A New Lower Boun o th Hilber Number for Quarti System 1,2 - - PowerPoint PPT Presentation
JNCF 2019, CIRM, Marseill A New Lower Boun o th Hilber Number for Quarti System 1,2 1 2
1,2 1 2 1,3
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)
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)
Hilbert’s 16th Problem
Hilbert’s 16th problem (second part) For a given integer n, what is the maximum number H(n) of limit cycles a polynomial vector field of degree at most n in the plane can have?
A New Lower Bound on H(4) 1/17
Hilbert’s 16th Problem
−3 −2 −1 1 2 3 −4 −2 −4 2 4 x y
Van der Pol oscillator: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = y ˙ y = −x + (1 − x2)y
A New Lower Bound on H(4) 1/17
Hilbert’s 16th Problem
−3 −2 −1 1 2 3 −4 −2 −4 2 4 s1 x y
Van der Pol oscillator: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = y ˙ y = −x + (1 − x2)y
A New Lower Bound on H(4) 1/17
Hilbert’s 16th Problem
−3 −2 −1 1 2 3 −4 −2 −4 2 4 s1 x y
Van der Pol oscillator: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = y ˙ y = −x + (1 − x2)y
A New Lower Bound on H(4) 1/17
Hilbert’s 16th Problem
−3 −2 −1 1 2 3 −4 −2 −4 2 4 s1 s2 x y
Van der Pol oscillator: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = y ˙ y = −x + (1 − x2)y
A New Lower Bound on H(4) 1/17
Hilbert’s 16th Problem
−3 −2 −1 1 2 3 −4 −2 −4 2 4 s1 s2 x y
Van der Pol oscillator: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = y ˙ y = −x + (1 − x2)y
A New Lower Bound on H(4) 1/17
Hilbert’s 16th Problem
−3 −2 −1 1 2 3 −4 −2 −4 2 4 s1 s2 x y
Van der Pol oscillator: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = y ˙ y = −x + (1 − x2)y
A New Lower Bound on H(4) 1/17
Hilbert’s 16th Problem
Hilbert’s 16th problem (second part) For a given integer n, what is the maximum number H(n) of limit cycles a polynomial vector field of degree at most n in the plane can have?
1923: P. Dulac (incorrectly) proved that a single polynomial vector field has a finite number of limit cycles
−3 −2 −1 1 2 3 −4 −2 −4 2 4 s1 s2 x y
A New Lower Bound on H(4) 1/17
Hilbert’s 16th Problem
Hilbert’s 16th problem (second part) For a given integer n, what is the maximum number H(n) of limit cycles a polynomial vector field of degree at most n in the plane can have?
1923: P. Dulac (incorrectly) proved that a single polynomial vector field has a finite number of limit cycles 1981: Y. S. Il’Yashenko found a major gap in Dulac’s proof 1991: New proofs of Dulac’s result by Y. S. Il’Yashenko and J. Écalle
−3 −2 −1 1 2 3 −4 −2 −4 2 4 s1 s2 x y
A New Lower Bound on H(4) 1/17
Hilbert’s 16th Problem
Hilbert’s 16th problem (second part) For a given integer n, what is the maximum number H(n) of limit cycles a polynomial vector field of degree at most n in the plane can have?
1923: P. Dulac (incorrectly) proved that a single polynomial vector field has a finite number of limit cycles 1981: Y. S. Il’Yashenko found a major gap in Dulac’s proof 1991: New proofs of Dulac’s result by Y. S. Il’Yashenko and J. Écalle But even H(2) < ∞ is open!
−3 −2 −1 1 2 3 −4 −2 −4 2 4 s1 s2 x y
A New Lower Bound on H(4) 1/17
Hilbert’s 16th Problem
Hilbert’s 16th problem (second part) For a given integer n, what is the maximum number H(n) of limit cycles a polynomial vector field of degree at most n in the plane can have?
1923: P. Dulac (incorrectly) proved that a single polynomial vector field has a finite number of limit cycles 1981: Y. S. Il’Yashenko found a major gap in Dulac’s proof 1991: New proofs of Dulac’s result by Y. S. Il’Yashenko and J. Écalle But even H(2) < ∞ is open! Some lower bounds: H(2) ⩾ 4, H(3) ⩾ 13, H(4) ⩾ 22. We prove H(4) ⩾ 24.
