bernstein sato polynomials and generalizations
play

Bernstein-Sato polynomials and generalizations Nero Budur (KU - PowerPoint PPT Presentation

Apr 17, 2023 •379 likes •1.57k views

Bernstein-Sato polynomials and generalizations Nero Budur (KU Leuven) Summer school Algebra, Algorithms, and Algebraic Analysis , Rolduc Abbey, Netherlands September 2-6, 2013 Nero Budur (KU Leuven) Bernstein-Sato polynomials and

  1. � � � � � Natural functors on constr. sh. extend to derived functors on D b c ( X ) . E.g. p : X → Y gives Rp ∗ s.t. exact 0 → F q 1 → F q 2 → F q 3 → 0 , gives exact long sequence .. → H i ( Rp ∗ F q 1 ) → H i ( Rp ∗ F q 2 ) → H i ( Rp ∗ F q 3 ) → H i + 1 ( Rp ∗ F q 1 ) → .. Let f : X → C holom. fc. Consider p f − 1 ( 0 ) � � � X X × C ˜ C ∗ i f C ∗ ˜ � � C C ∗ • Nearby cycles functor ψ f := i ∗ Rp ∗ p ∗ : D b c ( X ) → D b c ( f − 1 ( 0 )) . Eigenspace decomposition: ψ f = ⊕ λ ψ f ,λ . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  2. • [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  3. • [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  4. • [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . D X = sheaf of alg. diff. operators Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  5. • [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . D X = sheaf of alg. diff. operators = locally A n ( C ) . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  6. • [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . D X = sheaf of alg. diff. operators = locally A n ( C ) . D b rh ( D X ) = bdd der cat of complexes of D X -mods with regular holonomic cohomology. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  7. • [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . D X = sheaf of alg. diff. operators = locally A n ( C ) . D b rh ( D X ) = bdd der cat of complexes of D X -mods with regular holonomic cohomology. • [Malgrange, Kashiwara, Mebkhout]: Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  8. • [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . D X = sheaf of alg. diff. operators = locally A n ( C ) . D b rh ( D X ) = bdd der cat of complexes of D X -mods with regular holonomic cohomology. • [Malgrange, Kashiwara, Mebkhout]: Let X be smooth C alg. variety of dim n. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  9. • [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . D X = sheaf of alg. diff. operators = locally A n ( C ) . D b rh ( D X ) = bdd der cat of complexes of D X -mods with regular holonomic cohomology. • [Malgrange, Kashiwara, Mebkhout]: Let X be smooth C alg. variety of dim n. Then ∃ DR X : D b rh ( D X ) ↔ D b c ( X ) Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  10. • [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . D X = sheaf of alg. diff. operators = locally A n ( C ) . D b rh ( D X ) = bdd der cat of complexes of D X -mods with regular holonomic cohomology. • [Malgrange, Kashiwara, Mebkhout]: Let X be smooth C alg. variety of dim n. Then ∃ DR X : D b rh ( D X ) ↔ D b c ( X ) s.t. if M = D X -mod, then DR X M = (Ω q X ⊗ O X M )[ n ] . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  11. • [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . D X = sheaf of alg. diff. operators = locally A n ( C ) . D b rh ( D X ) = bdd der cat of complexes of D X -mods with regular holonomic cohomology. • [Malgrange, Kashiwara, Mebkhout]: Let X be smooth C alg. variety of dim n. Then ∃ DR X : D b rh ( D X ) ↔ D b c ( X ) s.t. if M = D X -mod, then DR X M = (Ω q X ⊗ O X M )[ n ] . So ∃ D -modules counterpart of ψ f : Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  12. • [Deligne]: X = C -anal. mfd., f : X → C holomorphic fc. ⇒ H i ( F f , x , C ) λ = H i ( i ∗ x ψ f ,λ C X ) . D X = sheaf of alg. diff. operators = locally A n ( C ) . D b rh ( D X ) = bdd der cat of complexes of D X -mods with regular holonomic cohomology. • [Malgrange, Kashiwara, Mebkhout]: Let X be smooth C alg. variety of dim n. Then ∃ DR X : D b rh ( D X ) ↔ D b c ( X ) s.t. if M = D X -mod, then DR X M = (Ω q X ⊗ O X M )[ n ] . So ∃ D -modules counterpart of ψ f : via V -filtration, to define soon. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  13. Let i f : X → X × C be x �→ ( x , f ( x )) . