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JNCF 2019, CIRM, Marseill A New Lower Boun o th Hilber Number for Quarti System 1,2 1 2

  1. + Infinitesimal Hilbert’s 16th Problem 1 . 4 1 . 2 H ( x , y ) = ( x 2 − 0 . 9 ) 2 + ( y 2 − 1 . 1 ) 2 y 1 ⎧ x = 4 y ( y 2 − 1 . 1 ) 0 . 8 ⎪ ˙ ⎪ ⎨ y = 4 x ( x 2 − 0 . 9 ) − 0 . 4 y + 0 . 46 x 2 y ⎪ ˙ ⎪ ⎩ 0 . 6 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 x T. Johnson, A quartic system with twenty-six limit cycles, Experimental Mathematics , 2011 A New Lower Bound on H(4) 2/17

  2. + Infinitesimal Hilbert’s 16th Problem 1 . 4 1 . 2 H ( x , y ) = ( x 2 − 0 . 9 ) 2 + ( y 2 − 1 . 1 ) 2 y 1 ⎧ x = 4 y ( y 2 − 1 . 1 ) 0 . 8 ⎪ ˙ ⎪ ⎨ y = 4 x ( x 2 − 0 . 9 ) − 0 . 4 y + 0 . 46 x 2 y ⎪ ˙ ⎪ ⎩ 0 . 6 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 x T. Johnson, A quartic system with twenty-six limit cycles, Experimental Mathematics , 2011 A New Lower Bound on H(4) 2/17

  3. + Infinitesimal Hilbert’s 16th Problem 1 . 4 1 . 2 H ( x , y ) = ( x 2 − 0 . 9 ) 2 + ( y 2 − 1 . 1 ) 2 y 1 ⎧ x = 4 y ( y 2 − 1 . 1 ) 0 . 8 ⎪ ˙ ⎪ ⎨ y = 4 x ( x 2 − 0 . 9 ) − 0 . 4 y + 0 . 46 x 2 y ⎪ ˙ ⎪ ⎩ 0 . 6 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 x T. Johnson, A quartic system with twenty-six limit cycles, Experimental Mathematics , 2011 A New Lower Bound on H(4) 2/17

  4. + Infinitesimal Hilbert’s 16th Problem 1 . 4 1 . 2 H ( x , y ) = ( x 2 − 0 . 9 ) 2 + ( y 2 − 1 . 1 ) 2 y 1 ⎧ x = 4 y ( y 2 − 1 . 1 ) 0 . 8 ⎪ ˙ ⎪ ⎨ y = 4 x ( x 2 − 0 . 9 ) − 0 . 4 y + 0 . 46 x 2 y ⎪ ˙ ⎪ ⎩ 0 . 6 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 x T. Johnson, A quartic system with twenty-six limit cycles, Experimental Mathematics , 2011 A New Lower Bound on H(4) 2/17

  5. + Infinitesimal Hilbert’s 16th Problem 1 . 4 1 . 2 H ( x , y ) = ( x 2 − 0 . 9 ) 2 + ( y 2 − 1 . 1 ) 2 y 1 ⎧ x = 4 y ( y 2 − 1 . 1 ) 0 . 8 ⎪ ˙ ⎪ ⎨ y = 4 x ( x 2 − 0 . 9 ) − 0 . 4 y + 0 . 46 x 2 y ⎪ ˙ ⎪ ⎩ 0 . 6 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 x T. Johnson, A quartic system with twenty-six limit cycles, Experimental Mathematics , 2011 A New Lower Bound on H(4) 2/17

  6. + Infinitesimal Hilbert’s 16th Problem 1 . 4 1 . 2 H ( x , y ) = ( x 2 − 0 . 9 ) 2 + ( y 2 − 1 . 1 ) 2 y 1 ⎧ x = 4 y ( y 2 − 1 . 1 ) 0 . 8 ⎪ ˙ ⎪ ⎨ y = 4 x ( x 2 − 0 . 9 ) − 0 . 4 y + 0 . 46 x 2 y ⎪ ˙ ⎪ ⎩ 0 . 6 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 x T. Johnson, A quartic system with twenty-six limit cycles, Experimental Mathematics , 2011 A New Lower Bound on H(4) 2/17

  7. + Infinitesimal Hilbert’s 16th Problem 1 . 4 1 . 2 H ( x , y ) = ( x 2 − 0 . 9 ) 2 + ( y 2 − 1 . 1 ) 2 y 1 ⎧ x = 4 y ( y 2 − 1 . 1 ) 0 . 8 ⎪ ˙ ⎪ ⎨ y = 4 x ( x 2 − 0 . 9 ) − 0 . 4 y + 0 . 46 x 2 y ⎪ ˙ ⎪ ⎩ 0 . 6 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 x T. Johnson, A quartic system with twenty-six limit cycles, Experimental Mathematics , 2011 A New Lower Bound on H(4) 2/17

  8. + Infinitesimal Hilbert’s 16th Problem 1 . 4 1 . 2 H ( x , y ) = ( x 2 − 0 . 9 ) 2 + ( y 2 − 1 . 1 ) 2 y 1 ⎧ x = 4 y ( y 2 − 1 . 1 ) 0 . 8 ⎪ ˙ ⎪ ⎨ y = 4 x ( x 2 − 0 . 9 ) − 0 . 4 y + 0 . 46 x 2 y ⎪ ˙ ⎪ ⎩ 0 . 6 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 x T. Johnson, A quartic system with twenty-six limit cycles, Experimental Mathematics , 2011 A New Lower Bound on H(4) 2/17

  9. + Infinitesimal Hilbert’s 16th Problem 1 . 4 1 . 2 H ( x , y ) = ( x 2 − 0 . 9 ) 2 + ( y 2 − 1 . 1 ) 2 y 1 ⎧ x = 4 y ( y 2 − 1 . 1 ) 0 . 8 ⎪ ˙ ⎪ ⎨ y = 4 x ( x 2 − 0 . 9 ) − 0 . 4 y + 0 . 46 x 2 y ⎪ ˙ ⎪ ⎩ 0 . 6 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 x T. Johnson, A quartic system with twenty-six limit cycles, Experimental Mathematics , 2011 A New Lower Bound on H(4) 2/17

