well behaved at infinity first integrals of polynomial
play

Well-behaved At Infinity First Integrals of Polynomial Vector Fields - PowerPoint PPT Presentation

Feb 19, 2023 •572 likes •1.72k views

Well-behaved At Infinity First Integrals of Polynomial Vector Fields Antoni Ferragut Universitat Jaume I Institut de Matem` atica i Aplicacions de Castell o Joint work with C. Galindo (UJI) and F . Monserrat (UPV) To appear in the J. of

  1. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Contents Introduction and objectives 1 Polynomial vector fields in CP 2 2 Reduction of singularities 3 Linear systems. Clusters 4 Results and algorithms 5 6 WAI Positive Darboux first integrals WAIFI A. Ferragut 9/38

  2. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI The blow-up technique Blowing-up a singular point P z = q ( x , xz ) − zp ( x , xz ) V P 1 : ˙ ˙ x = p ( x , xz ) , ; x z = p ( yz , y ) − zq ( yz , y ) V P 2 : ˙ ˙ , y = q ( yz , y ) . y The exceptional divisor E P : { x = 0 } (resp. { y = 0 } ). The projection map Π P : BL P ( M ) → M ( x , z ) �→ ( x , xz ) from which E P = Π − 1 P ( P ) . WAIFI A. Ferragut 10/38

  3. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities From ω = p dy − q dx we have ω m = p m dy − q m dx . Let Π O : Bl O ( C 2 ) → C 2 and charts ( V O i , φ i ) . The total transform by Π O of w in V O 1 is ω ∗ | V O 1 := x m � ( α ( 1 , z ) + x β ( x , z )) dx � + x ( p m ( 1 , z ) + x γ ( x , z )) dz , where α ( x , y ) = yp m ( x , y ) − xq m ( x , y ) . The strict transform by Π O of w in V O 1 is 1 / x m + 1 if α ≡ 0 (resp. = ω ∗ | V O 1 / x m if α �≡ 0). 1 := ω ∗ | V O ω | V O ˜ ω on Bl O ( C 2 ) . From ˜ ω | V O i we construct a 1-form ˜ From X , M , P we can obtain ˜ X in Bl P ( M ) . WAIFI A. Ferragut 11/38

  4. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities From ω = p dy − q dx we have ω m = p m dy − q m dx . Let Π O : Bl O ( C 2 ) → C 2 and charts ( V O i , φ i ) . The total transform by Π O of w in V O 1 is ω ∗ | V O 1 := x m � ( α ( 1 , z ) + x β ( x , z )) dx � + x ( p m ( 1 , z ) + x γ ( x , z )) dz , where α ( x , y ) = yp m ( x , y ) − xq m ( x , y ) . The strict transform by Π O of w in V O 1 is 1 / x m + 1 if α ≡ 0 (resp. = ω ∗ | V O 1 / x m if α �≡ 0). 1 := ω ∗ | V O ω | V O ˜ ω on Bl O ( C 2 ) . From ˜ ω | V O i we construct a 1-form ˜ From X , M , P we can obtain ˜ X in Bl P ( M ) . WAIFI A. Ferragut 11/38

  5. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities From ω = p dy − q dx we have ω m = p m dy − q m dx . Let Π O : Bl O ( C 2 ) → C 2 and charts ( V O i , φ i ) . The total transform by Π O of w in V O 1 is ω ∗ | V O 1 := x m � ( α ( 1 , z ) + x β ( x , z )) dx � + x ( p m ( 1 , z ) + x γ ( x , z )) dz , where α ( x , y ) = yp m ( x , y ) − xq m ( x , y ) . The strict transform by Π O of w in V O 1 is 1 / x m + 1 if α ≡ 0 (resp. = ω ∗ | V O 1 / x m if α �≡ 0). 1 := ω ∗ | V O ω | V O ˜ ω on Bl O ( C 2 ) . From ˜ ω | V O i we construct a 1-form ˜ From X , M , P we can obtain ˜ X in Bl P ( M ) . WAIFI A. Ferragut 11/38

  6. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities From ω = p dy − q dx we have ω m = p m dy − q m dx . Let Π O : Bl O ( C 2 ) → C 2 and charts ( V O i , φ i ) . The total transform by Π O of w in V O 1 is ω ∗ | V O 1 := x m � ( α ( 1 , z ) + x β ( x , z )) dx � + x ( p m ( 1 , z ) + x γ ( x , z )) dz , where α ( x , y ) = yp m ( x , y ) − xq m ( x , y ) . The strict transform by Π O of w in V O 1 is 1 / x m + 1 if α ≡ 0 (resp. = ω ∗ | V O 1 / x m if α �≡ 0). 1 := ω ∗ | V O ω | V O ˜ ω on Bl O ( C 2 ) . From ˜ ω | V O i we construct a 1-form ˜ From X , M , P we can obtain ˜ X in Bl P ( M ) . WAIFI A. Ferragut 11/38

  7. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities From ω = p dy − q dx we have ω m = p m dy − q m dx . Let Π O : Bl O ( C 2 ) → C 2 and charts ( V O i , φ i ) . The total transform by Π O of w in V O 1 is ω ∗ | V O 1 := x m � ( α ( 1 , z ) + x β ( x , z )) dx � + x ( p m ( 1 , z ) + x γ ( x , z )) dz , where α ( x , y ) = yp m ( x , y ) − xq m ( x , y ) . The strict transform by Π O of w in V O 1 is 1 / x m + 1 if α ≡ 0 (resp. = ω ∗ | V O 1 / x m if α �≡ 0). 1 := ω ∗ | V O ω | V O ˜ ω on Bl O ( C 2 ) . From ˜ ω | V O i we construct a 1-form ˜ From X , M , P we can obtain ˜ X in Bl P ( M ) . WAIFI A. Ferragut 11/38

  8. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities From ω = p dy − q dx we have ω m = p m dy − q m dx . Let Π O : Bl O ( C 2 ) → C 2 and charts ( V O i , φ i ) . The total transform by Π O of w in V O 1 is ω ∗ | V O 1 := x m � ( α ( 1 , z ) + x β ( x , z )) dx � + x ( p m ( 1 , z ) + x γ ( x , z )) dz , where α ( x , y ) = yp m ( x , y ) − xq m ( x , y ) . The strict transform by Π O of w in V O 1 is 1 / x m + 1 if α ≡ 0 (resp. = ω ∗ | V O 1 / x m if α �≡ 0). 1 := ω ∗ | V O ω | V O ˜ ω on Bl O ( C 2 ) . From ˜ ω | V O i we construct a 1-form ˜ From X , M , P we can obtain ˜ X in Bl P ( M ) . WAIFI A. Ferragut 11/38