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  14. Let i f : X → X × C be x �→ ( x , f ( x )) . • [Malgrange, Kashiwara]: Let X be smooth C alg. variety of dim n Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  15. Let i f : X → X × C be x �→ ( x , f ( x )) . • [Malgrange, Kashiwara]: Let X be smooth C alg. variety of dim n and f : X → C a regular function. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  16. Let i f : X → X × C be x �→ ( x , f ( x )) . • [Malgrange, Kashiwara]: Let X be smooth C alg. variety of dim n and f : X → C a regular function. For α ∈ ( 0 , 1 ] , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  17. Let i f : X → X × C be x �→ ( x , f ( x )) . • [Malgrange, Kashiwara]: Let X be smooth C alg. variety of dim n and f : X → C a regular function. For α ∈ ( 0 , 1 ] , ψ f ,λ C X [ n − 1 ] = DR X ( Gr α V ( i f ) + O X ) Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  18. Let i f : X → X × C be x �→ ( x , f ( x )) . • [Malgrange, Kashiwara]: Let X be smooth C alg. variety of dim n and f : X → C a regular function. For α ∈ ( 0 , 1 ] , ψ f ,λ C X [ n − 1 ] = DR X ( Gr α V ( i f ) + O X ) where λ = exp ( − 2 π i α ) Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  19. Let i f : X → X × C be x �→ ( x , f ( x )) . • [Malgrange, Kashiwara]: Let X be smooth C alg. variety of dim n and f : X → C a regular function. For α ∈ ( 0 , 1 ] , ψ f ,λ C X [ n − 1 ] = DR X ( Gr α V ( i f ) + O X ) where λ = exp ( − 2 π i α ) and ( i f ) + is the D -module direct image. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  20. Let i f : X → X × C be x �→ ( x , f ( x )) . • [Malgrange, Kashiwara]: Let X be smooth C alg. variety of dim n and f : X → C a regular function. For α ∈ ( 0 , 1 ] , ψ f ,λ C X [ n − 1 ] = DR X ( Gr α V ( i f ) + O X ) where λ = exp ( − 2 π i α ) and ( i f ) + is the D -module direct image. Next: V -filtrations, generalized Bernstein-Sato polynomials, more geometry. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  21. V -filtration Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  22. V -filtration X = C n , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  23. V -filtration X = C n , D X = A n ( C ) , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  24. V -filtration X = C n , D X = A n ( C ) , Y = X × C r , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  25. V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  26. V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  27. V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , with x = x 1 , . . . , x n , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  28. V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , with x = x 1 , . . . , x n , t = t 1 , . . . , t r . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  29. V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , with x = x 1 , . . . , x n , t = t 1 , . . . , t r . So the ideal I ⊂ O Y of X × 0 in Y is generated by t . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  30. V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , with x = x 1 , . . . , x n , t = t 1 , . . . , t r . So the ideal I ⊂ O Y of X × 0 in Y is generated by t . So D X = O X [ ∂ x ] , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  31. V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , with x = x 1 , . . . , x n , t = t 1 , . . . , t r . So the ideal I ⊂ O Y of X × 0 in Y is generated by t . So D X = O X [ ∂ x ] , D Y = O Y [ ∂ x , ∂ t ] . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  32. V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , with x = x 1 , . . . , x n , t = t 1 , . . . , t r . So the ideal I ⊂ O Y of X × 0 in Y is generated by t . So D X = O X [ ∂ x ] , D Y = O Y [ ∂ x , ∂ t ] . • Filtration V along X × 0 on D Y : Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  33. V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , with x = x 1 , . . . , x n , t = t 1 , . . . , t r . So the ideal I ⊂ O Y of X × 0 in Y is generated by t . So D X = O X [ ∂ x ] , D Y = O Y [ ∂ x , ∂ t ] . • Filtration V along X × 0 on D Y : V j D Y = { P ∈ D Y | PI i ⊂ I i + j for all i ∈ Z } , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  34. V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , with x = x 1 , . . . , x n , t = t 1 , . . . , t r . So the ideal I ⊂ O Y of X × 0 in Y is generated by t . So D X = O X [ ∂ x ] , D Y = O Y [ ∂ x , ∂ t ] . • Filtration V along X × 0 on D Y : V j D Y = { P ∈ D Y | PI i ⊂ I i + j for all i ∈ Z } , with j ∈ Z and I i = O Y for i ≤ 0. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  35. V -filtration X = C n , D X = A n ( C ) , Y = X × C r , O X = C [ x ] , O Y = C [ x , t ] , with x = x 1 , . . . , x n , t = t 1 , . . . , t r . So the ideal I ⊂ O Y of X × 0 in Y is generated by t . So D X = O X [ ∂ x ] , D Y = O Y [ ∂ x , ∂ t ] . • Filtration V along X × 0 on D Y : V j D Y = { P ∈ D Y | PI i ⊂ I i + j for all i ∈ Z } , with j ∈ Z and I i = O Y for i ≤ 0. So V j D Y is generated over D X by the monomials t β ∂ γ t with | β | − | γ | ≥ j . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  36. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  37. • Let M be a fin. gen. D Y -module. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  38. • Let M be a fin. gen. D Y -module. The filtration V along X × 0 on M is Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  39. • Let M be a fin. gen. D Y -module. The filtration V along X × 0 on M is an exhaustive decreasing filtr. of fin. gen. V 0 D Y -submods V α M s.t.: Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  40. • Let M be a fin. gen. D Y -module. The filtration V along X × 0 on M is an exhaustive decreasing filtr. of fin. gen. V 0 D Y -submods V α M s.t.: (i) { V α M } α is indexed left-continuously and discretely by rational numbers, Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  41. • Let M be a fin. gen. D Y -module. The filtration V along X × 0 on M is an exhaustive decreasing filtr. of fin. gen. V 0 D Y -submods V α M s.t.: (i) { V α M } α is indexed left-continuously and discretely by rational numbers, i.e. V α M = ∩ β<α V β M ; Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  42. • Let M be a fin. gen. D Y -module. The filtration V along X × 0 on M is an exhaustive decreasing filtr. of fin. gen. V 0 D Y -submods V α M s.t.: (i) { V α M } α is indexed left-continuously and discretely by rational numbers, i.e. V α M = ∩ β<α V β M ; (ii) ( V i D Y )( V α M ) ⊂ V α + i M ; Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  43. • Let M be a fin. gen. D Y -module. The filtration V along X × 0 on M is an exhaustive decreasing filtr. of fin. gen. V 0 D Y -submods V α M s.t.: (i) { V α M } α is indexed left-continuously and discretely by rational numbers, i.e. V α M = ∩ β<α V β M ; (ii) ( V i D Y )( V α M ) ⊂ V α + i M ; j t j V α M = V α + 1 M for α ≫ 0; (iii) � Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  44. • Let M be a fin. gen. D Y -module. The filtration V along X × 0 on M is an exhaustive decreasing filtr. of fin. gen. V 0 D Y -submods V α M s.t.: (i) { V α M } α is indexed left-continuously and discretely by rational numbers, i.e. V α M = ∩ β<α V β M ; (ii) ( V i D Y )( V α M ) ⊂ V α + i M ; j t j V α M = V α + 1 M for α ≫ 0; (iii) � j ∂ t j t j − α on Gr α V M = V α M / V >α M is nilpotent. (iv) the action of � Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  45. • Let M be a fin. gen. D Y -module. The filtration V along X × 0 on M is an exhaustive decreasing filtr. of fin. gen. V 0 D Y -submods V α M s.t.: (i) { V α M } α is indexed left-continuously and discretely by rational numbers, i.e. V α M = ∩ β<α V β M ; (ii) ( V i D Y )( V α M ) ⊂ V α + i M ; j t j V α M = V α + 1 M for α ≫ 0; (iii) � j ∂ t j t j − α on Gr α V M = V α M / V >α M is nilpotent. (iv) the action of � • [Malgrange, Kashiwara]: The filtration V along X × 0 on M exists if M is regular holonomic and quasi-unipotent. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  46. • Let M be a fin. gen. D Y -module. The filtration V along X × 0 on M is an exhaustive decreasing filtr. of fin. gen. V 0 D Y -submods V α M s.t.: (i) { V α M } α is indexed left-continuously and discretely by rational numbers, i.e. V α M = ∩ β<α V β M ; (ii) ( V i D Y )( V α M ) ⊂ V α + i M ; j t j V α M = V α + 1 M for α ≫ 0; (iii) � j ∂ t j t j − α on Gr α V M = V α M / V >α M is nilpotent. (iv) the action of � • [Malgrange, Kashiwara]: The filtration V along X × 0 on M exists if M is regular holonomic and quasi-unipotent. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations We see later how the proof of existence reduces to the case r 1.