−3 −2 −1 1 2 3 −4 −2 −4 2 4 s1 s2 x y
A New Lower Bound on H(4) 1/17
Infinitesimal Hilbert’s 16th Problem
−1 1 −1 1
y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
−1 1 −1 1
y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
−1 1 −1 1
y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
−1 1 −1 1
y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
−1 1 −1 1
y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
−1 1 −1 1
y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
−1 1 −1 1
y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
−1 1 −1 1
y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
−1 1 −1 1
y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = −∂yH(x, y) = 4y(y2 − 1.1) ˙ y = ∂xH(x, y) = 4x(x2 − 0.9)
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 x y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = 4y(y2 − 1.1) ˙ y = 4x(x2 − 0.9)
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 x y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = 4y(y2 − 1.1) ˙ y = 4x(x2 − 0.9)
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 x y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = 4y(y2 − 1.1) ˙ y = 4x(x2 − 0.9)
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 x y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = 4y(y2 − 1.1) ˙ y = 4x(x2 − 0.9)−0.4y + 0.46x2y
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 x y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = 4y(y2 − 1.1) ˙ y = 4x(x2 − 0.9)−0.4y + 0.46x2y
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 x y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = 4y(y2 − 1.1) ˙ y = 4x(x2 − 0.9)−0.4y + 0.46x2y
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 x y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = 4y(y2 − 1.1) ˙ y = 4x(x2 − 0.9)−0.4y + 0.46x2y
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 x y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = 4y(y2 − 1.1) ˙ y = 4x(x2 − 0.9)−0.4y + 0.46x2y
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 x y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = 4y(y2 − 1.1) ˙ y = 4x(x2 − 0.9)−0.4y + 0.46x2y
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 x y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = 4y(y2 − 1.1) ˙ y = 4x(x2 − 0.9)−0.4y + 0.46x2y
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 x y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = 4y(y2 − 1.1) ˙ y = 4x(x2 − 0.9)−0.4y + 0.46x2y
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 x y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = 4y(y2 − 1.1) ˙ y = 4x(x2 − 0.9)−0.4y + 0.46x2y
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 x y
H(x, y) = (x2 − 0.9)2 + (y2 − 1.1)2 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = 4y(y2 − 1.1) ˙ y = 4x(x2 − 0.9)−0.4y + 0.46x2y
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 x y
Infinitesimal Hilbert’s 16th problem For a given integer n, what is the maximal number Z(n) of limit cycles a perturbed Hamiltonian vector field of the form: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = −∂yH(x, y) + εf (x, y) ˙ y = ∂xH(x, y) + εg(x, y) can have when ε → 0, with: H(x, y) a polynomial potential function of degree n + 1 f , g polynomial perturbations of degree n
Experimental Mathematics, 2011 A New Lower Bound on H(4) 2/17
Infinitesimal Hilbert’s 16th Problem
0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 x y
Infinitesimal Hilbert’s 16th problem For a given integer n, what is the maximal number Z(n) of limit cycles a perturbed Hamiltonian vector field of the form: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = −∂yH(x, y) + εf (x, y) ˙ y = ∂xH(x, y) + εg(x, y) can have when ε → 0, with: H(x, y) a polynomial potential function of degree n + 1 f , g polynomial perturbations of degree n
Experimental Mathematics, 2011
Z(n) < ∞ for all n Pessimistic upper bounds
A New Lower Bound on H(4) 2/17
A Fundamental Tool: the Poincaré-Pontryagin Theorem
0.9 1 1.1 1.2 1.3 0.9 1 1.1 1.2
h
x y ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = −∂yH(x, y) + εf (x, y) ˙ y = ∂xH(x, y) + εg(x, y)
A New Lower Bound on H(4) 3/17
A Fundamental Tool: the Poincaré-Pontryagin Theorem
0.9 1 1.1 1.2 1.3 0.9 1 1.1 1.2
h P(h)
x y Poincaré first return map P(h) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = −∂yH(x, y) + εf (x, y) ˙ y = ∂xH(x, y) + εg(x, y)
A New Lower Bound on H(4) 3/17