  9. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities From ω = p dy − q dx we have ω m = p m dy − q m dx . Let Π O : Bl O ( C 2 ) → C 2 and charts ( V O i , φ i ) . The total transform by Π O of w in V O 1 is ω ∗ | V O 1 := x m � ( α ( 1 , z ) + x β ( x , z )) dx � + x ( p m ( 1 , z ) + x γ ( x , z )) dz , where α ( x , y ) = yp m ( x , y ) − xq m ( x , y ) . The strict transform by Π O of w in V O 1 is 1 / x m + 1 if α ≡ 0 (resp. = ω ∗ | V O 1 / x m if α �≡ 0). 1 := ω ∗ | V O ω | V O ˜ ω on Bl O ( C 2 ) . From ˜ ω | V O i we construct a 1-form ˜ From X , M , P we can obtain ˜ X in Bl P ( M ) . WAIFI A. Ferragut 11/38

  10. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Types of singular points O is a dicritical singular point if α ≡ 0. O is non-dicritical if and only if E O is invariant. � p 1 x � p 1 y O is simple if m = 1 and has EV λ 1 , λ 2 s.t. q 1 x q 1 y λ 1 = 0 � = λ 2 or λ 1 λ 2 �∈ Q + . O is ordinary if it is not simple (includes dicritical). Some observations Simple singular points cannot be reduced. Ordinary singular points can be reduced (by a finite sequence of BU) s.t. the strict transform X in the last obtained complex manifold has no ordinary singularities. WAIFI A. Ferragut 12/38

  11. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Types of singular points O is a dicritical singular point if α ≡ 0. O is non-dicritical if and only if E O is invariant. � p 1 x � p 1 y O is simple if m = 1 and has EV λ 1 , λ 2 s.t. q 1 x q 1 y λ 1 = 0 � = λ 2 or λ 1 λ 2 �∈ Q + . O is ordinary if it is not simple (includes dicritical). Some observations Simple singular points cannot be reduced. Ordinary singular points can be reduced (by a finite sequence of BU) s.t. the strict transform X in the last obtained complex manifold has no ordinary singularities. WAIFI A. Ferragut 12/38

  12. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Types of singular points O is a dicritical singular point if α ≡ 0. O is non-dicritical if and only if E O is invariant. � p 1 x � p 1 y O is simple if m = 1 and has EV λ 1 , λ 2 s.t. q 1 x q 1 y λ 1 = 0 � = λ 2 or λ 1 λ 2 �∈ Q + . O is ordinary if it is not simple (includes dicritical). Some observations Simple singular points cannot be reduced. Ordinary singular points can be reduced (by a finite sequence of BU) s.t. the strict transform X in the last obtained complex manifold has no ordinary singularities. WAIFI A. Ferragut 12/38

  13. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Types of singular points O is a dicritical singular point if α ≡ 0. O is non-dicritical if and only if E O is invariant. � p 1 x � p 1 y O is simple if m = 1 and has EV λ 1 , λ 2 s.t. q 1 x q 1 y λ 1 = 0 � = λ 2 or λ 1 λ 2 �∈ Q + . O is ordinary if it is not simple (includes dicritical). Some observations Simple singular points cannot be reduced. Ordinary singular points can be reduced (by a finite sequence of BU) s.t. the strict transform X in the last obtained complex manifold has no ordinary singularities. WAIFI A. Ferragut 12/38

  14. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Types of singular points O is a dicritical singular point if α ≡ 0. O is non-dicritical if and only if E O is invariant. � p 1 x � p 1 y O is simple if m = 1 and has EV λ 1 , λ 2 s.t. q 1 x q 1 y λ 1 = 0 � = λ 2 or λ 1 λ 2 �∈ Q + . O is ordinary if it is not simple (includes dicritical). Some observations Simple singular points cannot be reduced. Ordinary singular points can be reduced (by a finite sequence of BU) s.t. the strict transform X in the last obtained complex manifold has no ordinary singularities. WAIFI A. Ferragut 12/38

  15. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Types of singular points O is a dicritical singular point if α ≡ 0. O is non-dicritical if and only if E O is invariant. � p 1 x � p 1 y O is simple if m = 1 and has EV λ 1 , λ 2 s.t. q 1 x q 1 y λ 1 = 0 � = λ 2 or λ 1 λ 2 �∈ Q + . O is ordinary if it is not simple (includes dicritical). Some observations Simple singular points cannot be reduced. Ordinary singular points can be reduced (by a finite sequence of BU) s.t. the strict transform X in the last obtained complex manifold has no ordinary singularities. WAIFI A. Ferragut 12/38

  16. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Types of singular points O is a dicritical singular point if α ≡ 0. O is non-dicritical if and only if E O is invariant. � p 1 x � p 1 y O is simple if m = 1 and has EV λ 1 , λ 2 s.t. q 1 x q 1 y λ 1 = 0 � = λ 2 or λ 1 λ 2 �∈ Q + . O is ordinary if it is not simple (includes dicritical). Some observations Simple singular points cannot be reduced. Ordinary singular points can be reduced (by a finite sequence of BU) s.t. the strict transform X in the last obtained complex manifold has no ordinary singularities. WAIFI A. Ferragut 12/38

  17. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Types of singular points O is a dicritical singular point if α ≡ 0. O is non-dicritical if and only if E O is invariant. � p 1 x � p 1 y O is simple if m = 1 and has EV λ 1 , λ 2 s.t. q 1 x q 1 y λ 1 = 0 � = λ 2 or λ 1 λ 2 �∈ Q + . O is ordinary if it is not simple (includes dicritical). Some observations Simple singular points cannot be reduced. Ordinary singular points can be reduced (by a finite sequence of BU) s.t. the strict transform X in the last obtained complex manifold has no ordinary singularities. WAIFI A. Ferragut 12/38

  18. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Neighbors E P is the first infinitesimal neighborhood of P . The i -th infinitesimal neighborhood of P is formed by the points on the first infinitesimal neighborhood of some point in the ( i − 1 ) -th infinitesimal neighborhood of P . They are infinitely near to P . Q is proximate to P if it belongs to the strict transform of E P . Q is a satellite if it is proximate to two points. Otherwise it is free. R precedes Q if Q is infinitely near to R . WAIFI A. Ferragut 13/38

  19. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Neighbors E P is the first infinitesimal neighborhood of P . The i -th infinitesimal neighborhood of P is formed by the points on the first infinitesimal neighborhood of some point in the ( i − 1 ) -th infinitesimal neighborhood of P . They are infinitely near to P . Q is proximate to P if it belongs to the strict transform of E P . Q is a satellite if it is proximate to two points. Otherwise it is free. R precedes Q if Q is infinitely near to R . WAIFI A. Ferragut 13/38

  20. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Neighbors E P is the first infinitesimal neighborhood of P . The i -th infinitesimal neighborhood of P is formed by the points on the first infinitesimal neighborhood of some point in the ( i − 1 ) -th infinitesimal neighborhood of P . They are infinitely near to P . Q is proximate to P if it belongs to the strict transform of E P . Q is a satellite if it is proximate to two points. Otherwise it is free. R precedes Q if Q is infinitely near to R . WAIFI A. Ferragut 13/38

