x 2 + 1 4 y 2 + ε xy − 1 ˜ = = f 2 0 The intersection of Z (˜ f 1 ) and Z (˜ f 2 ) consists of four perturbed points � X close to the points in X . I = (˜ f 1 , ˜ The ideal ˜ f 2 ) has the reduced σ -Gröbner basis { x 2 − y 2 , xy + 5 4 ε y 2 − 1 ε , y 3 − 16 ε 20 16 ε 2 − 25 x + 16 ε 2 − 25 y } I ) = ( x 2 , xy , y 3 ) and T 2 \ LT σ { ˜ Moreover, we have LT σ (˜ I } = { 1 , x , y , y 2 } . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 12 / 31

  27. Two conics II Now consider the slightly perturbed polynomial system 4 x 2 + y 2 + ε xy − 1 ˜ 1 f 1 = = 0 x 2 + 1 4 y 2 + ε xy − 1 ˜ = = f 2 0 The intersection of Z (˜ f 1 ) and Z (˜ f 2 ) consists of four perturbed points � X close to the points in X . I = (˜ f 1 , ˜ The ideal ˜ f 2 ) has the reduced σ -Gröbner basis { x 2 − y 2 , xy + 5 4 ε y 2 − 1 ε , y 3 − 16 ε 20 16 ε 2 − 25 x + 16 ε 2 − 25 y } I ) = ( x 2 , xy , y 3 ) and T 2 \ LT σ { ˜ Moreover, we have LT σ (˜ I } = { 1 , x , y , y 2 } . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 12 / 31

  28. Two conics II Now consider the slightly perturbed polynomial system 4 x 2 + y 2 + ε xy − 1 ˜ 1 f 1 = = 0 x 2 + 1 4 y 2 + ε xy − 1 ˜ = = f 2 0 The intersection of Z (˜ f 1 ) and Z (˜ f 2 ) consists of four perturbed points � X close to the points in X . I = (˜ f 1 , ˜ The ideal ˜ f 2 ) has the reduced σ -Gröbner basis { x 2 − y 2 , xy + 5 4 ε y 2 − 1 ε , y 3 − 16 ε 20 16 ε 2 − 25 x + 16 ε 2 − 25 y } I ) = ( x 2 , xy , y 3 ) and T 2 \ LT σ { ˜ Moreover, we have LT σ (˜ I } = { 1 , x , y , y 2 } . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 12 / 31

  29. Two conics II Now consider the slightly perturbed polynomial system 4 x 2 + y 2 + ε xy − 1 ˜ 1 f 1 = = 0 x 2 + 1 4 y 2 + ε xy − 1 ˜ = = f 2 0 The intersection of Z (˜ f 1 ) and Z (˜ f 2 ) consists of four perturbed points � X close to the points in X . I = (˜ f 1 , ˜ The ideal ˜ f 2 ) has the reduced σ -Gröbner basis { x 2 − y 2 , xy + 5 4 ε y 2 − 1 ε , y 3 − 16 ε 20 16 ε 2 − 25 x + 16 ε 2 − 25 y } I ) = ( x 2 , xy , y 3 ) and T 2 \ LT σ { ˜ Moreover, we have LT σ (˜ I } = { 1 , x , y , y 2 } . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 12 / 31

  30. Two conics II Now consider the slightly perturbed polynomial system 4 x 2 + y 2 + ε xy − 1 ˜ 1 f 1 = = 0 x 2 + 1 4 y 2 + ε xy − 1 ˜ = = f 2 0 The intersection of Z (˜ f 1 ) and Z (˜ f 2 ) consists of four perturbed points � X close to the points in X . I = (˜ f 1 , ˜ The ideal ˜ f 2 ) has the reduced σ -Gröbner basis { x 2 − y 2 , xy + 5 4 ε y 2 − 1 ε , y 3 − 16 ε 20 16 ε 2 − 25 x + 16 ε 2 − 25 y } I ) = ( x 2 , xy , y 3 ) and T 2 \ LT σ { ˜ Moreover, we have LT σ (˜ I } = { 1 , x , y , y 2 } . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 12 / 31

  31. Two conics II Now consider the slightly perturbed polynomial system 4 x 2 + y 2 + ε xy − 1 ˜ 1 f 1 = = 0 x 2 + 1 4 y 2 + ε xy − 1 ˜ = = f 2 0 The intersection of Z (˜ f 1 ) and Z (˜ f 2 ) consists of four perturbed points � X close to the points in X . I = (˜ f 1 , ˜ The ideal ˜ f 2 ) has the reduced σ -Gröbner basis { x 2 − y 2 , xy + 5 4 ε y 2 − 1 ε , y 3 − 16 ε 20 16 ε 2 − 25 x + 16 ε 2 − 25 y } I ) = ( x 2 , xy , y 3 ) and T 2 \ LT σ { ˜ Moreover, we have LT σ (˜ I } = { 1 , x , y , y 2 } . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 12 / 31

  32. Border Bases The basic idea of border basis theory is to describe a zero-dimensional ring P / I by an order ideal of monomials O whose residue classes form a K -basis of P / I and by the multiplication matrices of this basis. Let K be a field, let P = K [ x 1 , . . . , x n ] , let T n be the monoid of terms, and let O ⊆ T n be an order ideal. Definition (Border Prebases) Let O have µ elements and ∂ O have ν elements. The border of O is the set ∂ O = T n · O \ O = ( x 1 O ∪ · · · ∪ x n O ) \ O . A set of polynomials G = { g 1 , . . . , g ν } in P is called an O -border prebasis if the polynomials have the form g j = b j − � µ i = 1 α ij t i with α ij ∈ K for 1 ≤ i ≤ µ , 1 ≤ j ≤ ν, b j ∈ ∂ O , t i ∈ O . Definition (Border Bases) Let G = { g 1 , . . . , g ν } be an O -border prebasis, and let I ⊆ P be an ideal containing G . The set G is called an O -border basis of I if the residue classes O = { ¯ t 1 , . . . , ¯ t µ } form a K -vector space basis of P / I . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 13 / 31