  10. + Infinitesimal Hilbert’s 16th Problem Infinitesimal Hilbert’s 16th problem For a given integer n , what is the maximal 1 . 4 number Z( n ) of limit cycles a perturbed Hamiltonian vector field of the form: 1 . 2 ⎧ ⎪ x = − ∂ y H ( x , y ) + ε f ( x , y ) ˙ ⎪ ⎨ y ⎪ ˙ y = ∂ x H ( x , y ) + ε g ( x , y ) 1 ⎪ ⎩ can have when ε → 0, with: 0 . 8 H ( x , y ) a polynomial potential function of degree n + 1 0 . 6 f , g polynomial perturbations of degree n 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 x T. Johnson, A quartic system with twenty-six limit cycles, Experimental Mathematics , 2011 A New Lower Bound on H(4) 2/17

  11. + Infinitesimal Hilbert’s 16th Problem Infinitesimal Hilbert’s 16th problem For a given integer n , what is the maximal 1 . 4 number Z ( n ) of limit cycles a perturbed Hamiltonian vector field of the form: 1 . 2 ⎧ ⎪ x = − ∂ y H ( x , y ) + ε f ( x , y ) ⎪ ˙ ⎨ ⎪ y = ∂ x H ( x , y ) + ε g ( x , y ) ⎪ y ⎩ ˙ 1 can have when ε → 0, with: 0 . 8 H ( x , y ) a polynomial potential function of degree n + 1 0 . 6 f , g polynomial perturbations of degree n 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 x Z ( n ) < ∞ for all n T. Johnson, A quartic system with twenty-six limit cycles, Pessimistic upper bounds Experimental Mathematics , 2011 A New Lower Bound on H(4) 2/17

  12. + A Fundamental Tool: the Poincaré-Pontryagin Theorem 1 . 2 1 . 1 y h 1 0 . 9 0 . 9 1 1 . 1 1 . 2 1 . 3 x x = − ∂ y H ( x , y ) + ε f ( x , y ) ⎧ ⎪ ˙ ⎪ ⎨ y = ∂ x H ( x , y ) + ε g ( x , y ) ⎪ ˙ ⎪ ⎩ A New Lower Bound on H(4) 3/17

  13. + A Fundamental Tool: the Poincaré-Pontryagin Theorem 1 . 2 Poincaré first return map P ( h ) 1 . 1 y P ( h ) h 1 0 . 9 0 . 9 1 1 . 1 1 . 2 1 . 3 x x = − ∂ y H ( x , y ) + ε f ( x , y ) ⎧ ⎪ ˙ ⎪ ⎨ y = ∂ x H ( x , y ) + ε g ( x , y ) ⎪ ˙ ⎪ ⎩ A New Lower Bound on H(4) 3/17

  14. + A Fundamental Tool: the Poincaré-Pontryagin Theorem 1 . 2 Poincaré first return map P ( h ) 1 . 1 y P 3 ( h ) P 2 ( h ) P ( h ) h 1 0 . 9 0 . 9 1 1 . 1 1 . 2 1 . 3 x x = − ∂ y H ( x , y ) + ε f ( x , y ) ⎧ ⎪ ˙ ⎪ ⎨ y = ∂ x H ( x , y ) + ε g ( x , y ) ⎪ ˙ ⎪ ⎩ A New Lower Bound on H(4) 3/17

  15. + A Fundamental Tool: the Poincaré-Pontryagin Theorem 1 . 2 Poincaré first return map P ( h ) 1 . 1 Displacement d ( h ) = P ( h ) − h d ( h ) y P 3 ( h ) P 2 ( h ) P ( h ) h 1 0 . 9 0 . 9 1 1 . 1 1 . 2 1 . 3 x x = − ∂ y H ( x , y ) + ε f ( x , y ) ⎧ ⎪ ˙ ⎪ ⎨ y = ∂ x H ( x , y ) + ε g ( x , y ) ⎪ ˙ ⎪ ⎩ A New Lower Bound on H(4) 3/17

  16. + A Fundamental Tool: the Poincaré-Pontryagin Theorem 1 . 2 Poincaré first return map P ( h ) 1 . 1 Displacement d ( h ) = P ( h ) − h d ( h ) y Limit cycle ⇔ isolated zero of d P 3 ( h ) P 2 ( h ) P ( h ) h 1 0 . 9 0 . 9 1 1 . 1 1 . 2 1 . 3 x x = − ∂ y H ( x , y ) + ε f ( x , y ) ⎧ ⎪ ˙ ⎪ ⎨ y = ∂ x H ( x , y ) + ε g ( x , y ) ⎪ ˙ ⎪ ⎩ A New Lower Bound on H(4) 3/17

  17. + A Fundamental Tool: the Poincaré-Pontryagin Theorem 1 . 2 Poincaré first return map P ( h ) 1 . 1 Displacement d ( h ) = P ( h ) − h d ( h ) y Limit cycle ⇔ isolated zero of d P 3 ( h ) P 2 ( h ) P ( h ) h Abelian integral I ( h ) : 1 ∮ H − 1 ( h ) f ( x , y ) d y − g ( x , y ) d x 0 . 9 0 . 9 1 1 . 1 1 . 2 1 . 3 x x = − ∂ y H ( x , y ) + ε f ( x , y ) ⎧ ⎪ ˙ ⎪ ⎨ y = ∂ x H ( x , y ) + ε g ( x , y ) ⎪ ˙ ⎪ ⎩ A New Lower Bound on H(4) 3/17

  18. + A Fundamental Tool: the Poincaré-Pontryagin Theorem 1 . 2 Poincaré first return map P ( h ) 1 . 1 Displacement d ( h ) = P ( h ) − h d ( h ) y Limit cycle ⇔ isolated zero of d P 3 ( h ) P 2 ( h ) P ( h ) h Abelian integral I ( h ) : 1 ∮ H − 1 ( h ) f ( x , y ) d y − g ( x , y ) d x 0 . 9 0 . 9 1 1 . 1 1 . 2 1 . 3 x Poincaré-Pontryagin theorem The Abelian integral I ( h ) approximates the x = − ∂ y H ( x , y ) + ε f ( x , y ) ⎧ ⎪ ˙ ⎪ displacement function d ( h ) for small ε : ⎨ y = ∂ x H ( x , y ) + ε g ( x , y ) ⎪ ˙ ⎪ d ( h ) = ε ( I ( h ) + O ( ε )) when ε → 0 ⎩ A New Lower Bound on H(4) 3/17