  21. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Neighbors E P is the first infinitesimal neighborhood of P . The i -th infinitesimal neighborhood of P is formed by the points on the first infinitesimal neighborhood of some point in the ( i − 1 ) -th infinitesimal neighborhood of P . They are infinitely near to P . Q is proximate to P if it belongs to the strict transform of E P . Q is a satellite if it is proximate to two points. Otherwise it is free. R precedes Q if Q is infinitely near to R . WAIFI A. Ferragut 13/38

  22. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Neighbors E P is the first infinitesimal neighborhood of P . The i -th infinitesimal neighborhood of P is formed by the points on the first infinitesimal neighborhood of some point in the ( i − 1 ) -th infinitesimal neighborhood of P . They are infinitely near to P . Q is proximate to P if it belongs to the strict transform of E P . Q is a satellite if it is proximate to two points. Otherwise it is free. R precedes Q if Q is infinitely near to R . WAIFI A. Ferragut 13/38

  23. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Neighbors E P is the first infinitesimal neighborhood of P . The i -th infinitesimal neighborhood of P is formed by the points on the first infinitesimal neighborhood of some point in the ( i − 1 ) -th infinitesimal neighborhood of P . They are infinitely near to P . Q is proximate to P if it belongs to the strict transform of E P . Q is a satellite if it is proximate to two points. Otherwise it is free. R precedes Q if Q is infinitely near to R . WAIFI A. Ferragut 13/38

  24. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Configurations of infinitely near points A configuration is C = { Q 0 , . . . , Q n } such that Q 0 ∈ X 0 = M , Q i ∈ Bl Q i − 1 ( X i − 1 ) =: X i → X i − 1 . We can construct the proximity graph Γ C . The singular configuration S ( X ) = � P S P ( X ) , P ordinary. The dicritical configuration D ( X ) = { P ∈ S ( X ) : ∃ Q ∈ S ( X ) infinitely near dicritical singularity } . WAIFI A. Ferragut 14/38

  25. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Configurations of infinitely near points A configuration is C = { Q 0 , . . . , Q n } such that Q 0 ∈ X 0 = M , Q i ∈ Bl Q i − 1 ( X i − 1 ) =: X i → X i − 1 . We can construct the proximity graph Γ C . The singular configuration S ( X ) = � P S P ( X ) , P ordinary. The dicritical configuration D ( X ) = { P ∈ S ( X ) : ∃ Q ∈ S ( X ) infinitely near dicritical singularity } . WAIFI A. Ferragut 14/38

  26. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Configurations of infinitely near points A configuration is C = { Q 0 , . . . , Q n } such that Q 0 ∈ X 0 = M , Q i ∈ Bl Q i − 1 ( X i − 1 ) =: X i → X i − 1 . We can construct the proximity graph Γ C . The singular configuration S ( X ) = � P S P ( X ) , P ordinary. The dicritical configuration D ( X ) = { P ∈ S ( X ) : ∃ Q ∈ S ( X ) infinitely near dicritical singularity } . WAIFI A. Ferragut 14/38

  27. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Configurations of infinitely near points A configuration is C = { Q 0 , . . . , Q n } such that Q 0 ∈ X 0 = M , Q i ∈ Bl Q i − 1 ( X i − 1 ) =: X i → X i − 1 . We can construct the proximity graph Γ C . The singular configuration S ( X ) = � P S P ( X ) , P ordinary. The dicritical configuration D ( X ) = { P ∈ S ( X ) : ∃ Q ∈ S ( X ) infinitely near dicritical singularity } . WAIFI A. Ferragut 14/38

  28. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Reduction of singularities Configurations of infinitely near points A configuration is C = { Q 0 , . . . , Q n } such that Q 0 ∈ X 0 = M , Q i ∈ Bl Q i − 1 ( X i − 1 ) =: X i → X i − 1 . We can construct the proximity graph Γ C . The singular configuration S ( X ) = � P S P ( X ) , P ordinary. The dicritical configuration D ( X ) = { P ∈ S ( X ) : ∃ Q ∈ S ( X ) infinitely near dicritical singularity } . WAIFI A. Ferragut 14/38

  29. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI An example! Example Let X be the vector field 2 XZ 4 dX + 5 Y 4 Z dY − ( 5 Y 5 + 2 X 2 Z 3 ) dZ , � P = ( 1 : 0 : 0 ) , with singularities Q = ( 0 : 0 : 1 ) . We have S ( X ) = { P , Q } ∪ { P i } 13 i = 1 ∪ { Q i } 3 i = 1 , D ( X ) = { P } ∪ { P i } 13 i = 1 . WAIFI A. Ferragut 15/38

  30. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Contents Introduction and objectives 1 Polynomial vector fields in CP 2 2 Reduction of singularities 3 Linear systems. Clusters 4 Results and algorithms 5 6 WAI Positive Darboux first integrals WAIFI A. Ferragut 16/38

  31. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Linear systems A linear system on CP 2 is the set of algebraic curves given by a linear subspace of C m [ X , Y , Z ] ∪ { 0 } . If it has dimension 1, then it is a pencil. A cluster of CP 2 is ( C , m ) where C = ( Q 0 , . . . , Q n ) is a configuration and m = ( m 0 , . . . , m n ) , m i ∈ N . WAIFI A. Ferragut 17/38

  32. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Linear systems A linear system on CP 2 is the set of algebraic curves given by a linear subspace of C m [ X , Y , Z ] ∪ { 0 } . If it has dimension 1, then it is a pencil. A cluster of CP 2 is ( C , m ) where C = ( Q 0 , . . . , Q n ) is a configuration and m = ( m 0 , . . . , m n ) , m i ∈ N . WAIFI A. Ferragut 17/38

  33. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Linear systems A linear system on CP 2 is the set of algebraic curves given by a linear subspace of C m [ X , Y , Z ] ∪ { 0 } . If it has dimension 1, then it is a pencil. A cluster of CP 2 is ( C , m ) where C = ( Q 0 , . . . , Q n ) is a configuration and m = ( m 0 , . . . , m n ) , m i ∈ N . WAIFI A. Ferragut 17/38

  34. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Linear systems constructed from clusters Virtual transform Set K = ( C , m ) a cluster and C : { f = 0 } an algebraic curve. If Q k ∈ C , let ℓ ( Q k ) = # { Q j ∈ C| Q k is infinitely near to Q j } . Case ℓ ( Q k ) = 1: the virtual transform C K Q k is f ( x , y ) = 0. C passes virtually through Q k if m Q k ( C K Q k ) ≥ m k . Case ℓ ( Q k ) > 1: Q k in the 1IN of Q j ∈ C and C passes virtually through Q j . Q j f ( x , y ) = 0 a local equation of C K Q j ; Q k = ( 0 , λ ) ∈ V 1 . The virtual transform C K Q k at Q k : x − m j f ( x , x ( t + λ )) = 0. C passes virtually through Q k if m Q k ( C K Q k ) ≥ m k . C passes virtually through K if it passes virtually through all Q i ∈ K . WAIFI A. Ferragut 18/38