  33. Border Bases The basic idea of border basis theory is to describe a zero-dimensional ring P / I by an order ideal of monomials O whose residue classes form a K -basis of P / I and by the multiplication matrices of this basis. Let K be a field, let P = K [ x 1 , . . . , x n ] , let T n be the monoid of terms, and let O ⊆ T n be an order ideal. Definition (Border Prebases) Let O have µ elements and ∂ O have ν elements. The border of O is the set ∂ O = T n · O \ O = ( x 1 O ∪ · · · ∪ x n O ) \ O . A set of polynomials G = { g 1 , . . . , g ν } in P is called an O -border prebasis if the polynomials have the form g j = b j − � µ i = 1 α ij t i with α ij ∈ K for 1 ≤ i ≤ µ , 1 ≤ j ≤ ν, b j ∈ ∂ O , t i ∈ O . Definition (Border Bases) Let G = { g 1 , . . . , g ν } be an O -border prebasis, and let I ⊆ P be an ideal containing G . The set G is called an O -border basis of I if the residue classes O = { ¯ t 1 , . . . , ¯ t µ } form a K -vector space basis of P / I . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 13 / 31

  34. Border Bases The basic idea of border basis theory is to describe a zero-dimensional ring P / I by an order ideal of monomials O whose residue classes form a K -basis of P / I and by the multiplication matrices of this basis. Let K be a field, let P = K [ x 1 , . . . , x n ] , let T n be the monoid of terms, and let O ⊆ T n be an order ideal. Definition (Border Prebases) Let O have µ elements and ∂ O have ν elements. The border of O is the set ∂ O = T n · O \ O = ( x 1 O ∪ · · · ∪ x n O ) \ O . A set of polynomials G = { g 1 , . . . , g ν } in P is called an O -border prebasis if the polynomials have the form g j = b j − � µ i = 1 α ij t i with α ij ∈ K for 1 ≤ i ≤ µ , 1 ≤ j ≤ ν, b j ∈ ∂ O , t i ∈ O . Definition (Border Bases) Let G = { g 1 , . . . , g ν } be an O -border prebasis, and let I ⊆ P be an ideal containing G . The set G is called an O -border basis of I if the residue classes O = { ¯ t 1 , . . . , ¯ t µ } form a K -vector space basis of P / I . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 13 / 31

  35. Border Bases The basic idea of border basis theory is to describe a zero-dimensional ring P / I by an order ideal of monomials O whose residue classes form a K -basis of P / I and by the multiplication matrices of this basis. Let K be a field, let P = K [ x 1 , . . . , x n ] , let T n be the monoid of terms, and let O ⊆ T n be an order ideal. Definition (Border Prebases) Let O have µ elements and ∂ O have ν elements. The border of O is the set ∂ O = T n · O \ O = ( x 1 O ∪ · · · ∪ x n O ) \ O . A set of polynomials G = { g 1 , . . . , g ν } in P is called an O -border prebasis if the polynomials have the form g j = b j − � µ i = 1 α ij t i with α ij ∈ K for 1 ≤ i ≤ µ , 1 ≤ j ≤ ν, b j ∈ ∂ O , t i ∈ O . Definition (Border Bases) Let G = { g 1 , . . . , g ν } be an O -border prebasis, and let I ⊆ P be an ideal containing G . The set G is called an O -border basis of I if the residue classes O = { ¯ t 1 , . . . , ¯ t µ } form a K -vector space basis of P / I . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 13 / 31

  36. Border Bases The basic idea of border basis theory is to describe a zero-dimensional ring P / I by an order ideal of monomials O whose residue classes form a K -basis of P / I and by the multiplication matrices of this basis. Let K be a field, let P = K [ x 1 , . . . , x n ] , let T n be the monoid of terms, and let O ⊆ T n be an order ideal. Definition (Border Prebases) Let O have µ elements and ∂ O have ν elements. The border of O is the set ∂ O = T n · O \ O = ( x 1 O ∪ · · · ∪ x n O ) \ O . A set of polynomials G = { g 1 , . . . , g ν } in P is called an O -border prebasis if the polynomials have the form g j = b j − � µ i = 1 α ij t i with α ij ∈ K for 1 ≤ i ≤ µ , 1 ≤ j ≤ ν, b j ∈ ∂ O , t i ∈ O . Definition (Border Bases) Let G = { g 1 , . . . , g ν } be an O -border prebasis, and let I ⊆ P be an ideal containing G . The set G is called an O -border basis of I if the residue classes O = { ¯ t 1 , . . . , ¯ t µ } form a K -vector space basis of P / I . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 13 / 31