  19. + A Pseudo-Hamiltonian Quartic System Hamiltonian system: x = − 4 y ( y 2 − 1 . 1 ) ⎧ ⎪ ˙ ⎪ y = 4 x ( x 2 − 0 . 9 ) ⎨ ⎪ ˙ ⎪ ⎩ 1 y 0 − 1 − 1 0 1 x A New Lower Bound on H(4) 4/17

  20. + A Pseudo-Hamiltonian Quartic System pseudo- Hamiltonian system: x = − 4 y y ( y 2 − 1 . 1 ) ⎧ ⎪ ˙ ⎪ y = 4 y x ( x 2 − 0 . 9 ) ⎨ ⎪ ˙ ⎪ ⎩ 1 y 0 − 1 − 1 0 1 x A New Lower Bound on H(4) 4/17

  21. + A Pseudo-Hamiltonian Quartic System pseudo- Hamiltonian system: x = − 4 y y ( y 2 − 1 . 1 ) ⎧ ⎪ ˙ ⎪ y = 4 y x ( x 2 − 0 . 9 ) ⎨ ⎪ ˙ ⎪ ⎩ • • 1 same geometric orbits after rescaling y 0 • − 1 • • − 1 0 1 x A New Lower Bound on H(4) 4/17

  22. + A Pseudo-Hamiltonian Quartic System pseudo- Hamiltonian system: x = − 4 y y ( y 2 − 1 . 1 ) + ε f ( x , y ) ⎧ ⎪ ˙ ⎪ y = 4 y x ( x 2 − 0 . 9 ) + ε g ( x , y ) ⎨ ⎪ ˙ ⎪ ⎩ • • 1 same geometric orbits after rescaling ≃ perturbations without rescaling: f ( x , y ) , g ( x , y ) ⟨ x i y j , i ⩾ 0 , j ⩾ − 1 , i + j ⩽ 3 ⟩ y 0 • ∈ y y − 1 • • − 1 0 1 x A New Lower Bound on H(4) 4/17

  23. + A Pseudo-Hamiltonian Quartic System pseudo- Hamiltonian system: x = − 4 y y ( y 2 − 1 . 1 ) + ε f ( x , y ) ⎧ ⎪ ˙ ⎪ y = 4 y x ( x 2 − 0 . 9 ) + ε g ( x , y ) ⎨ ⎪ ˙ ⎪ ⎩ • • 1 same geometric orbits after rescaling ≃ perturbations without rescaling: f ( x , y ) , g ( x , y ) ⟨ x i y j , i ⩾ 0 , j ⩾ − 1 , i + j ⩽ 3 ⟩ y 0 • ∈ y y Generalized Poincaré-Pontryagin theorem The generalized Abelian integral: − 1 • • f ( x , y ) d y − g ( x , y ) d x I ( h ) = ∮ H − 1 ( h ) y − 1 0 1 x approximates the displacement function d ( h ) for small ε : d ( h ) = ε ( I ( h ) + O ( ε )) when ε → 0 A New Lower Bound on H(4) 4/17

  24. + A Pseudo-Hamiltonian Quartic System pseudo- Hamiltonian system: x = − 4 y y ( y 2 − 1 . 1 ) + ε f ( x , y ) ⎧ ⎪ ˙ ⎪ y = 4 y x ( x 2 − 0 . 9 ) + ε g ( x , y ) ⎨ ⎪ ˙ ⎪ ⎩ 1 same geometric orbits after rescaling ≃ perturbations without rescaling: f ( x , y ) , g ( x , y ) ⟨ x i y j , i ⩾ 0 , j ⩾ − 1 , i + j ⩽ 3 ⟩ 4 × ? + 2 × ? = ? y 0 ∈ y y Generalized Poincaré-Pontryagin theorem − 1 The generalized Abelian integral: f ( x , y ) d y − g ( x , y ) d x I ( h ) = ∮ H − 1 ( h ) y − 1 0 1 x ⇒ The finiteness of Z ( 4 ) does not apply, approximates the displacement function but we still have some tools of the d ( h ) for small ε : Hamiltonian case! d ( h ) = ε ( I ( h ) + O ( ε )) when ε → 0 A New Lower Bound on H(4) 4/17

  25. + Choice of Perturbations x 2 f ( x , y ) = 1 x y xy y 2 x 3 x 2 y xy 2 y 3 x 4 x 3 y x 2 y 2 xy 3 y 4 x 2 g ( x , y ) = 1 x y xy y 2 x 3 x 2 y xy 2 y 3 x 4 x 3 y x 2 y 2 xy 3 y 4 A New Lower Bound on H(4) 5/17

  26. + Choice of Perturbations x 2 f ( x , y ) = 1 x y xy y 2 x 3 x 2 y xy 2 y 3 x 4 x 3 y x 2 y 2 xy 3 y 4 x 2 g ( x , y ) = 1 x y xy y 2 x 3 x 2 y xy 2 y 3 x 3 y xy 3 x 4 x 2 y 2 y 4 symmetry requirements f ( x , y ) g ( x , y ) linear relations from Green’s formula: ∂ x ∝ ∂ y y y α 00 + α 20 x 2 + α 22 x 2 y 2 + α 40 x 4 + α 04 y 4 I( h ) = ∮ H − 1 ( h ) d x y A New Lower Bound on H(4) 5/17

  27. + Numerically Optimizing the Number of Zeros ▸ Find coefficients of I( h ) = α 00 I 00 ( h ) + α 20 I 20 ( h ) + α 22 I 22 ( h ) + α 40 I 40 ( h ) + α 04 I 04 ( h ) . 4 3 2 1 0 − 1 − 2 − 3 x 2 y 2 − 4 y 4 1 x 2 − 5 x 4 − 6 − 0 . 5 − 0 . 45 − 0 . 4 − 0 . 35 − 0 . 3 − 0 . 25 h A New Lower Bound on H(4) 6/17