  35. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Linear systems constructed from clusters Virtual transform Set K = ( C , m ) a cluster and C : { f = 0 } an algebraic curve. If Q k ∈ C , let ℓ ( Q k ) = # { Q j ∈ C| Q k is infinitely near to Q j } . Case ℓ ( Q k ) = 1: the virtual transform C K Q k is f ( x , y ) = 0. C passes virtually through Q k if m Q k ( C K Q k ) ≥ m k . Case ℓ ( Q k ) > 1: Q k in the 1IN of Q j ∈ C and C passes virtually through Q j . Q j f ( x , y ) = 0 a local equation of C K Q j ; Q k = ( 0 , λ ) ∈ V 1 . The virtual transform C K Q k at Q k : x − m j f ( x , x ( t + λ )) = 0. C passes virtually through Q k if m Q k ( C K Q k ) ≥ m k . C passes virtually through K if it passes virtually through all Q i ∈ K . WAIFI A. Ferragut 18/38

  36. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Linear systems constructed from clusters Virtual transform Set K = ( C , m ) a cluster and C : { f = 0 } an algebraic curve. If Q k ∈ C , let ℓ ( Q k ) = # { Q j ∈ C| Q k is infinitely near to Q j } . Case ℓ ( Q k ) = 1: the virtual transform C K Q k is f ( x , y ) = 0. C passes virtually through Q k if m Q k ( C K Q k ) ≥ m k . Case ℓ ( Q k ) > 1: Q k in the 1IN of Q j ∈ C and C passes virtually through Q j . Q j f ( x , y ) = 0 a local equation of C K Q j ; Q k = ( 0 , λ ) ∈ V 1 . The virtual transform C K Q k at Q k : x − m j f ( x , x ( t + λ )) = 0. C passes virtually through Q k if m Q k ( C K Q k ) ≥ m k . C passes virtually through K if it passes virtually through all Q i ∈ K . WAIFI A. Ferragut 18/38

  37. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Linear systems constructed from clusters Virtual transform Set K = ( C , m ) a cluster and C : { f = 0 } an algebraic curve. If Q k ∈ C , let ℓ ( Q k ) = # { Q j ∈ C| Q k is infinitely near to Q j } . Case ℓ ( Q k ) = 1: the virtual transform C K Q k is f ( x , y ) = 0. C passes virtually through Q k if m Q k ( C K Q k ) ≥ m k . Case ℓ ( Q k ) > 1: Q k in the 1IN of Q j ∈ C and C passes virtually through Q j . Q j f ( x , y ) = 0 a local equation of C K Q j ; Q k = ( 0 , λ ) ∈ V 1 . The virtual transform C K Q k at Q k : x − m j f ( x , x ( t + λ )) = 0. C passes virtually through Q k if m Q k ( C K Q k ) ≥ m k . C passes virtually through K if it passes virtually through all Q i ∈ K . WAIFI A. Ferragut 18/38

  38. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Linear systems constructed from clusters Virtual transform Set K = ( C , m ) a cluster and C : { f = 0 } an algebraic curve. If Q k ∈ C , let ℓ ( Q k ) = # { Q j ∈ C| Q k is infinitely near to Q j } . Case ℓ ( Q k ) = 1: the virtual transform C K Q k is f ( x , y ) = 0. C passes virtually through Q k if m Q k ( C K Q k ) ≥ m k . Case ℓ ( Q k ) > 1: Q k in the 1IN of Q j ∈ C and C passes virtually through Q j . Q j f ( x , y ) = 0 a local equation of C K Q j ; Q k = ( 0 , λ ) ∈ V 1 . The virtual transform C K Q k at Q k : x − m j f ( x , x ( t + λ )) = 0. C passes virtually through Q k if m Q k ( C K Q k ) ≥ m k . C passes virtually through K if it passes virtually through all Q i ∈ K . WAIFI A. Ferragut 18/38

  39. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Linear systems constructed from clusters Virtual transform Set K = ( C , m ) a cluster and C : { f = 0 } an algebraic curve. If Q k ∈ C , let ℓ ( Q k ) = # { Q j ∈ C| Q k is infinitely near to Q j } . Case ℓ ( Q k ) = 1: the virtual transform C K Q k is f ( x , y ) = 0. C passes virtually through Q k if m Q k ( C K Q k ) ≥ m k . Case ℓ ( Q k ) > 1: Q k in the 1IN of Q j ∈ C and C passes virtually through Q j . Q j f ( x , y ) = 0 a local equation of C K Q j ; Q k = ( 0 , λ ) ∈ V 1 . The virtual transform C K Q k at Q k : x − m j f ( x , x ( t + λ )) = 0. C passes virtually through Q k if m Q k ( C K Q k ) ≥ m k . C passes virtually through K if it passes virtually through all Q i ∈ K . WAIFI A. Ferragut 18/38

  40. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Linear systems constructed from clusters Virtual transform Set K = ( C , m ) a cluster and C : { f = 0 } an algebraic curve. If Q k ∈ C , let ℓ ( Q k ) = # { Q j ∈ C| Q k is infinitely near to Q j } . Case ℓ ( Q k ) = 1: the virtual transform C K Q k is f ( x , y ) = 0. C passes virtually through Q k if m Q k ( C K Q k ) ≥ m k . Case ℓ ( Q k ) > 1: Q k in the 1IN of Q j ∈ C and C passes virtually through Q j . Q j f ( x , y ) = 0 a local equation of C K Q j ; Q k = ( 0 , λ ) ∈ V 1 . The virtual transform C K Q k at Q k : x − m j f ( x , x ( t + λ )) = 0. C passes virtually through Q k if m Q k ( C K Q k ) ≥ m k . C passes virtually through K if it passes virtually through all Q i ∈ K . WAIFI A. Ferragut 18/38

  41. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Linear systems constructed from clusters Virtual transform Set K = ( C , m ) a cluster and C : { f = 0 } an algebraic curve. If Q k ∈ C , let ℓ ( Q k ) = # { Q j ∈ C| Q k is infinitely near to Q j } . Case ℓ ( Q k ) = 1: the virtual transform C K Q k is f ( x , y ) = 0. C passes virtually through Q k if m Q k ( C K Q k ) ≥ m k . Case ℓ ( Q k ) > 1: Q k in the 1IN of Q j ∈ C and C passes virtually through Q j . Q j f ( x , y ) = 0 a local equation of C K Q j ; Q k = ( 0 , λ ) ∈ V 1 . The virtual transform C K Q k at Q k : x − m j f ( x , x ( t + λ )) = 0. C passes virtually through Q k if m Q k ( C K Q k ) ≥ m k . C passes virtually through K if it passes virtually through all Q i ∈ K . WAIFI A. Ferragut 18/38