  47. Lemma: V q M along X × 0 is unique if it exists. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  48. Lemma: V q M along X × 0 is unique if it exists. See proof in the notes. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  49. Lemma: V q M along X × 0 is unique if it exists. See proof in the notes. • For m ∈ M , the Bernstein-Sato polynomial b m ( s ) of m is Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  50. Lemma: V q M along X × 0 is unique if it exists. See proof in the notes. • For m ∈ M , the Bernstein-Sato polynomial b m ( s ) of m is the non-zero monic minimal polynomial of the action of s = − � j ∂ t j t j on V 0 D Y · m / V 1 D Y · m . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  51. Lemma: V q M along X × 0 is unique if it exists. See proof in the notes. • For m ∈ M , the Bernstein-Sato polynomial b m ( s ) of m is the non-zero monic minimal polynomial of the action of s = − � j ∂ t j t j on V 0 D Y · m / V 1 D Y · m . • [Sabbah]: If the filtration V along X × 0 on M exists, then Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  52. Lemma: V q M along X × 0 is unique if it exists. See proof in the notes. • For m ∈ M , the Bernstein-Sato polynomial b m ( s ) of m is the non-zero monic minimal polynomial of the action of s = − � j ∂ t j t j on V 0 D Y · m / V 1 D Y · m . • [Sabbah]: If the filtration V along X × 0 on M exists, then b m ( s ) exists for all m ∈ M, and has all roots rational. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  53. Lemma: V q M along X × 0 is unique if it exists. See proof in the notes. • For m ∈ M , the Bernstein-Sato polynomial b m ( s ) of m is the non-zero monic minimal polynomial of the action of s = − � j ∂ t j t j on V 0 D Y · m / V 1 D Y · m . • [Sabbah]: If the filtration V along X × 0 on M exists, then b m ( s ) exists for all m ∈ M, and has all roots rational. Moreover, V α M = { m ∈ M | α ≤ c if b m ( − c ) = 0 } . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  54. Lemma: V q M along X × 0 is unique if it exists. See proof in the notes. • For m ∈ M , the Bernstein-Sato polynomial b m ( s ) of m is the non-zero monic minimal polynomial of the action of s = − � j ∂ t j t j on V 0 D Y · m / V 1 D Y · m . • [Sabbah]: If the filtration V along X × 0 on M exists, then b m ( s ) exists for all m ∈ M, and has all roots rational. Moreover, V α M = { m ∈ M | α ≤ c if b m ( − c ) = 0 } . See proof in the notes. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  55. Next: geometry behind the V -filtration. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  56. Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  57. Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  58. Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  59. Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  60. Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image = O X ⊗ C C [ ∂ t ] Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  61. Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image = O X ⊗ C C [ ∂ t ] s.t. for g , h ∈ O X , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  62. Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image = O X ⊗ C C [ ∂ t ] s.t. for g , h ∈ O X , g ( h ⊗ ∂ ν t ) = gh ⊗ ∂ ν t , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  63. Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image = O X ⊗ C C [ ∂ t ] s.t. for g , h ∈ O X , ∂ f j g ( h ⊗ ∂ ν t ) = gh ⊗ ∂ ν t , ∂ x i ( h ⊗ ∂ ν t ) = ∂ x i h ⊗ ∂ ν ∂ x i h ⊗ ∂ t j ∂ ν t − � t , j Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  64. Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image = O X ⊗ C C [ ∂ t ] s.t. for g , h ∈ O X , ∂ f j g ( h ⊗ ∂ ν t ) = gh ⊗ ∂ ν t , ∂ x i ( h ⊗ ∂ ν t ) = ∂ x i h ⊗ ∂ ν ∂ x i h ⊗ ∂ t j ∂ ν t − � t , j ∂ t j ( h ⊗ ∂ ν t ) = h ⊗ ∂ t j ∂ ν t , Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  65. Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image = O X ⊗ C C [ ∂ t ] s.t. for g , h ∈ O X , ∂ f j g ( h ⊗ ∂ ν t ) = gh ⊗ ∂ ν t , ∂ x i ( h ⊗ ∂ ν t ) = ∂ x i h ⊗ ∂ ν ∂ x i h ⊗ ∂ t j ∂ ν t − � t , j t − ν j h ⊗ ( ∂ ν − e j ∂ t j ( h ⊗ ∂ ν t ) = h ⊗ ∂ t j ∂ ν t , t j ( h ⊗ ∂ ν t ) = f j h ⊗ ∂ ν ) , t Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  66. Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image = O X ⊗ C C [ ∂ t ] s.t. for g , h ∈ O X , ∂ f j g ( h ⊗ ∂ ν t ) = gh ⊗ ∂ ν t , ∂ x i ( h ⊗ ∂ ν t ) = ∂ x i h ⊗ ∂ ν ∂ x i h ⊗ ∂ t j ∂ ν t − � t , j t − ν j h ⊗ ( ∂ ν − e j ∂ t j ( h ⊗ ∂ ν t ) = h ⊗ ∂ t j ∂ ν t , t j ( h ⊗ ∂ ν t ) = f j h ⊗ ∂ ν ) , t Facts: O X = reg. holon. D X -mod Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  67. Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image = O X ⊗ C C [ ∂ t ] s.t. for g , h ∈ O X , ∂ f j g ( h ⊗ ∂ ν t ) = gh ⊗ ∂ ν t , ∂ x i ( h ⊗ ∂ ν t ) = ∂ x i h ⊗ ∂ ν ∂ x i h ⊗ ∂ t j ∂ ν t − � t , j t − ν j h ⊗ ( ∂ ν − e j ∂ t j ( h ⊗ ∂ ν t ) = h ⊗ ∂ t j ∂ ν t , t j ( h ⊗ ∂ ν t ) = f j h ⊗ ∂ ν ) , t Facts: O X = reg. holon. D X -mod ⇒ ( i f ) + O X = reg. holon. quasi-unip. D Y -mod. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  68. Next: geometry behind the V -filtration. Let X = C n , f = ( f 1 , . . . , f r ) , f i ∈ C [ x ] = O X , i f : X → X × C r = Y given by x �→ ( x , f ( x )) . Let t = ( t 1 , . . . , t r ) be the coords of C r . Let ( i f ) + O X = D -mod direct image = O X ⊗ C C [ ∂ t ] s.t. for g , h ∈ O X , ∂ f j g ( h ⊗ ∂ ν t ) = gh ⊗ ∂ ν t , ∂ x i ( h ⊗ ∂ ν t ) = ∂ x i h ⊗ ∂ ν ∂ x i h ⊗ ∂ t j ∂ ν t − � t , j t − ν j h ⊗ ( ∂ ν − e j ∂ t j ( h ⊗ ∂ ν t ) = h ⊗ ∂ t j ∂ ν t , t j ( h ⊗ ∂ ν t ) = f j h ⊗ ∂ ν ) , t Facts: O X = reg. holon. D X -mod ⇒ ( i f ) + O X = reg. holon. quasi-unip. D Y -mod. So ∃ V q ( i f ) + O X along X × 0. Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  69. • When r = 1, M-K thm says ψ f ,λ C X [ n − 1 ] = DR X ( Gr α V ( i f ) + O X ) . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