A Fundamental Tool: the Poincaré-Pontryagin Theorem
0.9 1 1.1 1.2 1.3 0.9 1 1.1 1.2
h P(h)
P 2(h) P 3(h)
x y Poincaré first return map P(h) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = −∂yH(x, y) + εf (x, y) ˙ y = ∂xH(x, y) + εg(x, y)
A New Lower Bound on H(4) 3/17
A Fundamental Tool: the Poincaré-Pontryagin Theorem
0.9 1 1.1 1.2 1.3 0.9 1 1.1 1.2
h P(h)
P 2(h) P 3(h)
d(h)
x y Poincaré first return map P(h) Displacement d(h) = P(h) − h ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = −∂yH(x, y) + εf (x, y) ˙ y = ∂xH(x, y) + εg(x, y)
A New Lower Bound on H(4) 3/17
A Fundamental Tool: the Poincaré-Pontryagin Theorem
0.9 1 1.1 1.2 1.3 0.9 1 1.1 1.2
h P(h)
P 2(h) P 3(h)
d(h)
x y Poincaré first return map P(h) Displacement d(h) = P(h) − h Limit cycle ⇔ isolated zero of d ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = −∂yH(x, y) + εf (x, y) ˙ y = ∂xH(x, y) + εg(x, y)
A New Lower Bound on H(4) 3/17
A Fundamental Tool: the Poincaré-Pontryagin Theorem
0.9 1 1.1 1.2 1.3 0.9 1 1.1 1.2
h P(h)
P 2(h) P 3(h)
d(h)
x y Poincaré first return map P(h) Displacement d(h) = P(h) − h Limit cycle ⇔ isolated zero of d Abelian integral I(h): ∮H−1(h) f (x, y)dy − g(x, y)dx ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = −∂yH(x, y) + εf (x, y) ˙ y = ∂xH(x, y) + εg(x, y)
A New Lower Bound on H(4) 3/17
A Fundamental Tool: the Poincaré-Pontryagin Theorem
0.9 1 1.1 1.2 1.3 0.9 1 1.1 1.2
h P(h)
P 2(h) P 3(h)
d(h)
x y Poincaré first return map P(h) Displacement d(h) = P(h) − h Limit cycle ⇔ isolated zero of d Abelian integral I(h): ∮H−1(h) f (x, y)dy − g(x, y)dx ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = −∂yH(x, y) + εf (x, y) ˙ y = ∂xH(x, y) + εg(x, y) Poincaré-Pontryagin theorem The Abelian integral I(h) 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) 3/17
A Pseudo-Hamiltonian Quartic System
Hamiltonian system: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = −4y(y2 − 1.1) ˙ y = 4x(x2 − 0.9)
−1 1 −1 1 x y
A New Lower Bound on H(4) 4/17
A Pseudo-Hamiltonian Quartic System
pseudo-Hamiltonian system: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = −4yy(y2 − 1.1) ˙ y = 4yx(x2 − 0.9)
−1 1 −1 1 x y
A New Lower Bound on H(4) 4/17
A Pseudo-Hamiltonian Quartic System
pseudo-Hamiltonian system: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = −4yy(y2 − 1.1) ˙ y = 4yx(x2 − 0.9) same geometric orbits after rescaling
−1 1 −1 1
y
A New Lower Bound on H(4) 4/17
A Pseudo-Hamiltonian Quartic System
pseudo-Hamiltonian system: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = −4yy(y2 − 1.1) + εf (x, y) ˙ y = 4yx(x2 − 0.9) + εg(x, y) same geometric orbits after rescaling ≃ perturbations without rescaling: f (x, y) y , g(x, y) y ∈ ⟨xiyj, i ⩾ 0, j ⩾ −1, i+j ⩽ 3⟩
−1 1 −1 1
y
A New Lower Bound on H(4) 4/17
A Pseudo-Hamiltonian Quartic System
pseudo-Hamiltonian system: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = −4yy(y2 − 1.1) + εf (x, y) ˙ y = 4yx(x2 − 0.9) + εg(x, y) same geometric orbits after rescaling ≃ perturbations without rescaling: f (x, y) y , g(x, y) y ∈ ⟨xiyj, i ⩾ 0, j ⩾ −1, i+j ⩽ 3⟩ Generalized Poincaré-Pontryagin theorem The generalized Abelian integral: I(h) = ∮H−1(h) f (x, y)dy − g(x, y)dx y approximates the displacement function d(h) for small ε: d(h) = ε(I(h) + O(ε)) when ε → 0
−1 1 −1 1
y
A New Lower Bound on H(4) 4/17
A Pseudo-Hamiltonian Quartic System
pseudo-Hamiltonian system: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ˙ x = −4yy(y2 − 1.1) + εf (x, y) ˙ y = 4yx(x2 − 0.9) + εg(x, y) same geometric orbits after rescaling ≃ perturbations without rescaling: f (x, y) y , g(x, y) y ∈ ⟨xiyj, i ⩾ 0, j ⩾ −1, i+j ⩽ 3⟩ Generalized Poincaré-Pontryagin theorem The generalized Abelian integral: I(h) = ∮H−1(h) f (x, y)dy − g(x, y)dx y approximates the displacement function d(h) for small ε: d(h) = ε(I(h) + O(ε)) when ε → 0
−1 1 −1 1
4×? + 2×? = ?
x y
⇒ The finiteness of Z(4) does not apply, but we still have some tools of the Hamiltonian case!
A New Lower Bound on H(4) 4/17
Choice of Perturbations
A New Lower Bound on H(4) 5/17
Choice of Perturbations
symmetry requirements linear relations from Green’s formula: ∂x
f (x,y) y
∝ ∂y
g(x,y) y
I(h) = ∮H−1(h) α00 + α20x2 + α22x2y2 + α40x4 + α04y4 y dx
A New Lower Bound on H(4) 5/17
Numerically Optimizing the Number of Zeros
▸ Find coefficients of I(h) = α00I00(h) + α20I20(h) + α22I22(h) + α40I40(h) + α04I04(h).