  28. + Numerically Optimizing the Number of Zeros = -0.78622148667854837664 α 00 = 0.87723523612653436051 α 20 = 1 α 22 = 0.23742713894293038223 α 40 = -0.21823846173078863753 α 04 2 ( × 10 − 4 ) 1 0 − 1 h − 2 − 0 . 5 − 0 . 45 − 0 . 4 − 0 . 35 − 0 . 3 − 0 . 25 A New Lower Bound on H(4) 6/17

  29. + Numerically Optimizing the Number of Zeros = -0.78622148667854837664 α 00 = 0.87723523612653436051 α 20 = 1 α 22 = 0.23742713894293038223 α 40 = -0.21823846173078863753 α 04 2 ( × 10 − 4 ) 1 0 − 1 h − 2 − 0 . 5 − 0 . 45 − 0 . 4 − 0 . 35 − 0 . 3 − 0 . 25 1 ( × 10 − 7 ) 0 . 5 0 − 0 . 5 h − 1 -0.315 -0.310 -0.305 A New Lower Bound on H(4) 6/17

  30. + Numerically Optimizing the Number of Zeros = -0.78622148667854837664 α 00 = 0.87723523612653436051 α 20 = 1 α 22 = 0.23742713894293038223 α 40 = -0.21823846173078863753 α 04 2 ( × 10 − 4 ) 1 0 − 1 h − 2 − 0 . 5 − 0 . 45 − 0 . 4 − 0 . 35 − 0 . 3 − 0 . 25 1 ( × 10 − 7 ) 0 . 5 0 − 0 . 5 h − 1 -0.315 -0.310 -0.305 4 × 5 + 2 × 2 = 24 A New Lower Bound on H(4) 6/17

  31. Outline 1 A quartic example for Hilbert 16th problem 2 Computing Abelian integrals with rigorous polynomial approximations 3 Wronskian and extended Chebyshev systems 4 Conclusion A New Lower Bound on H(4)

  32. + Computing Abelian Integrals √ 0 < r ( = h ) < 0 . 9 2 y 2 1 0 0 1 2 x 2 1 y 0 − 1 − 1 0 1 x A New Lower Bound on H(4) 7/17

  33. + Computing Abelian Integrals √ 0 < r ( = h ) < 0 . 9 2 x min = 0 . 9 − x max = 0 . 9 + √ r √ r 2 2 y 2 1 x min = 1 . 1 − √ r x max = 1 . 1 + √ r 2 2 0 0 1 2 x 2 1 y 0 − 1 − 1 0 1 x A New Lower Bound on H(4) 7/17

  34. + Computing Abelian Integrals √ 0 < r ( = h ) < 0 . 9 2 x min = 0 . 9 − x max = 0 . 9 + √ r √ r 2 2 y 2 1 x min = 1 . 1 − √ r x max = 1 . 1 + √ r 2 2 √ √ r 2 − ( x 2 − 0 . 9 ) 2 0 y up ( x ) = 1 . 1 + 0 1 2 √ x 2 √ r 2 − ( x 2 − 0 . 9 ) 2 y down ( x ) = 1 . 1 − √ √ r 2 − ( y 2 − 1 . 1 ) 2 x left ( y ) = 0 . 9 − √ √ 1 r 2 − ( y 2 − 1 . 1 ) 2 x right ( y ) = 0 . 9 + y 0 − 1 − 1 0 1 x A New Lower Bound on H(4) 7/17

  35. + Computing Abelian Integrals √ 0 < r ( = h ) < 0 . 9 2 x min = 0 . 9 − x max = 0 . 9 + √ r √ r 2 2 y 2 1 x min = 1 . 1 − √ r x max = 1 . 1 + √ r 2 2 √ √ r 2 − ( x 2 − 0 . 9 ) 2 0 y up ( x ) = 1 . 1 + 0 1 2 √ x 2 √ r 2 − ( x 2 − 0 . 9 ) 2 y down ( x ) = 1 . 1 − √ √ r 2 − ( y 2 − 1 . 1 ) 2 x left ( y ) = 0 . 9 − √ √ 1 r 2 − ( y 2 − 1 . 1 ) 2 x right ( y ) = 0 . 9 + y 0 g ( x , y ) ( g ( x , y up ( x )) − g ( x , y down ( x )) I( h ) = ∮ H − 1 ( h ) d x = ∫ ) d x max − 1 y up ( x ) y down ( x ) y x min y 2 − 1 . 1 ( g ( x left ( y ) , y ) + g ( x right ( y ) , y ) − 1 0 1 +∫ ) √ y max r 2 − ( y 2 − 1 . 1 ) 2 d y . x left ( y ) x right ( y ) x y min A New Lower Bound on H(4) 7/17

  36. + Computing Abelian Integrals √ 0 . 9 < r ( = h ) < 1 . 1 2 y 2 1 0 0 1 2 x 2 1 y 0 − 1 − 1 0 1 x A New Lower Bound on H(4) 7/17

  37. + Computing Abelian Integrals √ 0 . 9 < r ( = h ) < 1 . 1 2 x min = 0 . 9 − x max = 0 . 9 + √ r √ r 2 2 y 2 1 x min = 1 . 1 − √ r x max = 1 . 1 + √ r 2 2 √ √ r 2 − ( x 2 − 0 . 9 ) 2 0 y up ( x ) = 1 . 1 + 0 1 2 √ x 2 √ r 2 − ( x 2 − 0 . 9 ) 2 y down ( x ) = 1 . 1 − √ √ r 2 − ( y 2 − 1 . 1 ) 2 x left ( y ) = 0 . 9 − √ √ 1 r 2 − ( y 2 − 1 . 1 ) 2 x right ( y ) = 0 . 9 + y 0 g ( x , y ) ( g ( x , y up ( x )) − g ( x , y down ( x )) I( h ) = ∮ H − 1 ( h ) d x = ∫ ) d x max − 1 y up ( x ) y down ( x ) y − x max g ( x right ( y ) , y )( y 2 − 1 . 1 ) − 1 0 1 + 2 ∫ √ y max r 2 − ( y 2 − 1 . 1 ) 2 d y . x x right ( y ) y min A New Lower Bound on H(4) 7/17