  37. Border Bases The basic idea of border basis theory is to describe a zero-dimensional ring P / I by an order ideal of monomials O whose residue classes form a K -basis of P / I and by the multiplication matrices of this basis. Let K be a field, let P = K [ x 1 , . . . , x n ] , let T n be the monoid of terms, and let O ⊆ T n be an order ideal. Definition (Border Prebases) Let O have µ elements and ∂ O have ν elements. The border of O is the set ∂ O = T n · O \ O = ( x 1 O ∪ · · · ∪ x n O ) \ O . A set of polynomials G = { g 1 , . . . , g ν } in P is called an O -border prebasis if the polynomials have the form g j = b j − � µ i = 1 α ij t i with α ij ∈ K for 1 ≤ i ≤ µ , 1 ≤ j ≤ ν, b j ∈ ∂ O , t i ∈ O . Definition (Border Bases) Let G = { g 1 , . . . , g ν } be an O -border prebasis, and let I ⊆ P be an ideal containing G . The set G is called an O -border basis of I if the residue classes O = { ¯ t 1 , . . . , ¯ t µ } form a K -vector space basis of P / I . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 13 / 31

  38. Border Bases The basic idea of border basis theory is to describe a zero-dimensional ring P / I by an order ideal of monomials O whose residue classes form a K -basis of P / I and by the multiplication matrices of this basis. Let K be a field, let P = K [ x 1 , . . . , x n ] , let T n be the monoid of terms, and let O ⊆ T n be an order ideal. Definition (Border Prebases) Let O have µ elements and ∂ O have ν elements. The border of O is the set ∂ O = T n · O \ O = ( x 1 O ∪ · · · ∪ x n O ) \ O . A set of polynomials G = { g 1 , . . . , g ν } in P is called an O -border prebasis if the polynomials have the form g j = b j − � µ i = 1 α ij t i with α ij ∈ K for 1 ≤ i ≤ µ , 1 ≤ j ≤ ν, b j ∈ ∂ O , t i ∈ O . Definition (Border Bases) Let G = { g 1 , . . . , g ν } be an O -border prebasis, and let I ⊆ P be an ideal containing G . The set G is called an O -border basis of I if the residue classes O = { ¯ t 1 , . . . , ¯ t µ } form a K -vector space basis of P / I . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 13 / 31

  39. Border Bases The basic idea of border basis theory is to describe a zero-dimensional ring P / I by an order ideal of monomials O whose residue classes form a K -basis of P / I and by the multiplication matrices of this basis. Let K be a field, let P = K [ x 1 , . . . , x n ] , let T n be the monoid of terms, and let O ⊆ T n be an order ideal. Definition (Border Prebases) Let O have µ elements and ∂ O have ν elements. The border of O is the set ∂ O = T n · O \ O = ( x 1 O ∪ · · · ∪ x n O ) \ O . A set of polynomials G = { g 1 , . . . , g ν } in P is called an O -border prebasis if the polynomials have the form g j = b j − � µ i = 1 α ij t i with α ij ∈ K for 1 ≤ i ≤ µ , 1 ≤ j ≤ ν, b j ∈ ∂ O , t i ∈ O . Definition (Border Bases) Let G = { g 1 , . . . , g ν } be an O -border prebasis, and let I ⊆ P be an ideal containing G . The set G is called an O -border basis of I if the residue classes O = { ¯ t 1 , . . . , ¯ t µ } form a K -vector space basis of P / I . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 13 / 31

  40. Border Bases The basic idea of border basis theory is to describe a zero-dimensional ring P / I by an order ideal of monomials O whose residue classes form a K -basis of P / I and by the multiplication matrices of this basis. Let K be a field, let P = K [ x 1 , . . . , x n ] , let T n be the monoid of terms, and let O ⊆ T n be an order ideal. Definition (Border Prebases) Let O have µ elements and ∂ O have ν elements. The border of O is the set ∂ O = T n · O \ O = ( x 1 O ∪ · · · ∪ x n O ) \ O . A set of polynomials G = { g 1 , . . . , g ν } in P is called an O -border prebasis if the polynomials have the form g j = b j − � µ i = 1 α ij t i with α ij ∈ K for 1 ≤ i ≤ µ , 1 ≤ j ≤ ν, b j ∈ ∂ O , t i ∈ O . Definition (Border Bases) Let G = { g 1 , . . . , g ν } be an O -border prebasis, and let I ⊆ P be an ideal containing G . The set G is called an O -border basis of I if the residue classes O = { ¯ t 1 , . . . , ¯ t µ } form a K -vector space basis of P / I . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 13 / 31

  41. Two conics III What are the border bases in the two cases of the conics and the perturbed conics? Two conics { x 2 − 4 x 2 y − 4 5 , 5 y , xy 2 − 4 y 2 − 4 5 x , 5 } Two perturbed conics { x 2 + 4 x 2 y − 16 ε 5 ε xy − 4 20 5 , 16 ε 2 − 25 x + 16 ε 2 − 25 y , xy 2 + y 2 + 4 16 ε 20 5 ε xy − 4 16 ε 2 − 25 x + 16 ε 2 − 25 y , 5 } Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 14 / 31

  42. Two conics III What are the border bases in the two cases of the conics and the perturbed conics? Two conics { x 2 − 4 x 2 y − 4 5 , 5 y , xy 2 − 4 y 2 − 4 5 x , 5 } Two perturbed conics { x 2 + 4 x 2 y − 16 ε 5 ε xy − 4 20 5 , 16 ε 2 − 25 x + 16 ε 2 − 25 y , xy 2 + y 2 + 4 16 ε 20 5 ε xy − 4 16 ε 2 − 25 x + 16 ε 2 − 25 y , 5 } Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 14 / 31

  43. Two conics III What are the border bases in the two cases of the conics and the perturbed conics? Two conics { x 2 − 4 x 2 y − 4 5 , 5 y , xy 2 − 4 y 2 − 4 5 x , 5 } Two perturbed conics { x 2 + 4 x 2 y − 16 ε 5 ε xy − 4 20 5 , 16 ε 2 − 25 x + 16 ε 2 − 25 y , xy 2 + y 2 + 4 16 ε 20 5 ε xy − 4 16 ε 2 − 25 x + 16 ε 2 − 25 y , 5 } Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 14 / 31