−0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −6 −5 −4 −3 −2 −1 1 2 3 4 h x2y2 y4 1 x2 x4 A New Lower Bound on H(4) 6/17
Numerically Optimizing the Number of Zeros
α00 =
α20 = 0.87723523612653436051 α22 = 1 α40 = 0.23742713894293038223 α04 =
−0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −2 −1 1 2 (×10−4) h A New Lower Bound on H(4) 6/17
Numerically Optimizing the Number of Zeros
α00 =
α20 = 0.87723523612653436051 α22 = 1 α40 = 0.23742713894293038223 α04 =
−0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −2 −1 1 2 (×10−4) h
−1 −0.5 0.5 1 (×10−7) h
A New Lower Bound on H(4) 6/17
Numerically Optimizing the Number of Zeros
α00 =
α20 = 0.87723523612653436051 α22 = 1 α40 = 0.23742713894293038223 α04 =
−0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −2 −1 1 2 (×10−4) h
−1 −0.5 0.5 1 (×10−7) h
A New Lower Bound on H(4) 6/17
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)
Computing Abelian Integrals
1 2 1 2
x2 y2
−1 1 −1 1
x y
0 < r (= √ h) < 0.9
A New Lower Bound on H(4) 7/17
Computing Abelian Integrals
1 2 1 2
x2 y2
−1 1 −1 1
x y
0 < r (= √ h) < 0.9
xmin = 0.9 − r √ 2 xmax = 0.9 + r √ 2 xmin = 1.1 − r √ 2 xmax = 1.1 + r √ 2
A New Lower Bound on H(4) 7/17
Computing Abelian Integrals
1 2 1 2
x2 y2
−1 1 −1 1
x y
0 < r (= √ h) < 0.9
xmin = 0.9 − r √ 2 xmax = 0.9 + r √ 2 xmin = 1.1 − r √ 2 xmax = 1.1 + r √ 2 yup(x) = √ 1.1 + √ r 2 − (x2 − 0.9)2 ydown(x) = √ 1.1 − √ r 2 − (x2 − 0.9)2 xleft(y) = √ 0.9 − √ r 2 − (y2 − 1.1)2 xright(y) = √ 0.9 + √ r 2 − (y2 − 1.1)2
A New Lower Bound on H(4) 7/17
Computing Abelian Integrals
1 2 1 2
x2 y2
−1 1 −1 1
x y
0 < r (= √ h) < 0.9
xmin = 0.9 − r √ 2 xmax = 0.9 + r √ 2 xmin = 1.1 − r √ 2 xmax = 1.1 + r √ 2 yup(x) = √ 1.1 + √ r 2 − (x2 − 0.9)2 ydown(x) = √ 1.1 − √ r 2 − (x2 − 0.9)2 xleft(y) = √ 0.9 − √ r 2 − (y2 − 1.1)2 xright(y) = √ 0.9 + √ r 2 − (y2 − 1.1)2 I(h) = ∮H−1(h) g(x, y) y dx = ∫
xmax xmin
( g(x, yup(x)) yup(x) − g(x, ydown(x)) ydown(x) ) d +∫
ymax ymin
( g(xleft(y), y) xleft(y) + g(xright(y), y) xright(y) ) y2 − 1.1 √ r 2 − (y2 − 1.1)2 dy.
A New Lower Bound on H(4) 7/17
Computing Abelian Integrals
1 2 1 2
x2 y2
−1 1 −1 1
x y
0.9 < r (= √ h) < 1.1
A New Lower Bound on H(4) 7/17
Computing Abelian Integrals
1 2 1 2
x2 y2
−1 1 −1 1
x y
0.9 < r (= √ h) < 1.1
xmin = 0.9 − r √ 2 xmax = 0.9 + r √ 2 xmin = 1.1 − r √ 2 xmax = 1.1 + r √ 2 yup(x) = √ 1.1 + √ r 2 − (x2 − 0.9)2 ydown(x) = √ 1.1 − √ r 2 − (x2 − 0.9)2 xleft(y) = √ 0.9 − √ r 2 − (y2 − 1.1)2 xright(y) = √ 0.9 + √ r 2 − (y2 − 1.1)2 I(h) = ∮H−1(h) g(x, y) y dx = ∫
xmax −xmax
( g(x, yup(x)) yup(x) − g(x, ydown(x)) ydown(x) ) d + 2 ∫
ymax ymin
g(xright(y), y)(y2 − 1.1) xright(y) √ r 2 − (y2 − 1.1)2 dy.
A New Lower Bound on H(4) 7/17
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.
K − + F G H
A New Lower Bound on H(4) 8/17
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]
K − + F G H
A New Lower Bound on H(4) 8/17
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] Some elementary operations: (P, ε) + (Q, η) ∶= (P + Q, ε + η), Example: r(t) = f (t) + g(t)
K − + F G H
A New Lower Bound on H(4) 8/17
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] Some elementary operations: (P, ε) + (Q, η) ∶= (P + Q, ε + η), (P, ε) − (Q, η) ∶= (P − Q, ε + η), Example: r(t) = f (t) + g(t) − h(t)
K − + F G H
A New Lower Bound on H(4) 8/17
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] Some elementary operations: (P, ε) + (Q, η) ∶= (P + Q, ε + η), (P, ε) − (Q, η) ∶= (P − Q, ε + η), (P, ε) ⋅ (Q, η) ∶= (PQ, ∥Q∥η + ∥P∥ε + ηε) Example: r(t) = k(t)(f (t) + g(t) − h(t))
K − + F G H
A New Lower Bound on H(4) 8/17
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] Some elementary operations: (P, ε) + (Q, η) ∶= (P + Q, ε + η), (P, ε) − (Q, η) ∶= (P − Q, ε + η), (P, ε) ⋅ (Q, η) ∶= (PQ, ∥Q∥η + ∥P∥ε + ηε) ∫0(P, ε) ∶= (∫
t 0 P(s)ds, ε)
Example: r(t) = ∫
t 0 k(s)(f (s) + g(s) − h(s))ds
K − + F G H
A New Lower Bound on H(4) 8/17
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] Some elementary operations: (P, ε) + (Q, η) ∶= (P + Q, ε + η), (P, ε) − (Q, η) ∶= (P − Q, ε + η), (P, ε) ⋅ (Q, η) ∶= (PQ, ∥Q∥η + ∥P∥ε + ηε) ∫0(P, ε) ∶= (∫
t 0 P(s)ds, ε)
Example: r(t) = ∫
t 0 k(s)(f (s) + g(s) − h(s))ds ÷ ?
√ ?
K − + F G H
A New Lower Bound on H(4) 8/17
Banach Fixed-Point Theorem for A Posteriori Validation
▸ Fixed-point equation T ⋅ ϕ = ϕ with T contracting, General scheme ▸ Approximation ϕ○ to exact solution ϕ⋆, ▸ Compute a posteriori error bounds with Banach theorem.