  38. + Rigorous Polynomial Approximations Definition A pair ( P , ε ) ∈ R [ X ] × R + is a rigorous polynomial approximation (RPA) of f for a given norm ∥ ⋅ ∥ if ∥ f − P ∥ ⩽ ε . � 0 × K − + H F G A New Lower Bound on H(4) 8/17

  39. + Rigorous Polynomial Approximations Definition A pair ( P , ε ) ∈ R [ X ] × R + is a rigorous polynomial approximation (RPA) of f for a given norm ∥ ⋅ ∥ if ∥ f − P ∥ ⩽ ε . Example: sup-norm over [ − 1 , 1 ] : f ∈ ( P , ε ) ⇔ ∣ f ( t ) − P ( t )∣ ⩽ ε ∀ t ∈ [ − 1 , 1 ] � 0 × K − + H F G A New Lower Bound on H(4) 8/17

  40. + Rigorous Polynomial Approximations Definition A pair ( P , ε ) ∈ R [ X ] × R + is a rigorous polynomial approximation (RPA) of f for a given norm ∥ ⋅ ∥ if ∥ f − P ∥ ⩽ ε . Example: sup-norm over [ − 1 , 1 ] : Example: r ( t ) = f ( t ) + g ( t ) f ∈ ( P , ε ) ⇔ ∣ f ( t ) − P ( t )∣ ⩽ ε ∀ t ∈ [ − 1 , 1 ] � 0 Some elementary operations: × ( P , ε ) + ( Q , η ) ∶ = ( P + Q , ε + η ) , K − + H F G A New Lower Bound on H(4) 8/17

  41. + Rigorous Polynomial Approximations Definition A pair ( P , ε ) ∈ R [ X ] × R + is a rigorous polynomial approximation (RPA) of f for a given norm ∥ ⋅ ∥ if ∥ f − P ∥ ⩽ ε . Example: sup-norm over [ − 1 , 1 ] : Example: r ( t ) = f ( t ) + g ( t ) − h ( t ) f ∈ ( P , ε ) ⇔ ∣ f ( t ) − P ( t )∣ ⩽ ε ∀ t ∈ [ − 1 , 1 ] � 0 Some elementary operations: × ( P , ε ) + ( Q , η ) ∶ = ( P + Q , ε + η ) , ( P , ε ) − ( Q , η ) ∶ = ( P − Q , ε + η ) , K − + H F G A New Lower Bound on H(4) 8/17

  42. + Rigorous Polynomial Approximations Definition A pair ( P , ε ) ∈ R [ X ] × R + is a rigorous polynomial approximation (RPA) of f for a given norm ∥ ⋅ ∥ if ∥ f − P ∥ ⩽ ε . Example: sup-norm over [ − 1 , 1 ] : Example: r ( t ) = k ( t )( f ( t ) + g ( t ) − h ( t )) f ∈ ( P , ε ) ⇔ ∣ f ( t ) − P ( t )∣ ⩽ ε ∀ t ∈ [ − 1 , 1 ] � 0 Some elementary operations: × ( P , ε ) + ( Q , η ) ∶ = ( P + Q , ε + η ) , ( P , ε ) − ( Q , η ) ∶ = ( P − Q , ε + η ) , K − ( P , ε ) ⋅ ( Q , η ) ∶ = ( PQ , ∥ Q ∥ η + ∥ P ∥ ε + ηε ) + H F G A New Lower Bound on H(4) 8/17

  43. + Rigorous Polynomial Approximations Definition A pair ( P , ε ) ∈ R [ X ] × R + is a rigorous polynomial approximation (RPA) of f for a given norm ∥ ⋅ ∥ if ∥ f − P ∥ ⩽ ε . Example: sup-norm over [ − 1 , 1 ] : Example: r ( t ) = ∫ 0 k ( s )( f ( s ) + g ( s ) − h ( s )) d s t f ∈ ( P , ε ) ⇔ ∣ f ( t ) − P ( t )∣ ⩽ ε ∀ t ∈ [ − 1 , 1 ] � 0 Some elementary operations: × ( P , ε ) + ( Q , η ) ∶ = ( P + Q , ε + η ) , ( P , ε ) − ( Q , η ) ∶ = ( P − Q , ε + η ) , K − ( P , ε ) ⋅ ( Q , η ) ∶ = ( PQ , ∥ Q ∥ η + ∥ P ∥ ε + ηε ) ∫ 0 ( P , ε ) ∶ = ( ∫ 0 P ( s ) d s , ε ) t + H F G A New Lower Bound on H(4) 8/17

  44. + Rigorous Polynomial Approximations Definition A pair ( P , ε ) ∈ R [ X ] × R + is a rigorous polynomial approximation (RPA) of f for a given norm ∥ ⋅ ∥ if ∥ f − P ∥ ⩽ ε . Example: sup-norm over [ − 1 , 1 ] : Example: r ( t ) = ∫ 0 k ( s )( f ( s ) + g ( s ) − h ( s )) d s t f ∈ ( P , ε ) ⇔ ∣ f ( t ) − P ( t )∣ ⩽ ε ∀ t ∈ [ − 1 , 1 ] √ ? � ÷ ? 0 Some elementary operations: × ( P , ε ) + ( Q , η ) ∶ = ( P + Q , ε + η ) , ( P , ε ) − ( Q , η ) ∶ = ( P − Q , ε + η ) , K − ( P , ε ) ⋅ ( Q , η ) ∶ = ( PQ , ∥ Q ∥ η + ∥ P ∥ ε + ηε ) ∫ 0 ( P , ε ) ∶ = ( ∫ 0 P ( s ) d s , ε ) t + H F G A New Lower Bound on H(4) 8/17

  45. + Banach Fixed-Point Theorem for A Posteriori Validation ▸ Fixed-point equation T ⋅ ϕ = ϕ with T contracting, ▸ Approximation ϕ ○ to exact solution ϕ ⋆ , General scheme ▸ Compute a posteriori error bounds with Banach theorem. A New Lower Bound on H(4) 9/17