  44. Two conics III What are the border bases in the two cases of the conics and the perturbed conics? Two conics { x 2 − 4 x 2 y − 4 5 , 5 y , xy 2 − 4 y 2 − 4 5 x , 5 } Two perturbed conics { x 2 + 4 x 2 y − 16 ε 5 ε xy − 4 20 5 , 16 ε 2 − 25 x + 16 ε 2 − 25 y , xy 2 + y 2 + 4 16 ε 20 5 ε xy − 4 16 ε 2 − 25 x + 16 ε 2 − 25 y , 5 } Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 14 / 31

  45. Existence and Uniqueness of Border Bases Proposition Let O = { t 1 , . . . , t µ } be an order ideal, let I ⊆ P be a zero-dimensional ideal, and assume that the residue classes of the elements of O form a K -vector space basis of P / I . Then there exists a unique O -border basis of I . Proposition Let σ be a term ordering on T n , and let O σ ( I ) be the order ideal T n \ LT σ { I } . Then there exists a unique O σ ( I ) -border basis G of I , and the reduced σ -Gröbner basis of I is the subset of G corresponding to the corners of O σ ( I ) . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 15 / 31

  46. Existence and Uniqueness of Border Bases Proposition Let O = { t 1 , . . . , t µ } be an order ideal, let I ⊆ P be a zero-dimensional ideal, and assume that the residue classes of the elements of O form a K -vector space basis of P / I . Then there exists a unique O -border basis of I . Proposition Let σ be a term ordering on T n , and let O σ ( I ) be the order ideal T n \ LT σ { I } . Then there exists a unique O σ ( I ) -border basis G of I , and the reduced σ -Gröbner basis of I is the subset of G corresponding to the corners of O σ ( I ) . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 15 / 31

  47. Commuting matrices The following is a fundamental fact. B. Mourrain: A new criterion for normal form algorithms , AAECC Lecture Notes in Computer Science 1719 (1999), 430–443. Theorem (Border Bases and Commuting Matrices) Let O = { t 1 , . . . , t µ } be an order ideal, let G = { g 1 , . . . , g ν } be an O -border prebasis, and let I = ( g 1 , . . . , g ν ) . Then the following conditions are equivalent. The set G is an O -border basis of I . 1 The multiplication matrices of G are pairwise commuting. 2 In that case the multiplication matrices represent the multiplication endomorphisms of P / I with respect to the basis { ¯ t 1 , . . . , ¯ t µ } . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 16 / 31

  48. Commuting matrices The following is a fundamental fact. B. Mourrain: A new criterion for normal form algorithms , AAECC Lecture Notes in Computer Science 1719 (1999), 430–443. Theorem (Border Bases and Commuting Matrices) Let O = { t 1 , . . . , t µ } be an order ideal, let G = { g 1 , . . . , g ν } be an O -border prebasis, and let I = ( g 1 , . . . , g ν ) . Then the following conditions are equivalent. The set G is an O -border basis of I . 1 The multiplication matrices of G are pairwise commuting. 2 In that case the multiplication matrices represent the multiplication endomorphisms of P / I with respect to the basis { ¯ t 1 , . . . , ¯ t µ } . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 16 / 31

  49. Commuting matrices The following is a fundamental fact. B. Mourrain: A new criterion for normal form algorithms , AAECC Lecture Notes in Computer Science 1719 (1999), 430–443. Theorem (Border Bases and Commuting Matrices) Let O = { t 1 , . . . , t µ } be an order ideal, let G = { g 1 , . . . , g ν } be an O -border prebasis, and let I = ( g 1 , . . . , g ν ) . Then the following conditions are equivalent. The set G is an O -border basis of I . 1 The multiplication matrices of G are pairwise commuting. 2 In that case the multiplication matrices represent the multiplication endomorphisms of P / I with respect to the basis { ¯ t 1 , . . . , ¯ t µ } . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 16 / 31

  50. Commuting matrices The following is a fundamental fact. B. Mourrain: A new criterion for normal form algorithms , AAECC Lecture Notes in Computer Science 1719 (1999), 430–443. Theorem (Border Bases and Commuting Matrices) Let O = { t 1 , . . . , t µ } be an order ideal, let G = { g 1 , . . . , g ν } be an O -border prebasis, and let I = ( g 1 , . . . , g ν ) . Then the following conditions are equivalent. The set G is an O -border basis of I . 1 The multiplication matrices of G are pairwise commuting. 2 In that case the multiplication matrices represent the multiplication endomorphisms of P / I with respect to the basis { ¯ t 1 , . . . , ¯ t µ } . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 16 / 31

  51. Commuting matrices The following is a fundamental fact. B. Mourrain: A new criterion for normal form algorithms , AAECC Lecture Notes in Computer Science 1719 (1999), 430–443. Theorem (Border Bases and Commuting Matrices) Let O = { t 1 , . . . , t µ } be an order ideal, let G = { g 1 , . . . , g ν } be an O -border prebasis, and let I = ( g 1 , . . . , g ν ) . Then the following conditions are equivalent. The set G is an O -border basis of I . 1 The multiplication matrices of G are pairwise commuting. 2 In that case the multiplication matrices represent the multiplication endomorphisms of P / I with respect to the basis { ¯ t 1 , . . . , ¯ t µ } . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 16 / 31