A New Lower Bound on H(4) 9/17
Banach Fixed-Point Theorem for A Posteriori Validation
▸ Fixed-point equation T ⋅ ϕ = ϕ with T contracting, General scheme ▸ Approximation ϕ○ to exact solution ϕ⋆, ▸ 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○) 1 + λ ⩽ d(ϕ○, ϕ⋆) ⩽ d(ϕ○, T ⋅ ϕ○) 1 − λ .
A New Lower Bound on H(4) 9/17
Banach Fixed-Point Theorem for A Posteriori Validation
▸ Fixed-point equation T ⋅ ϕ = ϕ with T contracting, General scheme ▸ Approximation ϕ○ to exact solution ϕ⋆, ▸ 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○) 1 + λ ⩽ d(ϕ○, ϕ⋆) ⩽ d(ϕ○, T ⋅ ϕ○) 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
Banach Fixed-Point Theorem for A Posteriori Validation
▸ Fixed-point equation T ⋅ ϕ = ϕ with T contracting, General scheme ▸ Approximation ϕ○ to exact solution ϕ⋆, ▸ 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○) 1 + λ ⩽ d(ϕ○, ϕ⋆) ⩽ d(ϕ○, T ⋅ ϕ○) 1 − λ . ▸ Newton’s method = reformulate F ⋅ ϕ = 0 as T ⋅ ϕ = ϕ with: T ⋅ ϕ = ϕ − A ⋅ F ⋅ ϕ, A ≈ (DF(ϕ○))−1 and check T is contracting.
A New Lower Bound on H(4) 9/17
Banach Fixed-Point Theorem for A Posteriori Validation
▸ Fixed-point equation T ⋅ ϕ = ϕ with T contracting, General scheme ▸ Approximation ϕ○ to exact solution ϕ⋆, ▸ 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○) 1 + λ ⩽ d(ϕ○, ϕ⋆) ⩽ d(ϕ○, T ⋅ ϕ○) 1 − λ . ▸ Newton’s method = reformulate F ⋅ ϕ = 0 as T ⋅ ϕ = ϕ with: T ⋅ ϕ = ϕ − A ⋅ F ⋅ ϕ, A ≈ (DF(ϕ○))−1 and check T is contracting. ▸ Applications to numerous function space problems.
A New Lower Bound on H(4) 9/17
Division of RPAs
▸ Approximation ϕ○(x) of ϕ⋆ = x2/ydown(x) using Chebyshev interpolation:
0.6 0.7 0.8 0.9 1 1.1 1.2 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.6 0.7 0.8 0.9 1 1.1 1.2 0.5 1 1.5 2
A New Lower Bound on H(4) 10/17
Division of RPAs
▸ Approximation ϕ○(x) of ϕ⋆ = x2/ydown(x) using Chebyshev interpolation:
0.6 0.7 0.8 0.9 1 1.1 1.2 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.6 0.7 0.8 0.9 1 1.1 1.2 0.5 1 1.5 2
A New Lower Bound on H(4) 10/17
Division of RPAs
▸ Approximation ϕ○(x) of ϕ⋆ = x2/ydown(x) using Chebyshev interpolation:
0.6 0.7 0.8 0.9 1 1.1 1.2 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.6 0.7 0.8 0.9 1 1.1 1.2 0.5 1 1.5 2
A New Lower Bound on H(4) 10/17
Division of RPAs
▸ Approximation ϕ○(x) of ϕ⋆ = x2/ydown(x) using Chebyshev interpolation:
0.6 0.7 0.8 0.9 1 1.1 1.2 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.6 0.7 0.8 0.9 1 1.1 1.2 0.5 1 1.5 2
A New Lower Bound on H(4) 10/17
Division of RPAs
▸ Approximation ϕ○(x) of ϕ⋆ = x2/ydown(x) using Chebyshev interpolation:
0.6 0.7 0.8 0.9 1 1.1 1.2 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.6 0.7 0.8 0.9 1 1.1 1.2 0.5 1 1.5 2
▸ Newton-like operator T with unique fixed point ϕ⋆ =
x2 ydown(x):
T ⋅ ϕ = ϕ − ψ(ydownϕ − x2) ψ(x) ≈ 1/ydown(x)
A New Lower Bound on H(4) 10/17
Division of RPAs
▸ Approximation ϕ○(x) of ϕ⋆ = x2/ydown(x) using Chebyshev interpolation:
0.6 0.7 0.8 0.9 1 1.1 1.2 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.6 0.7 0.8 0.9 1 1.1 1.2 0.5 1 1.5 2
▸ Newton-like operator T with unique fixed point ϕ⋆ =
x2 ydown(x):