  46. + Banach Fixed-Point Theorem for A Posteriori Validation ▸ Fixed-point equation T ⋅ ϕ = ϕ with T contracting, ▸ Approximation ϕ ○ to exact solution ϕ ⋆ , General scheme ▸ Compute a posteriori error bounds with Banach theorem. Banach Fixed-Point Theorem If ( X , d ) is complete and T contracting of ratio λ < 1, ▸ T admits a unique fixed-point ϕ ⋆ , and ▸ For all ϕ ○ ∈ X , d ( ϕ ○ , T ⋅ ϕ ○ ) ⩽ d ( ϕ ○ , ϕ ⋆ ) ⩽ d ( ϕ ○ , T ⋅ ϕ ○ ) . 1 + λ 1 − λ A New Lower Bound on H(4) 9/17

  47. + Banach Fixed-Point Theorem for A Posteriori Validation ▸ Fixed-point equation T ⋅ ϕ = ϕ with T contracting, ▸ Approximation ϕ ○ to exact solution ϕ ⋆ , General scheme ▸ Compute a posteriori error bounds with Banach theorem. Banach Fixed-Point Theorem If ( X , d ) is complete and T contracting of ratio λ < 1, ▸ T admits a unique fixed-point ϕ ⋆ , and ▸ For all ϕ ○ ∈ X , d ( ϕ ○ , T ⋅ ϕ ○ ) ⩽ d ( ϕ ○ , ϕ ⋆ ) ⩽ d ( ϕ ○ , T ⋅ ϕ ○ ) . 1 + λ 1 − λ ▸ Newton’s method = reformulate F ⋅ ϕ = 0 as T ⋅ ϕ = ϕ with: T ⋅ ϕ = ϕ − A ⋅ F ⋅ ϕ, A New Lower Bound on H(4) 9/17

  48. + Banach Fixed-Point Theorem for A Posteriori Validation ▸ Fixed-point equation T ⋅ ϕ = ϕ with T contracting, ▸ Approximation ϕ ○ to exact solution ϕ ⋆ , General scheme ▸ Compute a posteriori error bounds with Banach theorem. Banach Fixed-Point Theorem If ( X , d ) is complete and T contracting of ratio λ < 1, ▸ T admits a unique fixed-point ϕ ⋆ , and ▸ For all ϕ ○ ∈ X , d ( ϕ ○ , T ⋅ ϕ ○ ) ⩽ d ( ϕ ○ , ϕ ⋆ ) ⩽ d ( ϕ ○ , T ⋅ ϕ ○ ) . 1 + λ 1 − λ ▸ Newton’s method = reformulate F ⋅ ϕ = 0 as T ⋅ ϕ = ϕ with: A ≈ ( D F ( ϕ ○ )) − 1 T ⋅ ϕ = ϕ − A ⋅ F ⋅ ϕ, and check T is contracting. A New Lower Bound on H(4) 9/17

  49. + Banach Fixed-Point Theorem for A Posteriori Validation ▸ Fixed-point equation T ⋅ ϕ = ϕ with T contracting, ▸ Approximation ϕ ○ to exact solution ϕ ⋆ , General scheme ▸ Compute a posteriori error bounds with Banach theorem. Banach Fixed-Point Theorem If ( X , d ) is complete and T contracting of ratio λ < 1, ▸ T admits a unique fixed-point ϕ ⋆ , and ▸ For all ϕ ○ ∈ X , d ( ϕ ○ , T ⋅ ϕ ○ ) ⩽ d ( ϕ ○ , ϕ ⋆ ) ⩽ d ( ϕ ○ , T ⋅ ϕ ○ ) . 1 + λ 1 − λ ▸ Newton’s method = reformulate F ⋅ ϕ = 0 as T ⋅ ϕ = ϕ with: A ≈ ( D F ( ϕ ○ )) − 1 T ⋅ ϕ = ϕ − A ⋅ F ⋅ ϕ, and check T is contracting. ▸ Applications to numerous function space problems. A New Lower Bound on H(4) 9/17

  50. + Division of RPAs ▸ Approximation ϕ ○ ( x ) of ϕ ⋆ = x 2 / y down ( x ) using Chebyshev interpolation: 0 . 8 2 0 . 75 0 . 7 1 . 5 0 . 65 1 0 . 6 0 . 55 0 . 5 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 A New Lower Bound on H(4) 10/17

  51. + Division of RPAs ▸ Approximation ϕ ○ ( x ) of ϕ ⋆ = x 2 / y down ( x ) using Chebyshev interpolation: 0 . 8 2 0 . 75 0 . 7 1 . 5 0 . 65 1 0 . 6 0 . 55 0 . 5 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 A New Lower Bound on H(4) 10/17

  52. + Division of RPAs ▸ Approximation ϕ ○ ( x ) of ϕ ⋆ = x 2 / y down ( x ) using Chebyshev interpolation: 0 . 8 2 0 . 75 0 . 7 1 . 5 0 . 65 1 0 . 6 0 . 55 0 . 5 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 A New Lower Bound on H(4) 10/17

  53. + Division of RPAs ▸ Approximation ϕ ○ ( x ) of ϕ ⋆ = x 2 / y down ( x ) using Chebyshev interpolation: 0 . 8 2 0 . 75 0 . 7 1 . 5 0 . 65 1 0 . 6 0 . 55 0 . 5 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 A New Lower Bound on H(4) 10/17

  54. + Division of RPAs ▸ Approximation ϕ ○ ( x ) of ϕ ⋆ = x 2 / y down ( x ) using Chebyshev interpolation: 0 . 8 2 0 . 75 0 . 7 1 . 5 0 . 65 1 0 . 6 0 . 55 0 . 5 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 ▸ Newton-like operator T with unique fixed point ϕ ⋆ = x 2 y down ( x ) : T ⋅ ϕ = ϕ − ψ ( y down ϕ − x 2 ) ψ ( x ) ≈ 1 / y down ( x ) A New Lower Bound on H(4) 10/17