  52. Commuting matrices The following is a fundamental fact. B. Mourrain: A new criterion for normal form algorithms , AAECC Lecture Notes in Computer Science 1719 (1999), 430–443. Theorem (Border Bases and Commuting Matrices) Let O = { t 1 , . . . , t µ } be an order ideal, let G = { g 1 , . . . , g ν } be an O -border prebasis, and let I = ( g 1 , . . . , g ν ) . Then the following conditions are equivalent. The set G is an O -border basis of I . 1 The multiplication matrices of G are pairwise commuting. 2 In that case the multiplication matrices represent the multiplication endomorphisms of P / I with respect to the basis { ¯ t 1 , . . . , ¯ t µ } . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 16 / 31

  53. Commuting matrices The following is a fundamental fact. B. Mourrain: A new criterion for normal form algorithms , AAECC Lecture Notes in Computer Science 1719 (1999), 430–443. Theorem (Border Bases and Commuting Matrices) Let O = { t 1 , . . . , t µ } be an order ideal, let G = { g 1 , . . . , g ν } be an O -border prebasis, and let I = ( g 1 , . . . , g ν ) . Then the following conditions are equivalent. The set G is an O -border basis of I . 1 The multiplication matrices of G are pairwise commuting. 2 In that case the multiplication matrices represent the multiplication endomorphisms of P / I with respect to the basis { ¯ t 1 , . . . , ¯ t µ } . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 16 / 31

  54. Commuting matrices The following is a fundamental fact. B. Mourrain: A new criterion for normal form algorithms , AAECC Lecture Notes in Computer Science 1719 (1999), 430–443. Theorem (Border Bases and Commuting Matrices) Let O = { t 1 , . . . , t µ } be an order ideal, let G = { g 1 , . . . , g ν } be an O -border prebasis, and let I = ( g 1 , . . . , g ν ) . Then the following conditions are equivalent. The set G is an O -border basis of I . 1 The multiplication matrices of G are pairwise commuting. 2 In that case the multiplication matrices represent the multiplication endomorphisms of P / I with respect to the basis { ¯ t 1 , . . . , ¯ t µ } . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 16 / 31

  55. PART 3 Border Bases and the Hilbert Scheme Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 17 / 31

  56. A glimpse at punctual Hilbert schemes Punctual Hilbert schemes are schemes which parametrize all the zero-dimensional projective subschemes of P n which share the same multiplicity. Every zero-dimensional sub-scheme of P n is contained in a standard open set which is an affine space, say A n ⊂ P n . There is a one-to-one correspondence between zero-dimensional ideals in P = K [ x 1 , . . . , x n ] and zero-dimensional saturated homogeneous ideals in P = K [ x 0 , x 1 , . . . , x n ] . The correspondence is set via homogenization and dehomogenization. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 18 / 31

  57. A glimpse at punctual Hilbert schemes Punctual Hilbert schemes are schemes which parametrize all the zero-dimensional projective subschemes of P n which share the same multiplicity. Every zero-dimensional sub-scheme of P n is contained in a standard open set which is an affine space, say A n ⊂ P n . There is a one-to-one correspondence between zero-dimensional ideals in P = K [ x 1 , . . . , x n ] and zero-dimensional saturated homogeneous ideals in P = K [ x 0 , x 1 , . . . , x n ] . The correspondence is set via homogenization and dehomogenization. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 18 / 31

  58. A glimpse at punctual Hilbert schemes Punctual Hilbert schemes are schemes which parametrize all the zero-dimensional projective subschemes of P n which share the same multiplicity. Every zero-dimensional sub-scheme of P n is contained in a standard open set which is an affine space, say A n ⊂ P n . There is a one-to-one correspondence between zero-dimensional ideals in P = K [ x 1 , . . . , x n ] and zero-dimensional saturated homogeneous ideals in P = K [ x 0 , x 1 , . . . , x n ] . The correspondence is set via homogenization and dehomogenization. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 18 / 31

  59. A glimpse at punctual Hilbert schemes Punctual Hilbert schemes are schemes which parametrize all the zero-dimensional projective subschemes of P n which share the same multiplicity. Every zero-dimensional sub-scheme of P n is contained in a standard open set which is an affine space, say A n ⊂ P n . There is a one-to-one correspondence between zero-dimensional ideals in P = K [ x 1 , . . . , x n ] and zero-dimensional saturated homogeneous ideals in P = K [ x 0 , x 1 , . . . , x n ] . The correspondence is set via homogenization and dehomogenization. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 18 / 31

  60. An Example: Hilbert Polynomial = 4 Zero-dimensional subschemes of P 2 with Hilbert polynomial 4 correspond to saturated homogeneous ideals I such that if P denotes the polynomial ring K [ x , y , z ] , then the Hilbert function of P / I is either HF P / I = 1 , 2 , 3 , 4 , 4 , . . . or HF P / I = 1 , 3 , 4 , 4 , . . . . The difference function is HF P / I = 1 , 1 , 1 , 1 , 0 , . . . or HF P / I = 1 , 2 , 1 , 0 , . . . . either What are the possible good bases? Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 19 / 31

  61. An Example: Hilbert Polynomial = 4 Zero-dimensional subschemes of P 2 with Hilbert polynomial 4 correspond to saturated homogeneous ideals I such that if P denotes the polynomial ring K [ x , y , z ] , then the Hilbert function of P / I is either HF P / I = 1 , 2 , 3 , 4 , 4 , . . . or HF P / I = 1 , 3 , 4 , 4 , . . . . The difference function is HF P / I = 1 , 1 , 1 , 1 , 0 , . . . or HF P / I = 1 , 2 , 1 , 0 , . . . . either What are the possible good bases? Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 19 / 31