T ⋅ ϕ = ϕ − ψ(ydownϕ − x2) ψ(x) ≈ 1/ydown(x) ▸ Is T contracting? ∥DT∥ = ∥1 − ψydown∥ = λ < 1
A New Lower Bound on H(4) 10/17
Division of RPAs
▸ Approximation ϕ○(x) of ϕ⋆ = x2/ydown(x) using Chebyshev interpolation:
0.6 0.7 0.8 0.9 1 1.1 1.2 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.6 0.7 0.8 0.9 1 1.1 1.2 0.5 1 1.5 2
▸ Newton-like operator T with unique fixed point ϕ⋆ =
x2 ydown(x):
T ⋅ ϕ = ϕ − ψ(ydownϕ − x2) ψ(x) ≈ 1/ydown(x) ▸ Is T contracting? ∥DT∥ = ∥1 − ψydown∥ = λ < 1 ▸ Apply the Banach fixed-point theorem: ∥ϕ○ − T ⋅ ϕ○∥ = ∥ψ(ydownϕ○ − x2)∥ ⩽ η
A New Lower Bound on H(4) 10/17
Division of RPAs
▸ Approximation ϕ○(x) of ϕ⋆ = x2/ydown(x) using Chebyshev interpolation:
0.6 0.7 0.8 0.9 1 1.1 1.2 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.6 0.7 0.8 0.9 1 1.1 1.2 0.5 1 1.5 2
▸ Newton-like operator T with unique fixed point ϕ⋆ =
x2 ydown(x):
T ⋅ ϕ = ϕ − ψ(ydownϕ − x2) ψ(x) ≈ 1/ydown(x) ▸ Is T contracting? ∥DT∥ = ∥1 − ψydown∥ = λ < 1 ▸ Apply the Banach fixed-point theorem: ∥ϕ○ − T ⋅ ϕ○∥ = ∥ψ(ydownϕ○ − x2)∥ ⩽ η ⇒ ∥ϕ○ − ϕ⋆∥ ⩽ η/(1 − λ) = ε+
A New Lower Bound on H(4) 10/17
Square Root of a RPA
▸ ϕ○(x) ≈ √ f (x) where f (x) = 0.8 − (x2 − 0.9)2.
A New Lower Bound on H(4) 11/17
Square Root of a RPA
▸ ϕ○(x) ≈ √ f (x) where f (x) = 0.8 − (x2 − 0.9)2. ▸ ϕ⋆ = √ f unique fixed point of: T ⋅ ϕ = ϕ − ψ 2 (ϕ2 − f ) ψ(x) ≈ 1/ϕ○(x) ≈ 1/ √ f (x)
A New Lower Bound on H(4) 11/17
Square Root of a RPA
▸ ϕ○(x) ≈ √ f (x) where f (x) = 0.8 − (x2 − 0.9)2. ▸ ϕ⋆ = √ f unique fixed point of: T ⋅ ϕ = ϕ − ψ 2 (ϕ2 − f ) ψ(x) ≈ 1/ϕ○(x) ≈ 1/ √ f (x) ▸ Is T contracting? ∥DT(ϕ)∥ = ∥1−ψϕ∥ ⩽ ∥1−ψϕ○∥+∥ψ∥∥ϕ−ϕ○∥
A New Lower Bound on H(4) 11/17
Square Root of a RPA
▸ ϕ○(x) ≈ √ f (x) where f (x) = 0.8 − (x2 − 0.9)2. ▸ ϕ⋆ = √ f unique fixed point of: T ⋅ ϕ = ϕ − ψ 2 (ϕ2 − f ) ψ(x) ≈ 1/ϕ○(x) ≈ 1/ √ f (x) ▸ Is T contracting? λ = sup
∥ϕ−ϕ○∥⩽r
∥DT(ϕ)∥ ⩽ ∥1 − ψϕ○∥ + ∥ψ∥r
0.2 0.4 0.6 0.8 1 0.5 1 1.5 r 0.2 0.4 0.6 0.8 1 0.5 1 1.5 r
A New Lower Bound on H(4) 11/17
Square Root of a RPA
▸ ϕ○(x) ≈ √ f (x) where f (x) = 0.8 − (x2 − 0.9)2. ▸ ϕ⋆ = √ f unique fixed point of: T ⋅ ϕ = ϕ − ψ 2 (ϕ2 − f ) ψ(x) ≈ 1/ϕ○(x) ≈ 1/ √ f (x) ▸ Is T contracting? ▸ Stable neighborhood for ϕ○: λ = sup
∥ϕ−ϕ○∥⩽r
∥DT(ϕ)∥ ⩽ ∥1 − ψϕ○∥ + ∥ψ∥r ∥ϕ○ − T ⋅ ϕ○∥ + λr ⩽ r
0.2 0.4 0.6 0.8 1 0.5 1 1.5 r 0.2 0.4 0.6 0.8 1 0.5 1 1.5 r
A New Lower Bound on H(4) 11/17
Square Root of a RPA
▸ ϕ○(x) ≈ √ f (x) where f (x) = 0.8 − (x2 − 0.9)2. ▸ ϕ⋆ = √ f unique fixed point of: T ⋅ ϕ = ϕ − ψ 2 (ϕ2 − f ) ψ(x) ≈ 1/ϕ○(x) ≈ 1/ √ f (x) ▸ Is T contracting? ▸ Stable neighborhood for ϕ○: λ = sup
∥ϕ−ϕ○∥⩽r
∥DT(ϕ)∥ ⩽ ∥1 − ψϕ○∥ + ∥ψ∥r ∥ψ(ϕ○2 − f )/2∥ + r(∥1 − ψϕ○∥ + ∥ψ∥r) ⩽ r
0.2 0.4 0.6 0.8 1 0.5 1 1.5 r 0.2 0.4 0.6 0.8 1 0.5 1 1.5 r
A New Lower Bound on H(4) 11/17
Square Root of a RPA
▸ ϕ○(x) ≈ √ f (x) where f (x) = 0.8 − (x2 − 0.9)2. ▸ ϕ⋆ = √ f unique fixed point of: T ⋅ ϕ = ϕ − ψ 2 (ϕ2 − f ) ψ(x) ≈ 1/ϕ○(x) ≈ 1/ √ f (x) ▸ Is T contracting? ▸ Stable neighborhood for ϕ○: λ = sup
∥ϕ−ϕ○∥⩽r
∥DT(ϕ)∥ ⩽ ∥1 − ψϕ○∥ + ∥ψ∥r ∥ψ(ϕ○2 − f )/2∥ + r(∥1 − ψϕ○∥ + ∥ψ∥r) ⩽ r
0.2 0.4 0.6 0.8 1 0.5 1 1.5 r 0.2 0.4 0.6 0.8 1 0.5 1 1.5 r
▸ Apply the Banach fixed-point theorem!