  70. • When r = 1, M-K thm says ψ f ,λ C X [ n − 1 ] = DR X ( Gr α V ( i f ) + O X ) . Moreover, s = − ∂ t t on V > 0 ( i f ) + O X / V 1 ( i f ) + O X corresponds to logarithm of unipotent part T u of monodromy T = T s T u . Nero Budur (KU Leuven) Bernstein-Sato polynomials and generalizations

Recommend

BERNSTEIN-SATO POLYNOMIALS AND GENERALIZATIONS (LECTURE NOTES, ROLDUC ABBEY, 2013) NERO BUDUR

BERNSTEIN-SATO POLYNOMIALS AND GENERALIZATIONS (LECTURE NOTES, ROLDUC ABBEY, 2013) NERO BUDUR

BERNSTEIN-SATO POLYNOMIALS AND GENERALIZATIONS (LECTURE NOTES, ROLDUC ABBEY, 2013) NERO BUDUR Abstract. These are lecture notes from a series of lectures at the Summer school Algebra, Algorithms, and Algebraic Analysis , Rolduc Abbey, Netherlands,

216 views • 21 slides
Introduction to D -module Theory. Algorithms for Computing Bernstein-Sato Polynomials Jorge

Introduction to D -module Theory. Algorithms for Computing Bernstein-Sato Polynomials Jorge

Introduction to D -module Theory. Algorithms for Computing Bernstein-Sato Polynomials Jorge Mart n-Morales Centro Universitario de la Defensa de Zaragoza Academia General Militar Differential Algebra and Related Topics October 27-30,

866 views • 42 slides
Why Fuzzy Interpretation of . . . Bernstein Polynomials Fuzzy Interpretation . . . How Can We .

Why Fuzzy Interpretation of . . . Bernstein Polynomials Fuzzy Interpretation . . . How Can We .

Functional . . . Need for Polynomial . . . How to Represent . . . Bernstein . . . Why Fuzzy Interpretation of . . . Bernstein Polynomials Fuzzy Interpretation . . . How Can We . . . Are Better: Fuzzy Analysis (cont-d) Why Bernstein . . .

420 views • 18 slides
Properties and applications of the constrained dual Bernstein polynomials Stanisaw Lewanowicz

Properties and applications of the constrained dual Bernstein polynomials Stanisaw Lewanowicz

International Conference on Scientific Computing Properties and applications of the constrained dual Bernstein polynomials Stanisaw Lewanowicz &amp; Pawe Wony Institute of Computer Science University of Wrocaw, POLAND e-mail:

861 views • 69 slides
AN INTRODUCTION TO TRANSITION POLYNOMIALS Jo Ellis-Monaghan The story There are

AN INTRODUCTION TO TRANSITION POLYNOMIALS Jo Ellis-Monaghan The story There are

AN INTRODUCTION TO TRANSITION POLYNOMIALS Jo Ellis-Monaghan The story There are deletion-contraction polynomials. The Tutte polynomial assimilates them all. There are many generalizations of the Tutte polynomial to adapt it to

471 views • 23 slides
Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING Lecture 16 Job Shop 1. Job Shop

Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING Lecture 16 Job Shop 1. Job Shop

Job Shop Generalizations Outline DMP204 SCHEDULING, TIMETABLING AND ROUTING Lecture 16 Job Shop 1. Job Shop Generalizations The Alternative Graph Model Marco Chiarandini 2 Job Shop Generalizations Job Shop Generalizations Resume

509 views • 5 slides
Generalizations are driven by semantics and constrained by statistical preemption New evidence

Generalizations are driven by semantics and constrained by statistical preemption New evidence

Generalizations are driven by semantics and constrained by statistical preemption New evidence from artificial language experiments Florent Perek &amp; Adele Goldberg University of Birmingham &amp; Princeton University Generalizations Previous

360 views • 25 slides
The Computational and Logical Nature of Phonological Generalizations Jeffrey Heinz , Jane

The Computational and Logical Nature of Phonological Generalizations Jeffrey Heinz , Jane

Intro Generalizations Stringsets String mappings The Computational and Logical Nature of Phonological Generalizations Jeffrey Heinz , Jane Chandlee , Bill Idsardi , and Jim Rogers University of Delaware, University of

1.15k views • 86 slides
More Zeroes of Polynomials In this lecture we look more carefully at zeroes of polynomials.