  42. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Linear systems constructed from clusters Virtual transform Set K = ( C , m ) a cluster and C : { f = 0 } an algebraic curve. If Q k ∈ C , let ℓ ( Q k ) = # { Q j ∈ C| Q k is infinitely near to Q j } . Case ℓ ( Q k ) = 1: the virtual transform C K Q k is f ( x , y ) = 0. C passes virtually through Q k if m Q k ( C K Q k ) ≥ m k . Case ℓ ( Q k ) > 1: Q k in the 1IN of Q j ∈ C and C passes virtually through Q j . Q j f ( x , y ) = 0 a local equation of C K Q j ; Q k = ( 0 , λ ) ∈ V 1 . The virtual transform C K Q k at Q k : x − m j f ( x , x ( t + λ )) = 0. C passes virtually through Q k if m Q k ( C K Q k ) ≥ m k . C passes virtually through K if it passes virtually through all Q i ∈ K . WAIFI A. Ferragut 18/38

  43. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Linear systems constructed from clusters The strict transform ˜ C of C is the global curve given by the virtual transform through the cluster of points and multiplicities defined by the curve. The linear system L m ( K ) determined by m ∈ N and K is the linear system on CP 2 given by those curves defined by polynomials in C m [ X , Y , Z ] ∪ { 0 } that pass virtually through K . WAIFI A. Ferragut 19/38

  44. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI An example Example Consider the cluster K = ( C , m ) , where C = { Q , P , P 1 , P 2 } , m = ( 2 , 2 , 1 , 1 ) ; P = ( 0 : 0 : 1 ) , Q = ( 1 : 0 : 1 ) ; or ( 0 , 0 ) , ( 1 , 0 ) in Z � = 0. 1 , P 2 = ( 1 , 0 ) ∈ V P 1 P 1 = ( 0 , 3 ) ∈ V P infinitely near to P . 2 Let us compute L 3 ( K ) . WAIFI A. Ferragut 20/38

  45. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI An example Example Let C ∈ L 3 ( K ) be aX 3 + bX 2 Y + cX 2 Z + dXY 2 + eXYZ + fXZ 2 + gY 3 + hY 2 Z + iYZ 2 + kZ 3 , Consider it in the local chart Z � = 0. The multiplicity of C at P must be at least 2, then f = i = k = 0. The multiplicity of C at Q must be at least 2, so a = c = 0 and b = − e . WAIFI A. Ferragut 21/38

  46. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI An example Example The local equation defining the virtual transform of C at P 1 , C K P 1 , is 3 ( e + 3 h ) + ( 9 d − 3 e + 27 g ) x 1 + ( e + 6 h ) y 1 + ( 6 d − e + 27 g ) x 1 y 1 + hy 2 1 + ( d + 9 g ) x 1 y 2 1 + gx 1 y 3 1 = 0 in coordinates ( x 1 = x , y 1 = y / x ) . The multiplicity of C K P 1 at P 1 must be at least 1, then e = − 3 h . WAIFI A. Ferragut 22/38

  47. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI An example Example The local equation of the virtual transform of C at P 2 is 3 h + ( 9 d + 27 g + 9 h ) x 2 + hy 2 + ( 6 d + 27 g + 3 h ) x 2 y 2 + ( d + 9 g ) x 2 y 2 2 + gx 2 y 3 2 = 0 , where x 2 = x 1 / y 1 and y 2 = y 1 . C K P 2 passes virtually through P 2 if and only if h = 0. Hence the curves in L 3 ( K ) are defined by Y 2 ( α X + β Y ) = 0, for ( α, β ) ∈ C 2 \ { ( 0 , 0 ) } . WAIFI A. Ferragut 23/38

  48. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Cluster of base points Let BP ( L ) be the configuration of points such that all the generic curves of L have the same multiplicities mult Q ( L ) at every point Q ∈ BP ( L ) and empty intersection at the manifold obtained by blowing-up these points. Let m = ( mult Q ( L )) Q ∈BP ( L ) . We have the cluster of base points ( BP ( L ) , m ) . WAIFI A. Ferragut 24/38

  49. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI An example Example Back to 2 XZ 4 dX + 5 Y 4 Z dY − ( 5 Y 5 + 2 X 2 Z 3 ) dZ , consider L defined by α ( X 2 Z 3 + Y 5 ) + β Z 5 = 0 . The cluster of base points of L is � � D ( X ) , ( 3 , 2 , 1 , . . . , 1 ) . WAIFI A. Ferragut 25/38

  50. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Cluster of base points Proposition If L is a pencil, then BP ( L ) = D ( X L ) , where X L is the vector field with invariant curves given by L . Let P X = P � F n 1 1 · · · F n r r , Z n � ( ⇔ ¯ H ). We have P X = L n ( BP X ) . We can compute H from BP X and n . WAIFI A. Ferragut 26/38

  51. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Cluster of base points Proposition If L is a pencil, then BP ( L ) = D ( X L ) , where X L is the vector field with invariant curves given by L . Let P X = P � F n 1 1 · · · F n r r , Z n � ( ⇔ ¯ H ). We have P X = L n ( BP X ) . We can compute H from BP X and n . WAIFI A. Ferragut 26/38

  52. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Cluster of base points Proposition If L is a pencil, then BP ( L ) = D ( X L ) , where X L is the vector field with invariant curves given by L . Let P X = P � F n 1 1 · · · F n r r , Z n � ( ⇔ ¯ H ). We have P X = L n ( BP X ) . We can compute H from BP X and n . WAIFI A. Ferragut 26/38

  53. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Cluster of base points Proposition If L is a pencil, then BP ( L ) = D ( X L ) , where X L is the vector field with invariant curves given by L . Let P X = P � F n 1 1 · · · F n r r , Z n � ( ⇔ ¯ H ). We have P X = L n ( BP X ) . We can compute H from BP X and n . WAIFI A. Ferragut 26/38

  54. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Cluster of base points Proposition If L is a pencil, then BP ( L ) = D ( X L ) , where X L is the vector field with invariant curves given by L . Let P X = P � F n 1 1 · · · F n r r , Z n � ( ⇔ ¯ H ). We have P X = L n ( BP X ) . We can compute H from BP X and n . WAIFI A. Ferragut 26/38

  55. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Contents Introduction and objectives 1 Polynomial vector fields in CP 2 2 Reduction of singularities 3 Linear systems. Clusters 4 Results and algorithms 5 6 WAI Positive Darboux first integrals WAIFI A. Ferragut 27/38

  56. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI First main result Theorem i = 1 f n i Consider X having a WAI PFI H = � r i . D ( X ) = BP ( P X ) . D ( X ) has exactly r maximal points R i . They are the unique dicritical singularities of X . The set Fr ( D ( X )) of free points of D ( X ) has exactly r maximal elements M i . Moreover, R i is infinitely near to M i . The degree of F i can be obtained from: M i and the points of D ( X ) to which M i is infinitely near. A convenient set of multiplicities. WAIFI A. Ferragut 28/38