A New Lower Bound on H(4) 11/17
Rigorous Computation of an Abelian Integral
Using Degree N = 10
▸ √ 0.8 − (x2 − 0.9)2:
0.6 0.8 1 1.2 −1 1 ·10−5
A New Lower Bound on H(4) 12/17
Rigorous Computation of an Abelian Integral
Using Degree N = 10
▸ √ 0.8 − (x2 − 0.9)2:
0.6 0.8 1 1.2 −1 1 ·10−5
▸ ydown(x) = √ 1.1 − √ 0.8 − (x2 − 0.9)2:
0.6 0.8 1 1.2 −1 1 ·10−5
A New Lower Bound on H(4) 12/17
Rigorous Computation of an Abelian Integral
Using Degree N = 10
▸ √ 0.8 − (x2 − 0.9)2:
0.6 0.8 1 1.2 −1 1 ·10−5
▸ ydown(x) = √ 1.1 − √ 0.8 − (x2 − 0.9)2:
0.6 0.8 1 1.2 −1 1 ·10−5
▸ x2/ydown(x) = x2/ydown(x) = √ 1.1 − √ 0.8 − (x2 − 0.9)2:
0.6 0.8 1 1.2 −1 −0.5 0.5 1 ·10−4
A New Lower Bound on H(4) 12/17
Validation of Our Result
−0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −2 −1 1 2 (×10−4) h
−1 −0.5 0.5 1 (×10−7) h
A New Lower Bound on H(4) 13/17
Validation of Our Result
−0.5 −0.45 −0.4 −0.35 −0.3 −0.25 −2 −1 1 2 (×10−4) N = 12 N = 16 N = 22 N = 24 h
−1 −0.5 0.5 1 (×10−7) N = 80 N = 100 N = 130 N = 240 h
A New Lower Bound on H(4) 13/17
Validation of Our Result
A New Lower Bound on H(4) 13/17
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)
Wronskian and Extended Chebyshev Systems
(f0, f1, . . . , fn) extended Chebyshev system if all combination α0f0 + ⋅ ⋅ ⋅ + αifi has at most i zeros, for 0 ⩽ i ⩽ n.
A New Lower Bound on H(4) 14/17
Wronskian and Extended Chebyshev Systems
(f0, f1, . . . , fn) extended Chebyshev system if all combination α0f0 + ⋅ ⋅ ⋅ + αifi has at most i zeros, for 0 ⩽ i ⩽ n. ⇔ W0(x), . . . , Wn(x) ≠ 0 for all x: Wi(x) =
f1(x) . . . fi(x) f ′
0 (x)
f ′
1 (x)
. . . f ′
i (x)
⋮ ⋮ ⋱ ⋮ f (i) (x) f (i)
1
(x) . . . f (i)
i
(x)
14/17
Wronskian and Extended Chebyshev Systems
(f0, f1, . . . , fn) extended Chebyshev system if all combination α0f0 + ⋅ ⋅ ⋅ + αifi has at most i zeros, for 0 ⩽ i ⩽ n. ⇔ W0(x), . . . , Wn(x) ≠ 0 for all x: Wi(x) =
f1(x) . . . fi(x) f ′
0 (x)
f ′
1 (x)
. . . f ′
i (x)
⋮ ⋮ ⋱ ⋮ f (i) (x) f (i)
1
(x) . . . f (i)
i
(x)
⇒ α0f0 + ⋅ ⋅ ⋅ + αnfn has at most n + 1 zeros.
A New Lower Bound on H(4) 14/17
Wronskian and Extended Chebyshev Systems
(f0, f1, . . . , fn) extended Chebyshev system if all combination α0f0 + ⋅ ⋅ ⋅ + αifi has at most i zeros, for 0 ⩽ i ⩽ n. ⇔ W0(x), . . . , Wn(x) ≠ 0 for all x: Wi(x) =
f1(x) . . . fi(x) f ′
0 (x)
f ′
1 (x)
. . . f ′
i (x)
⋮ ⋮ ⋱ ⋮ f (i) (x) f (i)
1
(x) . . . f (i)
i
(x)
⇒ α0f0 + ⋅ ⋅ ⋅ + αnfn has at most n + 1 zeros. In our case, W0, W1, W2, W3 never vanish on small ovals and W4 does once. ⇒ at most 5 zeros on small ovals.