More Zeroes of Polynomials In this lecture we look more carefully at zeroes of polynomials.

More Zeroes of Polynomials In this lecture we look more carefully at zeroes of polynomials. (Recall: a zero of a polynomial is sometimes called a root.) Our goal in the next few presentations is to set up a strategy for Elementary

331 views • 9 slides
28 4x 81y 4 z rt 3 2 , 4 5 6 7 2 17x mn 3 We usually write the variables in exponential

28 4x 81y 4 z rt 3 2 , 4 5 6 7 2 17x mn 3 We usually write the variables in exponential

Slide 1 / 216 Slide 2 / 216 Algebra I Polynomials 2015-11-02 www.njctl.org Slide 3 / 216 Click on the topic to go to that section Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials

964 views • 72 slides
Negative Churchs Thesis and Russian Constructivism SATO Kentaro sato@inf.unibe.ch

Negative Churchs Thesis and Russian Constructivism SATO Kentaro sato@inf.unibe.ch

Negative Churchs Thesis and Russian Constructivism SATO Kentaro sato@inf.unibe.ch Institution for Computer Science, University of Bern * partially supported by John Templeton Foundation Negative Churchs Thesis and Russian Constructivism

1.02k views • 66 slides
Remarks on Gdels Incomplentess Theorems SATO Kentaro * sato@inf.unibe.ch SGSLPS Autumn 2016

Remarks on Gdels Incomplentess Theorems SATO Kentaro * sato@inf.unibe.ch SGSLPS Autumn 2016

Remarks on Gdels Incomplentess Theorems SATO Kentaro * sato@inf.unibe.ch SGSLPS Autumn 2016 - *His research is supported by John Templeton Foundation

1.58k views • 144 slides
Classification of Combinatorial Polynomials (in particular, Ehrhart Polynomials of Zonotopes)

Classification of Combinatorial Polynomials (in particular, Ehrhart Polynomials of Zonotopes)

Classification of Combinatorial Polynomials (in particular, Ehrhart Polynomials of Zonotopes) Matthias Beck San Francisco State University Katharina Jochemko Kungliga Tekniska H ogskolan Emily McCullough University of San Francisco

244 views • 20 slides
Generalizations of Gowers Theorem Dana Barto sov a (USP) Aleksandra Kwiatkowska (UCLA)

Generalizations of Gowers Theorem Dana Barto sov a (USP) Aleksandra Kwiatkowska (UCLA)

Generalizations of Gowers Theorem Dana Barto sov a (USP) Aleksandra Kwiatkowska (UCLA) BWB 2014 Maresias August 25-29, 2014 This work was supported by the grant FAPESP 2013/14458-9. Dana Barto sov a Generalizations of

832 views • 70 slides
Unimodality of q -Eulerian polynomials and q , p -Eulerian polynomials Michelle Wachs University

Unimodality of q -Eulerian polynomials and q , p -Eulerian polynomials Michelle Wachs University

Unimodality of q -Eulerian polynomials and q , p -Eulerian polynomials Michelle Wachs University of Miami Joint work with John Shareshian and with Anthony Henderson Eulerian numbers and Eulerian polynomials Eulerian numbers: a n , j := |{

710 views • 43 slides
Multivariate generalizations of the Foata-Sch utzenberger equidistribution Fourth Colloquium

Multivariate generalizations of the Foata-Sch utzenberger equidistribution Fourth Colloquium

Generalizations of the Foata-Sch utzenberger equidistribution MathInfo 2006 1 de 26 Multivariate generalizations of the Foata-Sch utzenberger equidistribution Fourth Colloquium in Mathematics and Computer Science F. Hivert, J.-C.