  57. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI First main result Theorem i = 1 f n i Consider X having a WAI PFI H = � r i . D ( X ) = BP ( P X ) . D ( X ) has exactly r maximal points R i . They are the unique dicritical singularities of X . The set Fr ( D ( X )) of free points of D ( X ) has exactly r maximal elements M i . Moreover, R i is infinitely near to M i . The degree of F i can be obtained from: M i and the points of D ( X ) to which M i is infinitely near. A convenient set of multiplicities. WAIFI A. Ferragut 28/38

  58. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI First main result Theorem i = 1 f n i Consider X having a WAI PFI H = � r i . D ( X ) = BP ( P X ) . D ( X ) has exactly r maximal points R i . They are the unique dicritical singularities of X . The set Fr ( D ( X )) of free points of D ( X ) has exactly r maximal elements M i . Moreover, R i is infinitely near to M i . The degree of F i can be obtained from: M i and the points of D ( X ) to which M i is infinitely near. A convenient set of multiplicities. WAIFI A. Ferragut 28/38

  59. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI First main result Theorem i = 1 f n i Consider X having a WAI PFI H = � r i . D ( X ) = BP ( P X ) . D ( X ) has exactly r maximal points R i . They are the unique dicritical singularities of X . The set Fr ( D ( X )) of free points of D ( X ) has exactly r maximal elements M i . Moreover, R i is infinitely near to M i . The degree of F i can be obtained from: M i and the points of D ( X ) to which M i is infinitely near. A convenient set of multiplicities. WAIFI A. Ferragut 28/38

  60. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI First main result Theorem i = 1 f n i Consider X having a WAI PFI H = � r i . D ( X ) = BP ( P X ) . D ( X ) has exactly r maximal points R i . They are the unique dicritical singularities of X . The set Fr ( D ( X )) of free points of D ( X ) has exactly r maximal elements M i . Moreover, R i is infinitely near to M i . The degree of F i can be obtained from: M i and the points of D ( X ) to which M i is infinitely near. A convenient set of multiplicities. WAIFI A. Ferragut 28/38

  61. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI First main result Theorem i = 1 f n i Consider X having a WAI PFI H = � r i . D ( X ) = BP ( P X ) . D ( X ) has exactly r maximal points R i . They are the unique dicritical singularities of X . The set Fr ( D ( X )) of free points of D ( X ) has exactly r maximal elements M i . Moreover, R i is infinitely near to M i . The degree of F i can be obtained from: M i and the points of D ( X ) to which M i is infinitely near. A convenient set of multiplicities. WAIFI A. Ferragut 28/38

  62. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI First main result Theorem i = 1 f n i Consider X having a WAI PFI H = � r i . D ( X ) = BP ( P X ) . D ( X ) has exactly r maximal points R i . They are the unique dicritical singularities of X . The set Fr ( D ( X )) of free points of D ( X ) has exactly r maximal elements M i . Moreover, R i is infinitely near to M i . The degree of F i can be obtained from: M i and the points of D ( X ) to which M i is infinitely near. A convenient set of multiplicities. WAIFI A. Ferragut 28/38

  63. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Second main result Theorem L is invariant and contains D ( X ) ∩ P 2 . R i are the unique IN dicritical singularities of X . MFr ( D ( X )) = { M 1 , . . . , M r } . For each i there exists C i associated to M i of degree d i computable. After some computations (skipped), n i ∈ N are obtained. If C i : { f i ( x , y ) = 0 } then � r i = 1 f n i is a WAI PFI. i WAIFI A. Ferragut 29/38

  64. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Second main result Theorem L is invariant and contains D ( X ) ∩ P 2 . R i are the unique IN dicritical singularities of X . MFr ( D ( X )) = { M 1 , . . . , M r } . For each i there exists C i associated to M i of degree d i computable. After some computations (skipped), n i ∈ N are obtained. If C i : { f i ( x , y ) = 0 } then � r i = 1 f n i is a WAI PFI. i WAIFI A. Ferragut 29/38

  65. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Second main result Theorem L is invariant and contains D ( X ) ∩ P 2 . R i are the unique IN dicritical singularities of X . MFr ( D ( X )) = { M 1 , . . . , M r } . For each i there exists C i associated to M i of degree d i computable. After some computations (skipped), n i ∈ N are obtained. If C i : { f i ( x , y ) = 0 } then � r i = 1 f n i is a WAI PFI. i WAIFI A. Ferragut 29/38

  66. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Second main result Theorem L is invariant and contains D ( X ) ∩ P 2 . R i are the unique IN dicritical singularities of X . MFr ( D ( X )) = { M 1 , . . . , M r } . For each i there exists C i associated to M i of degree d i computable. After some computations (skipped), n i ∈ N are obtained. If C i : { f i ( x , y ) = 0 } then � r i = 1 f n i is a WAI PFI. i WAIFI A. Ferragut 29/38

  67. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Second main result Theorem L is invariant and contains D ( X ) ∩ P 2 . R i are the unique IN dicritical singularities of X . MFr ( D ( X )) = { M 1 , . . . , M r } . For each i there exists C i associated to M i of degree d i computable. After some computations (skipped), n i ∈ N are obtained. If C i : { f i ( x , y ) = 0 } then � r i = 1 f n i is a WAI PFI. i WAIFI A. Ferragut 29/38

  68. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Second main result Theorem L is invariant and contains D ( X ) ∩ P 2 . R i are the unique IN dicritical singularities of X . MFr ( D ( X )) = { M 1 , . . . , M r } . For each i there exists C i associated to M i of degree d i computable. After some computations (skipped), n i ∈ N are obtained. If C i : { f i ( x , y ) = 0 } then � r i = 1 f n i is a WAI PFI. i WAIFI A. Ferragut 29/38

  69. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Second main result Theorem L is invariant and contains D ( X ) ∩ P 2 . R i are the unique IN dicritical singularities of X . MFr ( D ( X )) = { M 1 , . . . , M r } . For each i there exists C i associated to M i of degree d i computable. After some computations (skipped), n i ∈ N are obtained. If C i : { f i ( x , y ) = 0 } then � r i = 1 f n i is a WAI PFI. i WAIFI A. Ferragut 29/38

  70. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Second main result Corollary n and n i can be computed from the proximity graph of D ( X ) and the points in D ( X ) through which the strict transform of the infinity line passes. The proximity graph of D ( X ) determines a bound for the degree of the (minimal) WAI polynomial first integral. WAIFI A. Ferragut 30/38

  71. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Second main result Corollary n and n i can be computed from the proximity graph of D ( X ) and the points in D ( X ) through which the strict transform of the infinity line passes. The proximity graph of D ( X ) determines a bound for the degree of the (minimal) WAI polynomial first integral. WAIFI A. Ferragut 30/38

  72. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI Second main result Corollary n and n i can be computed from the proximity graph of D ( X ) and the points in D ( X ) through which the strict transform of the infinity line passes. The proximity graph of D ( X ) determines a bound for the degree of the (minimal) WAI polynomial first integral. WAIFI A. Ferragut 30/38