A New Lower Bound on H(4) 14/17
Wronskian and Extended Chebyshev Systems
(f0, f1, . . . , fn) extended Chebyshev system if all combination α0f0 + ⋅ ⋅ ⋅ + αifi has at most i zeros, for 0 ⩽ i ⩽ n. ⇔ W0(x), . . . , Wn(x) ≠ 0 for all x: Wi(x) =
f1(x) . . . fi(x) f ′
0 (x)
f ′
1 (x)
. . . f ′
i (x)
⋮ ⋮ ⋱ ⋮ f (i) (x) f (i)
1
(x) . . . f (i)
i
(x)
⇒ α0f0 + ⋅ ⋅ ⋅ + αnfn has at most n + 1 zeros. In our case, W0, W1, W2, W3 never vanish on small ovals and W4 does once. ⇒ at most 5 zeros on small ovals. but no rigorous proof!
A New Lower Bound on H(4) 14/17
Creative Telescoping for Abelian Integrals
∮H=h f (x, y)dy − g(x, y)dx y = ∬H⩽h (∂x f (x, y) y + ∂y g(x, y) y ) dxdy
A New Lower Bound on H(4) 15/17
Creative Telescoping for Abelian Integrals
∮H=h f (x, y)dy − g(x, y)dx y = ∬R2 (∂x f (x, y) y + ∂y g(x, y) y ) 1h−H(x, y, h)dxdy
A New Lower Bound on H(4) 15/17
Creative Telescoping for Abelian Integrals
∮H=h f (x, y)dy − g(x, y)dx y = ∬R2 (∂x f (x, y) y + ∂y g(x, y) y ) 1h−H(x, y, h)dxdy ∂x
f (x,y) y
+ ∂y
g(x,y) y
satisfies a holonomic system (= PDEs with polynomial coefficients)
A New Lower Bound on H(4) 15/17
Creative Telescoping for Abelian Integrals
∮H=h f (x, y)dy − g(x, y)dx y = ∬R2 (∂x f (x, y) y + ∂y g(x, y) y ) 1h−H(x, y, h)dxdy ∂x
f (x,y) y
+ ∂y
g(x,y) y
satisfies a holonomic system (= PDEs with polynomial coefficients) Hence, so does (∂x
f (x,y) y
+ ∂y
g(x,y) y
) 1h−H(x, y, h).
A New Lower Bound on H(4) 15/17
Creative Telescoping for Abelian Integrals
∮H=h f (x, y)dy − g(x, y)dx y = ∬R2 (∂x f (x, y) y + ∂y g(x, y) y ) 1h−H(x, y, h)dxdy ∂x
f (x,y) y
+ ∂y
g(x,y) y
satisfies a holonomic system (= PDEs with polynomial coefficients) Hence, so does (∂x
f (x,y) y
+ ∂y
g(x,y) y
) 1h−H(x, y, h). By creative telescoping, one finds a differential equation for I(h).
(X0 = 0.9, Y0 = 1.1)
A New Lower Bound on H(4) 15/17
Creative Telescoping for Abelian Integrals
Singularities
Rigorous Polynomial Approximations at ordinary points1
Linear Ordinary Differential Equations. ACM Transactions on Mathematical Software (TOMS), 2018. A New Lower Bound on H(4) 16/17
Creative Telescoping for Abelian Integrals
Singularities
Rigorous Polynomial Approximations at ordinary points1 But singularities!
Linear Ordinary Differential Equations. ACM Transactions on Mathematical Software (TOMS), 2018. A New Lower Bound on H(4) 16/17
Creative Telescoping for Abelian Integrals
Singularities
Rigorous Polynomial Approximations at ordinary points1 But singularities! Analytic at h = 0 ⇒ Initial conditions with Laplace transform
Linear Ordinary Differential Equations. ACM Transactions on Mathematical Software (TOMS), 2018. A New Lower Bound on H(4) 16/17
Creative Telescoping for Abelian Integrals
Singularities
Rigorous Polynomial Approximations at ordinary points1 But singularities! Analytic at h = 0 ⇒ Initial conditions with Laplace transform log singularities at h = X 2
0 ...
⇒ Other rigorous approximation tools.
Linear Ordinary Differential Equations. ACM Transactions on Mathematical Software (TOMS), 2018. A New Lower Bound on H(4) 16/17
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)
Conclusion and Future Work
A rigorous proof of H(4) ⩾ 24. Exploring properties of the Wronskian using symbolic-numeric tools and Creative Telescoping.
A New Lower Bound on H(4) 17/17
Conclusion and Future Work
A rigorous proof of H(4) ⩾ 24. Exploring properties of the Wronskian using symbolic-numeric tools and Creative Telescoping. Ongoing work: formal proof in COQ, with D. Pous. Ongoing work: Rigorous uniform approximation of Abelian integrals.
A New Lower Bound on H(4) 17/17
Conclusion and Future Work
A rigorous proof of H(4) ⩾ 24. Exploring properties of the Wronskian using symbolic-numeric tools and Creative Telescoping. Ongoing work: formal proof in COQ, with D. Pous. Ongoing work: Rigorous uniform approximation of Abelian integrals. Future work: Generalization to other systems?
A New Lower Bound on H(4) 17/17