  62. An Example: Hilbert Polynomial = 4 Zero-dimensional subschemes of P 2 with Hilbert polynomial 4 correspond to saturated homogeneous ideals I such that if P denotes the polynomial ring K [ x , y , z ] , then the Hilbert function of P / I is either HF P / I = 1 , 2 , 3 , 4 , 4 , . . . or HF P / I = 1 , 3 , 4 , 4 , . . . . The difference function is HF P / I = 1 , 1 , 1 , 1 , 0 , . . . or HF P / I = 1 , 2 , 1 , 0 , . . . . either What are the possible good bases? Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 19 / 31

  63. An Example: Hilbert Polynomial = 4 Zero-dimensional subschemes of P 2 with Hilbert polynomial 4 correspond to saturated homogeneous ideals I such that if P denotes the polynomial ring K [ x , y , z ] , then the Hilbert function of P / I is either HF P / I = 1 , 2 , 3 , 4 , 4 , . . . or HF P / I = 1 , 3 , 4 , 4 , . . . . The difference function is HF P / I = 1 , 1 , 1 , 1 , 0 , . . . or HF P / I = 1 , 2 , 1 , 0 , . . . . either What are the possible good bases? Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 19 / 31

  64. Good bases Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 20 / 31

  65. Good bases Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 20 / 31

  66. Good bases Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 20 / 31

  67. Good bases Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 20 / 31

  68. Good bases Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 20 / 31

  69. Border Basis Schemes • Let O = { t 1 , . . . , t µ } be an order ideal in T n , and let ∂ O = { b 1 , . . . , b ν } be its border. Definition (The Border Basis Scheme) Let { c ij | 1 ≤ i ≤ µ, 1 ≤ j ≤ ν } be a set of further indeterminates. The generic O -border prebasis is the set of polynomials G = { g 1 , . . . , g ν } 1 in Q = K [ x 1 , . . . , x n , c 11 , . . . , c µν ] given by g j = b j − � µ i = 1 c ij t i . For k = 1 , . . . , n , let A k ∈ Mat µ ( K [ c ij ]) be the k th formal multiplication 2 matrix associated to G . Then the affine scheme B O ⊆ K µν defined by the ideal I ( B O ) generated by the entries of the matrices A k A ℓ − A ℓ A k with 1 ≤ k < ℓ ≤ n is called the O -border basis scheme. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 21 / 31

  70. Border Basis Schemes • Let O = { t 1 , . . . , t µ } be an order ideal in T n , and let ∂ O = { b 1 , . . . , b ν } be its border. Definition (The Border Basis Scheme) Let { c ij | 1 ≤ i ≤ µ, 1 ≤ j ≤ ν } be a set of further indeterminates. The generic O -border prebasis is the set of polynomials G = { g 1 , . . . , g ν } 1 in Q = K [ x 1 , . . . , x n , c 11 , . . . , c µν ] given by g j = b j − � µ i = 1 c ij t i . For k = 1 , . . . , n , let A k ∈ Mat µ ( K [ c ij ]) be the k th formal multiplication 2 matrix associated to G . Then the affine scheme B O ⊆ K µν defined by the ideal I ( B O ) generated by the entries of the matrices A k A ℓ − A ℓ A k with 1 ≤ k < ℓ ≤ n is called the O -border basis scheme. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 21 / 31

  71. Border Basis Schemes • Let O = { t 1 , . . . , t µ } be an order ideal in T n , and let ∂ O = { b 1 , . . . , b ν } be its border. Definition (The Border Basis Scheme) Let { c ij | 1 ≤ i ≤ µ, 1 ≤ j ≤ ν } be a set of further indeterminates. The generic O -border prebasis is the set of polynomials G = { g 1 , . . . , g ν } 1 in Q = K [ x 1 , . . . , x n , c 11 , . . . , c µν ] given by g j = b j − � µ i = 1 c ij t i . For k = 1 , . . . , n , let A k ∈ Mat µ ( K [ c ij ]) be the k th formal multiplication 2 matrix associated to G . Then the affine scheme B O ⊆ K µν defined by the ideal I ( B O ) generated by the entries of the matrices A k A ℓ − A ℓ A k with 1 ≤ k < ℓ ≤ n is called the O -border basis scheme. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 21 / 31

  72. Border Basis Schemes • Let O = { t 1 , . . . , t µ } be an order ideal in T n , and let ∂ O = { b 1 , . . . , b ν } be its border. Definition (The Border Basis Scheme) Let { c ij | 1 ≤ i ≤ µ, 1 ≤ j ≤ ν } be a set of further indeterminates. The generic O -border prebasis is the set of polynomials G = { g 1 , . . . , g ν } 1 in Q = K [ x 1 , . . . , x n , c 11 , . . . , c µν ] given by g j = b j − � µ i = 1 c ij t i . For k = 1 , . . . , n , let A k ∈ Mat µ ( K [ c ij ]) be the k th formal multiplication 2 matrix associated to G . Then the affine scheme B O ⊆ K µν defined by the ideal I ( B O ) generated by the entries of the matrices A k A ℓ − A ℓ A k with 1 ≤ k < ℓ ≤ n is called the O -border basis scheme. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 21 / 31