  55. + Division of RPAs ▸ Approximation ϕ ○ ( x ) of ϕ ⋆ = x 2 / y down ( x ) using Chebyshev interpolation: 0 . 8 2 0 . 75 0 . 7 1 . 5 0 . 65 1 0 . 6 0 . 55 0 . 5 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 ▸ Newton-like operator T with unique fixed point ϕ ⋆ = x 2 y down ( x ) : T ⋅ ϕ = ϕ − ψ ( y down ϕ − x 2 ) ψ ( x ) ≈ 1 / y down ( x ) ▸ Is T contracting? ∥ D T ∥ = ∥ 1 − ψ y down ∥ = λ < 1 A New Lower Bound on H(4) 10/17

  56. + Division of RPAs ▸ Approximation ϕ ○ ( x ) of ϕ ⋆ = x 2 / y down ( x ) using Chebyshev interpolation: 0 . 8 2 0 . 75 0 . 7 1 . 5 0 . 65 1 0 . 6 0 . 55 0 . 5 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 ▸ Newton-like operator T with unique fixed point ϕ ⋆ = x 2 y down ( x ) : T ⋅ ϕ = ϕ − ψ ( y down ϕ − x 2 ) ψ ( x ) ≈ 1 / y down ( x ) ▸ Is T contracting? ∥ D T ∥ = ∥ 1 − ψ y down ∥ = λ < 1 ▸ Apply the Banach fixed-point theorem: ∥ ϕ ○ − T ⋅ ϕ ○ ∥ = ∥ ψ ( y down ϕ ○ − x 2 )∥ ⩽ η A New Lower Bound on H(4) 10/17

  57. + Division of RPAs ▸ Approximation ϕ ○ ( x ) of ϕ ⋆ = x 2 / y down ( x ) using Chebyshev interpolation: 0 . 8 2 0 . 75 0 . 7 1 . 5 0 . 65 1 0 . 6 0 . 55 0 . 5 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 0 . 6 0 . 7 0 . 8 0 . 9 1 1 . 1 1 . 2 ▸ Newton-like operator T with unique fixed point ϕ ⋆ = x 2 y down ( x ) : T ⋅ ϕ = ϕ − ψ ( y down ϕ − x 2 ) ψ ( x ) ≈ 1 / y down ( x ) ▸ Is T contracting? ∥ D T ∥ = ∥ 1 − ψ y down ∥ = λ < 1 ▸ Apply the Banach fixed-point theorem: ∥ ϕ ○ − T ⋅ ϕ ○ ∥ = ∥ ψ ( y down ϕ ○ − x 2 )∥ ⩽ η ⇒ ∥ ϕ ○ − ϕ ⋆ ∥ ⩽ η /( 1 − λ ) = ε + A New Lower Bound on H(4) 10/17

  58. + Square Root of a RPA √ f ( x ) where f ( x ) = 0 . 8 − ( x 2 − 0 . 9 ) 2 . ▸ ϕ ○ ( x ) ≈ A New Lower Bound on H(4) 11/17

  59. + Square Root of a RPA √ f ( x ) where f ( x ) = 0 . 8 − ( x 2 − 0 . 9 ) 2 . ▸ ϕ ○ ( x ) ≈ √ ▸ ϕ ⋆ = f unique fixed point of: √ 2 ( ϕ 2 − f ) ψ ( x ) ≈ 1 / ϕ ○ ( x ) ≈ 1 / f ( x ) T ⋅ ϕ = ϕ − ψ A New Lower Bound on H(4) 11/17

  60. + Square Root of a RPA √ f ( x ) where f ( x ) = 0 . 8 − ( x 2 − 0 . 9 ) 2 . ▸ ϕ ○ ( x ) ≈ √ ▸ ϕ ⋆ = f unique fixed point of: √ 2 ( ϕ 2 − f ) ψ ( x ) ≈ 1 / ϕ ○ ( x ) ≈ 1 / f ( x ) T ⋅ ϕ = ϕ − ψ ▸ Is T contracting? ∥ D T ( ϕ )∥ = ∥ 1 − ψ ϕ ∥ ⩽ ∥ 1 − ψϕ ○ ∥ + ∥ ψ ∥∥ ϕ − ϕ ○ ∥ A New Lower Bound on H(4) 11/17

  61. + Square Root of a RPA √ f ( x ) where f ( x ) = 0 . 8 − ( x 2 − 0 . 9 ) 2 . ▸ ϕ ○ ( x ) ≈ √ ▸ ϕ ⋆ = f unique fixed point of: √ 2 ( ϕ 2 − f ) ψ ( x ) ≈ 1 / ϕ ○ ( x ) ≈ 1 / f ( x ) T ⋅ ϕ = ϕ − ψ ▸ Is T contracting? ∥ D T ( ϕ )∥ ⩽ ∥ 1 − ψϕ ○ ∥ + ∥ ψ ∥ r sup λ = ∥ ϕ − ϕ ○ ∥⩽ r 1 . 5 1 . 5 1 1 0 . 5 0 . 5 0 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 0 0 . 2 0 . 4 0 . 6 0 . 8 1 r r A New Lower Bound on H(4) 11/17

  62. + Square Root of a RPA √ f ( x ) where f ( x ) = 0 . 8 − ( x 2 − 0 . 9 ) 2 . ▸ ϕ ○ ( x ) ≈ √ ▸ ϕ ⋆ = f unique fixed point of: √ 2 ( ϕ 2 − f ) ψ ( x ) ≈ 1 / ϕ ○ ( x ) ≈ 1 / f ( x ) T ⋅ ϕ = ϕ − ψ ▸ Is T contracting? ▸ Stable neighborhood for ϕ ○ : ∥ D T ( ϕ )∥ ⩽ ∥ 1 − ψϕ ○ ∥ + ∥ ψ ∥ r ∥ ϕ ○ − T ⋅ ϕ ○ ∥ + λ r ⩽ r sup λ = ∥ ϕ − ϕ ○ ∥⩽ r 1 . 5 1 . 5 1 1 0 . 5 0 . 5 0 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 0 0 . 2 0 . 4 0 . 6 0 . 8 1 r r A New Lower Bound on H(4) 11/17