478 views • 26 slides
O N G E O D E S I C M A P P I N G S and THEIR GENERALIZATIONS Josef MIKE S Department of

O N G E O D E S I C M A P P I N G S and THEIR GENERALIZATIONS Josef MIKE S Department of

O N G E O D E S I C M A P P I N G S and THEIR GENERALIZATIONS Josef MIKE S Department of Algebra and Geometry Palacky University Olomouc Czech Republic mikes@inf.upol.cz VARNA (ODESSOS) 2 0 0 7 1. Introduction Diffeomorphisms and

587 views • 32 slides
Advances in Knot Polynomials 21 October 2016 Advances in Knot Polynomials 21 October 2016 1 /

Advances in Knot Polynomials 21 October 2016 Advances in Knot Polynomials 21 October 2016 1 /

Advances in Knot Polynomials 21 October 2016 Advances in Knot Polynomials 21 October 2016 1 / 49 ABSTRACT Review of achievements and problems in the theory of colored knot polynomials. Accent is on the current mystery around the differential

761 views • 49 slides
Robust Algorithms for Chebyshev Polynomials and Related Approximations Miroslav Vl cek

Robust Algorithms for Chebyshev Polynomials and Related Approximations Miroslav Vl cek

Introduction Chebyshev polynomials Symmetrical Zolotarev polynomials Overview Application in Filter Design Conclusion Robust Algorithms for Chebyshev Polynomials and Related Approximations Miroslav Vl cek vlcek@fd.cvut.cz Department of

484 views • 37 slides
MS&amp;AD Ventures Building long term value - Jon Soberg &amp; Tak Sato MS&amp;AD Ventures

MS&amp;AD Ventures Building long term value - Jon Soberg &amp; Tak Sato MS&amp;AD Ventures

MS&amp;AD Ventures Building long term value - Jon Soberg &amp; Tak Sato MS&amp;AD Ventures Inc. September, 2019 Self Introduction Mr. Tak Sato Mr. Jon Soberg 1 Silicon Valley:The basis of the concept may change Data from Cloud to device

405 views • 11 slides
Communicating generalizations (in computational terms) Michael Henry Tessler Stanford University

Communicating generalizations (in computational terms) Michael Henry Tessler Stanford University

Communicating generalizations (in computational terms) Michael Henry Tessler Stanford University A Generic Workshop (CSLI) May 20, 2017 What do generalizations in language mean? Dogs bark. Metric: P ( F | K ) = prevalence Some dogs bark.

1.08k views • 78 slides
From Eulerian Polynomials and Chromatic Polynomials to Hessenberg Varieties Michelle Wachs

From Eulerian Polynomials and Chromatic Polynomials to Hessenberg Varieties Michelle Wachs

From Eulerian Polynomials and Chromatic Polynomials to Hessenberg Varieties Michelle Wachs University of Miami 1 1 1 1 4 1 1 11 11 1 1 26 66 26 1 Eulerian polynomials - Eulers definition t t i = 1 t i 1 Eulerian

985 views • 69 slides
Algorithms for multiquadratic number fields D. J. Bernstein Jens Bauch, Daniel J. Bernstein,

Algorithms for multiquadratic number fields D. J. Bernstein Jens Bauch, Daniel J. Bernstein,

1 Algorithms for multiquadratic number fields D. J. Bernstein Jens Bauch, Daniel J. Bernstein, Henry de Valence, Tanja Lange, Christine van Vredendaal. Short generators without quantum computers: the case of multiquadratics. Eurocrypt

1.13k views • 97 slides
Correlation bounds for polynomials, and the disproof of a conjecture on Gowers' norm using Ramsey

Correlation bounds for polynomials, and the disproof of a conjecture on Gowers' norm using Ramsey

Correlation bounds for polynomials, and the disproof of a conjecture on Gowers' norm using Ramsey theory Emanuele Viola Northeastern University May 2011 Polynomials Polynomials: degree d, n variables over F 2 = {0,1} E.g., p = x 1 + x 5 +

411 views • 17 slides

More recommend