  73. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI The algorithm Compute D ( X ) . 1 Let r be the number of maximal points of D ( X ) . It must 2 happen # Fr ( D ( X )) = r . Compute f i = 0 for the unique curve C i defined by the 3 Theorem. Compute n i . 4 Check whether � r i = 1 f n i is a first integral of X . 5 i WAIFI A. Ferragut 31/38

  74. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI The algorithm Compute D ( X ) . 1 Let r be the number of maximal points of D ( X ) . It must 2 happen # Fr ( D ( X )) = r . Compute f i = 0 for the unique curve C i defined by the 3 Theorem. Compute n i . 4 Check whether � r i = 1 f n i is a first integral of X . 5 i WAIFI A. Ferragut 31/38

  75. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI The algorithm Compute D ( X ) . 1 Let r be the number of maximal points of D ( X ) . It must 2 happen # Fr ( D ( X )) = r . Compute f i = 0 for the unique curve C i defined by the 3 Theorem. Compute n i . 4 Check whether � r i = 1 f n i is a first integral of X . 5 i WAIFI A. Ferragut 31/38

  76. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI The algorithm Compute D ( X ) . 1 Let r be the number of maximal points of D ( X ) . It must 2 happen # Fr ( D ( X )) = r . Compute f i = 0 for the unique curve C i defined by the 3 Theorem. Compute n i . 4 Check whether � r i = 1 f n i is a first integral of X . 5 i WAIFI A. Ferragut 31/38

  77. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI The algorithm Compute D ( X ) . 1 Let r be the number of maximal points of D ( X ) . It must 2 happen # Fr ( D ( X )) = r . Compute f i = 0 for the unique curve C i defined by the 3 Theorem. Compute n i . 4 Check whether � r i = 1 f n i is a first integral of X . 5 i WAIFI A. Ferragut 31/38

  78. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI The algorithm Compute D ( X ) . 1 Let r be the number of maximal points of D ( X ) . It must 2 happen # Fr ( D ( X )) = r . Compute f i = 0 for the unique curve C i defined by the 3 Theorem. Compute n i . 4 Check whether � r i = 1 f n i is a first integral of X . 5 i WAIFI A. Ferragut 31/38

  79. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI An example Example ( 10 x 7 − 9 x 6 + 6 x 5 y + 9 x 4 y − 6 x 3 y + 6 x 2 y 2 + 2 xy 2 ) dx +( 2 x 6 − x 4 + 6 x 3 y − x 2 y + 4 y 2 ) dy . We have D ( X ) = { P i } 28 i = 0 . r = 3, R 1 = M 1 = P 13 , R 2 = M 2 = P 23 , R 3 = M 3 = P 28 . Figure: Γ D ( X ) . WAIFI A. Ferragut 32/38

  80. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI An example 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   3 2 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0   2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1     0 − 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 − 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0     0 0 0 − 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0    0 0 0 0 − 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 − 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 − 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 − 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 − 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 − 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0    0 0 0 0 0 0 0 0 0 0 − 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 − 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 − 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 1 0 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 1 0 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 1 0 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 1 0 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 1 0 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 1 0 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 1 0 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 1 0 0 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 1 0 0 0     0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 1 0 0    0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 1 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 1 WAIFI A. Ferragut 33/38

  81. PVF in CP 2 Intro Blow-up. Singularities Linear systems. Clusters Results. Algorithms WAI DFI An example Example After some technical stuff we compute R = ( 10 ; 6 , 4 , 2 , 2 , 1 , . . . , 1 , 2 , 2 , 2 , 2 , 2 ) . After this we know that n = 10. From the three first rows we can compute the three curves X 3 − X 2 Z + YZ 2 = 0, X 3 + YZ 2 = 0, X 2 + YZ = 0. Moreover, R = c 1 + c 2 + 2 c 3 , where c i is the i -th row of the matrix. H = ( y − x 2 + x 3 )( y + x 3 )( x 2 + y ) 2 is a first integral of X . WAIFI A. Ferragut 34/38

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Well- -behaved objects behaved objects Well Improving your coding skills 1.0 Main concepts to

Well- -behaved objects behaved objects Well Improving your coding skills 1.0 Main concepts to

Well- -behaved objects behaved objects Well Improving your coding skills 1.0 Main concepts to be covered Main concepts to be covered Testing Debugging Test automation Writing for maintainability 10/11/2005 Lecture 6:

509 views • 48 slides
TRIPLE INTEGRALS MATH 200 GOALS Be able to set up and evaluate triple integrals using

TRIPLE INTEGRALS MATH 200 GOALS Be able to set up and evaluate triple integrals using

MATH 200 WEEK 9 - WEDNESDAY TRIPLE INTEGRALS MATH 200 GOALS Be able to set up and evaluate triple integrals using rectangular, cylindrical, and spherical coordinates MATH 200 TRIPLE INTEGRALS We integrate functions of three variables

359 views • 24 slides
Lecture 17 : Double Integrals 0/ 15 Some of you have not learned how to do double integrals. In

Lecture 17 : Double Integrals 0/ 15 Some of you have not learned how to do double integrals. In

Lecture 17 : Double Integrals 0/ 15 Some of you have not learned how to do double integrals. In this course you will need to do double integrals over rectangles and I will now explain how to do such calculations. Partial (Indefinite)

389 views • 16 slides
Polynomials in two variables and their trees at infinity Pierrette Cassou-Nogu` es, Daniel

Polynomials in two variables and their trees at infinity Pierrette Cassou-Nogu` es, Daniel

Polynomials in two variables and their trees at infinity Pierrette Cassou-Nogu` es, Daniel Daigle IMB (Universit e de Bordeaux ), Universit e dOttawa Warsaw, May 2018 Let f : C 2 C be a polynomial map. Let C 2 X be a

1.07k views • 41 slides
INFINITY ECM PLATFORM Unique multifunctional Cloud SW platform that every World Company needs.

INFINITY ECM PLATFORM Unique multifunctional Cloud SW platform that every World Company needs.