  73. Border Basis Schemes • Let O = { t 1 , . . . , t µ } be an order ideal in T n , and let ∂ O = { b 1 , . . . , b ν } be its border. Definition (The Border Basis Scheme) Let { c ij | 1 ≤ i ≤ µ, 1 ≤ j ≤ ν } be a set of further indeterminates. The generic O -border prebasis is the set of polynomials G = { g 1 , . . . , g ν } 1 in Q = K [ x 1 , . . . , x n , c 11 , . . . , c µν ] given by g j = b j − � µ i = 1 c ij t i . For k = 1 , . . . , n , let A k ∈ Mat µ ( K [ c ij ]) be the k th formal multiplication 2 matrix associated to G . Then the affine scheme B O ⊆ K µν defined by the ideal I ( B O ) generated by the entries of the matrices A k A ℓ − A ℓ A k with 1 ≤ k < ℓ ≤ n is called the O -border basis scheme. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 21 / 31

  74. Border Basis Schemes • Let O = { t 1 , . . . , t µ } be an order ideal in T n , and let ∂ O = { b 1 , . . . , b ν } be its border. Definition (The Border Basis Scheme) Let { c ij | 1 ≤ i ≤ µ, 1 ≤ j ≤ ν } be a set of further indeterminates. The generic O -border prebasis is the set of polynomials G = { g 1 , . . . , g ν } 1 in Q = K [ x 1 , . . . , x n , c 11 , . . . , c µν ] given by g j = b j − � µ i = 1 c ij t i . For k = 1 , . . . , n , let A k ∈ Mat µ ( K [ c ij ]) be the k th formal multiplication 2 matrix associated to G . Then the affine scheme B O ⊆ K µν defined by the ideal I ( B O ) generated by the entries of the matrices A k A ℓ − A ℓ A k with 1 ≤ k < ℓ ≤ n is called the O -border basis scheme. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 21 / 31

  75. Border Basis and Gröbner Basis Schemes The Four Points Let O = { 1 , x , y , xy } . We observe that t 1 = 1 , t 2 = x , t 3 = y , t 4 = xy , b 1 = x 2 , b 2 = y 2 , b 3 = x 2 y , b 4 = xy 2 . Let σ = DegRevLex , so that x > σ y . x 2 − c 11 1 − c 21 x − c 31 y − c 41 xy = g 1 y 2 − c 12 1 − c 22 x − c 32 y − c 42 xy = g 2 x 2 y − c 13 1 − c 23 x − c 33 y − c 43 xy = g 3 xy 2 − c 14 1 − c 24 x − c 34 y − c 44 xy = g 4 If we do the Gröbner computation via critical pairs, then necessarily c 42 = 0 , so that g 2 is replaced by 2 = y 2 − c 12 1 − c 22 x − c 32 y g ∗ and we get a seven-dimensional scheme Y . If we use the commutativity criterion to get the border basis scheme we get an eigth-dimensional scheme X such that Y is an hyperplane section. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 22 / 31

  76. Border Basis and Gröbner Basis Schemes The Four Points Let O = { 1 , x , y , xy } . We observe that t 1 = 1 , t 2 = x , t 3 = y , t 4 = xy , b 1 = x 2 , b 2 = y 2 , b 3 = x 2 y , b 4 = xy 2 . Let σ = DegRevLex , so that x > σ y . x 2 − c 11 1 − c 21 x − c 31 y − c 41 xy = g 1 y 2 − c 12 1 − c 22 x − c 32 y − c 42 xy = g 2 x 2 y − c 13 1 − c 23 x − c 33 y − c 43 xy = g 3 xy 2 − c 14 1 − c 24 x − c 34 y − c 44 xy = g 4 If we do the Gröbner computation via critical pairs, then necessarily c 42 = 0 , so that g 2 is replaced by 2 = y 2 − c 12 1 − c 22 x − c 32 y g ∗ and we get a seven-dimensional scheme Y . If we use the commutativity criterion to get the border basis scheme we get an eigth-dimensional scheme X such that Y is an hyperplane section. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 22 / 31

  77. Border Basis and Gröbner Basis Schemes The Four Points Let O = { 1 , x , y , xy } . We observe that t 1 = 1 , t 2 = x , t 3 = y , t 4 = xy , b 1 = x 2 , b 2 = y 2 , b 3 = x 2 y , b 4 = xy 2 . Let σ = DegRevLex , so that x > σ y . x 2 − c 11 1 − c 21 x − c 31 y − c 41 xy = g 1 y 2 − c 12 1 − c 22 x − c 32 y − c 42 xy = g 2 x 2 y − c 13 1 − c 23 x − c 33 y − c 43 xy = g 3 xy 2 − c 14 1 − c 24 x − c 34 y − c 44 xy = g 4 If we do the Gröbner computation via critical pairs, then necessarily c 42 = 0 , so that g 2 is replaced by 2 = y 2 − c 12 1 − c 22 x − c 32 y g ∗ and we get a seven-dimensional scheme Y . If we use the commutativity criterion to get the border basis scheme we get an eigth-dimensional scheme X such that Y is an hyperplane section. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 22 / 31

  78. Border Basis and Gröbner Basis Schemes The Four Points Let O = { 1 , x , y , xy } . We observe that t 1 = 1 , t 2 = x , t 3 = y , t 4 = xy , b 1 = x 2 , b 2 = y 2 , b 3 = x 2 y , b 4 = xy 2 . Let σ = DegRevLex , so that x > σ y . x 2 − c 11 1 − c 21 x − c 31 y − c 41 xy = g 1 y 2 − c 12 1 − c 22 x − c 32 y − c 42 xy = g 2 x 2 y − c 13 1 − c 23 x − c 33 y − c 43 xy = g 3 xy 2 − c 14 1 − c 24 x − c 34 y − c 44 xy = g 4 If we do the Gröbner computation via critical pairs, then necessarily c 42 = 0 , so that g 2 is replaced by 2 = y 2 − c 12 1 − c 22 x − c 32 y g ∗ and we get a seven-dimensional scheme Y . If we use the commutativity criterion to get the border basis scheme we get an eigth-dimensional scheme X such that Y is an hyperplane section. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 22 / 31