  63. + Square Root of a RPA √ f ( x ) where f ( x ) = 0 . 8 − ( x 2 − 0 . 9 ) 2 . ▸ ϕ ○ ( x ) ≈ √ ▸ ϕ ⋆ = f unique fixed point of: √ 2 ( ϕ 2 − f ) ψ ( x ) ≈ 1 / ϕ ○ ( x ) ≈ 1 / f ( x ) T ⋅ ϕ = ϕ − ψ ▸ Is T contracting? ▸ Stable neighborhood for ϕ ○ : ∥ D T ( ϕ )∥ ⩽ ∥ 1 − ψϕ ○ ∥ + ∥ ψ ∥ r ∥ ψ ( ϕ ○ 2 − f )/ 2 ∥ + r (∥ 1 − ψϕ ○ ∥ + ∥ ψ ∥ r ) ⩽ r sup λ = ∥ ϕ − ϕ ○ ∥⩽ r 1 . 5 1 . 5 1 1 0 . 5 0 . 5 0 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 0 0 . 2 0 . 4 0 . 6 0 . 8 1 r r A New Lower Bound on H(4) 11/17

  64. + Square Root of a RPA √ f ( x ) where f ( x ) = 0 . 8 − ( x 2 − 0 . 9 ) 2 . ▸ ϕ ○ ( x ) ≈ √ ▸ ϕ ⋆ = f unique fixed point of: √ 2 ( ϕ 2 − f ) ψ ( x ) ≈ 1 / ϕ ○ ( x ) ≈ 1 / f ( x ) T ⋅ ϕ = ϕ − ψ ▸ Is T contracting? ▸ Stable neighborhood for ϕ ○ : ∥ D T ( ϕ )∥ ⩽ ∥ 1 − ψϕ ○ ∥ + ∥ ψ ∥ r ∥ ψ ( ϕ ○ 2 − f )/ 2 ∥ + r (∥ 1 − ψϕ ○ ∥ + ∥ ψ ∥ r ) ⩽ r sup λ = ∥ ϕ − ϕ ○ ∥⩽ r 1 . 5 1 . 5 1 1 0 . 5 0 . 5 0 0 0 0 . 2 0 . 4 0 . 6 0 . 8 1 0 0 . 2 0 . 4 0 . 6 0 . 8 1 r r ▸ Apply the Banach fixed-point theorem! A New Lower Bound on H(4) 11/17

  65. + Rigorous Computation of an Abelian Integral Using Degree N = 10 √ 0 . 8 − ( x 2 − 0 . 9 ) 2 : ▸ · 10 − 5 1 0 − 1 0 . 6 0 . 8 1 1 . 2 A New Lower Bound on H(4) 12/17

  66. + Rigorous Computation of an Abelian Integral Using Degree N = 10 √ 0 . 8 − ( x 2 − 0 . 9 ) 2 : ▸ · 10 − 5 1 0 − 1 √ √ 0 . 6 0 . 8 1 1 . 2 0 . 8 − ( x 2 − 0 . 9 ) 2 : ▸ y down ( x ) = 1 . 1 − · 10 − 5 1 0 − 1 0 . 6 0 . 8 1 1 . 2 A New Lower Bound on H(4) 12/17

  67. + Rigorous Computation of an Abelian Integral Using Degree N = 10 √ 0 . 8 − ( x 2 − 0 . 9 ) 2 : ▸ · 10 − 5 1 0 − 1 √ √ 0 . 6 0 . 8 1 1 . 2 0 . 8 − ( x 2 − 0 . 9 ) 2 : ▸ y down ( x ) = 1 . 1 − · 10 − 5 1 0 − 1 √ √ 0 . 6 0 . 8 1 1 . 2 0 . 8 − ( x 2 − 0 . 9 ) 2 : ▸ x 2 / y down ( x ) = x 2 / y down ( x ) = 1 . 1 − 1 · 10 − 4 0 . 5 0 − 0 . 5 − 1 0 . 6 0 . 8 1 1 . 2 A New Lower Bound on H(4) 12/17

  68. + Validation of Our Result 2 ( × 10 − 4 ) 1 0 − 1 h − 2 − 0 . 5 − 0 . 45 − 0 . 4 − 0 . 35 − 0 . 3 − 0 . 25 1 ( × 10 − 7 ) 0 . 5 0 − 0 . 5 h − 1 -0.315 -0.310 -0.305 A New Lower Bound on H(4) 13/17

  69. + Validation of Our Result 2 ( × 10 − 4 ) N = 12 N = 22 1 0 N = 24 − 1 N = 16 h − 2 − 0 . 5 − 0 . 45 − 0 . 4 − 0 . 35 − 0 . 3 − 0 . 25 1 ( × 10 − 7 ) N = 80 0 . 5 0 N = 130 − 0 . 5 N = 100 N = 240 h − 1 -0.315 -0.310 -0.305 A New Lower Bound on H(4) 13/17

  70. + Validation of Our Result 4 × 5 + 2 × 2 = 24 A New Lower Bound on H(4) 13/17

  71. Outline 1 A quartic example for Hilbert 16th problem 2 Computing Abelian integrals with rigorous polynomial approximations 3 Wronskian and extended Chebyshev systems 4 Conclusion A New Lower Bound on H(4)

  72. + Wronskian and Extended Chebyshev Systems ( f 0 , f 1 , . . . , f n ) extended Chebyshev system if all combination α 0 f 0 + ⋅ ⋅ ⋅ + α i f i has at most i zeros, for 0 ⩽ i ⩽ n . A New Lower Bound on H(4) 14/17

  73. + Wronskian and Extended Chebyshev Systems ( f 0 , f 1 , . . . , f n ) extended Chebyshev system if all combination α 0 f 0 + ⋅ ⋅ ⋅ + α i f i has at most i zeros, for 0 ⩽ i ⩽ n . ⇔ W 0 ( x ) , . . . , W n ( x ) ≠ 0 for all x : � � � f 0 ( x ) f 1 ( x ) f i ( x ) � � � � � � . . . � � � � 0 ( x ) 1 ( x ) i ( x ) � � � � f ′ f ′ f ′ � . . . W i ( x ) = � � � � ⋮ ⋮ ⋱ ⋮ � � � � � � � � � � � � ( x ) ( x ) ( x ) � � f ( i ) f ( i ) f ( i ) � � . . . 0 1 i A New Lower Bound on H(4) 14/17

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