INFINITY ECM PLATFORM Unique multifunctional Cloud SW platform that every World Company needs. INFINITY ECM Short Info 1 INFINITY ECM is a modern multi-functional Enterprise Content Management platform for horizontal markets that includes a

204 views • 8 slides
... Enough with derivatives. Let us move on to integrals! 85 / 158 Integrals Recall the notion

... Enough with derivatives. Let us move on to integrals! 85 / 158 Integrals Recall the notion

... Enough with derivatives. Let us move on to integrals! 85 / 158 Integrals Recall the notion of integration in one variable... b n f ( x ) d x = lim f ( x i )( x i x i 1 ) n a i =1 We call this the Riemann integral of

371 views • 11 slides
Actual infinity, potential infinity, objectivity, and reverse mathematics Stephen G. Simpson

Actual infinity, potential infinity, objectivity, and reverse mathematics Stephen G. Simpson

Actual infinity, potential infinity, objectivity, and reverse mathematics Stephen G. Simpson Vanderbilt University Nashville, TN 37240, USA http://www.math.psu.edu/simpson/ sgslogic@gmail.com PhilMath Intersem University of Paris 7 Diderot

521 views • 9 slides
= P ( t ) f ( x , y ) ds ( , t ) line A set of line integrals form

= P ( t ) f ( x , y ) ds ( , t ) line A set of line integrals form

Moscow-Bavarian Joint Advanced Student School 2006 / Medical Imaging Principles of Computerized Tomographic Imaging and Cone-Beam Reconstruction Line Integrals Line integrals represent the integral of some parameter of the object along the

658 views • 16 slides
Double integrals First, let's recall Riemann sums for integrals with a single variable... 1 2

Double integrals First, let's recall Riemann sums for integrals with a single variable... 1 2

Double integrals First, let's recall Riemann sums for integrals with a single variable... 1 2 3 4 5 Some comments: (1) Note that we can estimate double integrals using double Riemann sums. While this is tedious by hand, computers do this

208 views • 19 slides
20. Line integrals Lets look more at line integrals. Lets suppose we want to compute the

20. Line integrals Lets look more at line integrals. Lets suppose we want to compute the

20. Line integrals Lets look more at line integrals. Lets suppose we want to compute the line integral of F = y + x around the curve C which is the sector of the unit circle whose angle is / 4, starting and ending at the

57 views • 4 slides
Logarithms Are Not Infinity: This Infinity Problem . . . A Rational Physics-Related In Reality,

Logarithms Are Not Infinity: This Infinity Problem . . . A Rational Physics-Related In Reality,

Physicists Use Intuition Can We Formalize . . . An Example of . . . Why Are Infinities . . . Logarithms Are Not Infinity: This Infinity Problem . . . A Rational Physics-Related In Reality, Infinities . . . What Should We Do Explanation of

270 views • 14 slides
Surfaces, surface area and surface integrals Our main objective here is the construction of surface

Surfaces, surface area and surface integrals Our main objective here is the construction of surface

Surfaces, surface area and surface integrals Our main objective here is the construction of surface integrals . Surface integrals are essential for stating in completely general terms the basic laws of electromagnetism. Also, the theorems of

472 views • 43 slides
Math 233 - November 16, 2009 Exam 3: 8.1-10.1 1. Curve integrals 2. Reverse Path 3. Curve

Math 233 - November 16, 2009 Exam 3: 8.1-10.1 1. Curve integrals 2. Reverse Path 3. Curve

Math 233 - November 16, 2009 Exam 3: 8.1-10.1 1. Curve integrals 2. Reverse Path 3. Curve integrals and potential functions 4. Dependence of path, special vector field 5. Double integrals (setting up, changing order or integration, etc)

148 views • 11 slides
Abelian Integrals and Categoricity Martin Bays April 2, 2012 Abelian integrals w dz where

Abelian Integrals and Categoricity Martin Bays April 2, 2012 Abelian integrals w dz where

Abelian Integrals and Categoricity Martin Bays April 2, 2012 Abelian integrals w dz where w acl ( C ( z )) . i.e. where is a meromorphic differential form on a Riemann surface C . e.g. dz dz = z 3 + az + b

661 views • 27 slides
Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of

Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of

Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length Olivier Bournez, Daniel Graa and Amaury Pouly July 13, 2015 Main result and consequences Theorem (Informal) PTIME = PIVP of

633 views • 47 slides
Indefinite Integrals Return to Table of Contents Slide 5 / 91 Indefinite Integrals do not have

Indefinite Integrals Return to Table of Contents Slide 5 / 91 Indefinite Integrals do not have

Slide 1 / 91 Slide 2 / 91 AP Calculus Differential Equations 2015-11-23 www.njctl.org Slide 3 / 91 Table of Contents Indefinite Integrals Slope Fields U-Substitution U-Substitution & Definite Integrals Differential Equations

509 views • 31 slides
Gram-Schmidt process with a non-standard inner product and its application to the approximate

Gram-Schmidt process with a non-standard inner product and its application to the approximate

Gram-Schmidt process with a non-standard inner product and its application to the approximate inverse preconditioning Miro Rozlo zn k joint work with Ji r Kopal, Alicja Smoktunowicz and Miroslav T uma Institute of Computer

610 views • 26 slides
rt t

rt t

rt t r rtr stt rtt ts r

742 views • 22 slides
Bridging the UV and the IR at the loop level ERC workshop on Effective Field Theories for

Bridging the UV and the IR at the loop level ERC workshop on Effective Field Theories for

c e r n eth zurich Adrin Carmona Supported by a Marie Skodowska-Curie Individual Fellowship MSCA-IF-EF-2014 September 13, 2016 Slide 1/26 Bridging the UV and the IR at the loop level ERC workshop on Effective Field Theories for Collider

434 views • 29 slides
Safety control with performance guarantees of cooperative systems using compositional abstractions

Safety control with performance guarantees of cooperative systems using compositional abstractions

Cooperative systems Centralized symbolic control Compositional approach Safety control with performance guarantees of cooperative systems using compositional abstractions Pierre-Jean Meyer Antoine Girard Emmanuel Witrant Universit e

849 views • 45 slides
Energy efficiency Motivation Doing nothing to save energy? Why at Ena-HPC then? September 11,

Energy efficiency Motivation Doing nothing to save energy? Why at Ena-HPC then? September 11,

Doing Nothing to Save Energy in Matrix Computations Enrique S. Quintana-Ort quintana@icc.uji.es eeClust Workshop September 11, 2012, Hamburg, Germany Energy efficiency Motivation Doing nothing to save energy? Why at Ena-HPC then?

809 views • 53 slides
Lepton universality in K decays JHEP02(2013)048 / arXiv:1211.3052 C edric Weiland in

Lepton universality in K decays JHEP02(2013)048 / arXiv:1211.3052 C edric Weiland in

Lepton universality in K decays JHEP02(2013)048 / arXiv:1211.3052 C edric Weiland in collaboration with A. Abada, D. Das, A.M. Teixeira and A. Vicente Laboratoire de Physique Th eorique dOrsay, Universit e Paris-Sud 11, France

642 views • 10 slides
Duality in generalized Ising models Franz J. Wegner Institut f ur Theoretische Physik

Duality in generalized Ising models Franz J. Wegner Institut f ur Theoretische Physik

Duality in generalized Ising models Franz J. Wegner Institut f ur Theoretische Physik Universit at Heidelberg August 21, 2014 Contribution to the summer school Topological aspects of condensed matter physics Les Houches August

425 views • 18 slides
A Mathematical Proof When referring to a proof in logic we usually mean: 1. A sequence of

A Mathematical Proof When referring to a proof in logic we usually mean: 1. A sequence of

A Mathematical Proof When referring to a proof in logic we usually mean: 1. A sequence of statements. 2. Based on axioms. 3. Each statement is derived via the derivation rules. Zero Knowledge Protocols 4. The proof is fixed, i.e, in any time,

642 views • 14 slides

More recommend