  79. Border Basis and Gröbner Basis Schemes The Four Points Let O = { 1 , x , y , xy } . We observe that t 1 = 1 , t 2 = x , t 3 = y , t 4 = xy , b 1 = x 2 , b 2 = y 2 , b 3 = x 2 y , b 4 = xy 2 . Let σ = DegRevLex , so that x > σ y . x 2 − c 11 1 − c 21 x − c 31 y − c 41 xy = g 1 y 2 − c 12 1 − c 22 x − c 32 y − c 42 xy = g 2 x 2 y − c 13 1 − c 23 x − c 33 y − c 43 xy = g 3 xy 2 − c 14 1 − c 24 x − c 34 y − c 44 xy = g 4 If we do the Gröbner computation via critical pairs, then necessarily c 42 = 0 , so that g 2 is replaced by 2 = y 2 − c 12 1 − c 22 x − c 32 y g ∗ and we get a seven-dimensional scheme Y . If we use the commutativity criterion to get the border basis scheme we get an eigth-dimensional scheme X such that Y is an hyperplane section. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 22 / 31

  80. Border Basis and Gröbner Basis Schemes The Four Points Let O = { 1 , x , y , xy } . We observe that t 1 = 1 , t 2 = x , t 3 = y , t 4 = xy , b 1 = x 2 , b 2 = y 2 , b 3 = x 2 y , b 4 = xy 2 . Let σ = DegRevLex , so that x > σ y . x 2 − c 11 1 − c 21 x − c 31 y − c 41 xy = g 1 y 2 − c 12 1 − c 22 x − c 32 y − c 42 xy = g 2 x 2 y − c 13 1 − c 23 x − c 33 y − c 43 xy = g 3 xy 2 − c 14 1 − c 24 x − c 34 y − c 44 xy = g 4 If we do the Gröbner computation via critical pairs, then necessarily c 42 = 0 , so that g 2 is replaced by 2 = y 2 − c 12 1 − c 22 x − c 32 y g ∗ and we get a seven-dimensional scheme Y . If we use the commutativity criterion to get the border basis scheme we get an eigth-dimensional scheme X such that Y is an hyperplane section. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 22 / 31

  81. Border Basis and Gröbner Basis Schemes Philosophy A border basis of an ideal of points I in P is intrinsically related to a basis O of the quotient ring. If we move the points slightly, O is still a basis of the perturbed ideal ˜ I , since the evaluation matrix of the elements of O at the points has determinant different from zero. Moving the points moves the border basis, and the movement traces a path inside the border basis scheme. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 23 / 31

  82. Border Basis and Gröbner Basis Schemes Philosophy A border basis of an ideal of points I in P is intrinsically related to a basis O of the quotient ring. If we move the points slightly, O is still a basis of the perturbed ideal ˜ I , since the evaluation matrix of the elements of O at the points has determinant different from zero. Moving the points moves the border basis, and the movement traces a path inside the border basis scheme. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 23 / 31

  83. Border Basis and Gröbner Basis Schemes Philosophy A border basis of an ideal of points I in P is intrinsically related to a basis O of the quotient ring. If we move the points slightly, O is still a basis of the perturbed ideal ˜ I , since the evaluation matrix of the elements of O at the points has determinant different from zero. Moving the points moves the border basis, and the movement traces a path inside the border basis scheme. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 23 / 31

  84. Border Basis and Gröbner Basis Schemes Philosophy A border basis of an ideal of points I in P is intrinsically related to a basis O of the quotient ring. If we move the points slightly, O is still a basis of the perturbed ideal ˜ I , since the evaluation matrix of the elements of O at the points has determinant different from zero. Moving the points moves the border basis, and the movement traces a path inside the border basis scheme. Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 23 / 31

  85. Border Basis and Gröbner Basis Schemes The Gröbner Scheme and the Universal Family Gröbner basis schemes and their associated universal families can be viewed as weighted projective schemes. Gröbner basis schemes can be obtained as sections of border basis schemes with suitable linear spaces. The process of construction Gröbner basis schemes via Buchberger’s Algorithm turns out to be canonical. Let O be an order ideal and σ a term ordering on T n . If the order ideal O is a σ -cornercut then there is a natural isomorphism of schemes between G O ,σ and B O . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 24 / 31

  86. Border Basis and Gröbner Basis Schemes The Gröbner Scheme and the Universal Family Gröbner basis schemes and their associated universal families can be viewed as weighted projective schemes. Gröbner basis schemes can be obtained as sections of border basis schemes with suitable linear spaces. The process of construction Gröbner basis schemes via Buchberger’s Algorithm turns out to be canonical. Let O be an order ideal and σ a term ordering on T n . If the order ideal O is a σ -cornercut then there is a natural isomorphism of schemes between G O ,σ and B O . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 24 / 31

  87. Border Basis and Gröbner Basis Schemes The Gröbner Scheme and the Universal Family Gröbner basis schemes and their associated universal families can be viewed as weighted projective schemes. Gröbner basis schemes can be obtained as sections of border basis schemes with suitable linear spaces. The process of construction Gröbner basis schemes via Buchberger’s Algorithm turns out to be canonical. Let O be an order ideal and σ a term ordering on T n . If the order ideal O is a σ -cornercut then there is a natural isomorphism of schemes between G O ,σ and B O . Lorenzo Robbiano (Università di Genova) Factorial Designs and Hilbert Schemes Genova, June 2015 24 / 31

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