files / problems / hilbert16-limit-cycles / compute / q1 / c-chebyshev / certs / nine_rectangles.json
{
"rho2": [
1,
4
],
"T3_prime_factorization": "3*(2*t - 1)*(2*t + 1)",
"T3_prime_equals_3_(2t-1)(2t+1)": true,
"evaluations": {
"T3(1)": [
1,
1
],
"T3(-1)": [
-1,
1
],
"T3(1/2)": [
-1,
1
],
"T3(-1/2)": [
1,
1
],
"T3(0)": [
0,
1
],
"T3'(1/2)": [
0,
1
],
"T3'(-1/2)": [
0,
1
],
"T3'(0)": [
-3,
1
],
"T3'(3/4)": [
15,
4
],
"T3'(1/4)": [
-9,
4
],
"T3'(-3/4)": [
15,
4
]
},
"intervals": [
{
"name": "I1",
"left": [
1,
2
],
"right": [
1,
1
],
"T3_left": [
-1,
1
],
"T3_right": [
1,
1
],
"T3_prime_sign": 1,
"T3_prime_roots_in_open": 0,
"T3_pm_half_sign_changes": true,
"preimage_of_[-1/2,1/2]_strictly_inside": true,
"diffeo_onto_(-1,1)": true
},
{
"name": "I2",
"left": [
-1,
2
],
"right": [
1,
2
],
"T3_left": [
1,
1
],
"T3_right": [
-1,
1
],
"T3_prime_sign": -1,
"T3_prime_roots_in_open": 0,
"T3_pm_half_sign_changes": true,
"preimage_of_[-1/2,1/2]_strictly_inside": true,
"diffeo_onto_(-1,1)": true
},
{
"name": "I3",
"left": [
-1,
1
],
"right": [
-1,
2
],
"T3_left": [
-1,
1
],
"T3_right": [
1,
1
],
"T3_prime_sign": 1,
"T3_prime_roots_in_open": 0,
"T3_pm_half_sign_changes": true,
"preimage_of_[-1/2,1/2]_strictly_inside": true,
"diffeo_onto_(-1,1)": true
}
],
"level_curve": {
"equation": "T3(u)^2 + T3(v)^2 - 1/4 = 0",
"cleared": "4 T3(u)^2 + 4 T3(v)^2 - 1",
"total_degree": 6,
"factor_over_Q": "64*u**6 - 96*u**4 + 36*u**2 + 64*v**6 - 96*v**4 + 36*v**2 - 1",
"n_Q_factors": 1,
"nine_oval_factors_over_Q": false
},
"rectangles": [
{
"i": 1,
"j": 1,
"Iu": "I1",
"Iv": "I1",
"Phi_diffeo_onto_(-1,1)^2": true,
"jacobian_det": "T3'(u) T3'(v)",
"jacobian_nonzero": true,
"circle_in_[-1/2,1/2]^2": true,
"circle_compactly_in_(-1,1)^2": true,
"one_compact_oval": true,
"oval_strictly_inside_open_rectangle": true
},
{
"i": 1,
"j": 2,
"Iu": "I1",
"Iv": "I2",
"Phi_diffeo_onto_(-1,1)^2": true,
"jacobian_det": "T3'(u) T3'(v)",
"jacobian_nonzero": true,
"circle_in_[-1/2,1/2]^2": true,
"circle_compactly_in_(-1,1)^2": true,
"one_compact_oval": true,
"oval_strictly_inside_open_rectangle": true
},
{
"i": 1,
"j": 3,
"Iu": "I1",
"Iv": "I3",
"Phi_diffeo_onto_(-1,1)^2": true,
"jacobian_det": "T3'(u) T3'(v)",
"jacobian_nonzero": true,
"circle_in_[-1/2,1/2]^2": true,
"circle_compactly_in_(-1,1)^2": true,
"one_compact_oval": true,
"oval_strictly_inside_open_rectangle": true
},
{
"i": 2,
"j": 1,
"Iu": "I2",
"Iv": "I1",
"Phi_diffeo_onto_(-1,1)^2": true,
"jacobian_det": "T3'(u) T3'(v)",
"jacobian_nonzero": true,
"circle_in_[-1/2,1/2]^2": true,
"circle_compactly_in_(-1,1)^2": true,
"one_compact_oval": true,
"oval_strictly_inside_open_rectangle": true
},
{
"i": 2,
"j": 2,
"Iu": "I2",
"Iv": "I2",
"Phi_diffeo_onto_(-1,1)^2": true,
"jacobian_det": "T3'(u) T3'(v)",
"jacobian_nonzero": true,
"circle_in_[-1/2,1/2]^2": true,
"circle_compactly_in_(-1,1)^2": true,
"one_compact_oval": true,
"oval_strictly_inside_open_rectangle": true
},
{
"i": 2,
"j": 3,
"Iu": "I2",
"Iv": "I3",
"Phi_diffeo_onto_(-1,1)^2": true,
"jacobian_det": "T3'(u) T3'(v)",
"jacobian_nonzero": true,
"circle_in_[-1/2,1/2]^2": true,
"circle_compactly_in_(-1,1)^2": true,
"one_compact_oval": true,
"oval_strictly_inside_open_rectangle": true
},
{
"i": 3,
"j": 1,
"Iu": "I3",
"Iv": "I1",
"Phi_diffeo_onto_(-1,1)^2": true,
"jacobian_det": "T3'(u) T3'(v)",
"jacobian_nonzero": true,
"circle_in_[-1/2,1/2]^2": true,
"circle_compactly_in_(-1,1)^2": true,
"one_compact_oval": true,
"oval_strictly_inside_open_rectangle": true
},
{
"i": 3,
"j": 2,
"Iu": "I3",
"Iv": "I2",
"Phi_diffeo_onto_(-1,1)^2": true,
"jacobian_det": "T3'(u) T3'(v)",
"jacobian_nonzero": true,
"circle_in_[-1/2,1/2]^2": true,
"circle_compactly_in_(-1,1)^2": true,
"one_compact_oval": true,
"oval_strictly_inside_open_rectangle": true
},
{
"i": 3,
"j": 3,
"Iu": "I3",
"Iv": "I3",
"Phi_diffeo_onto_(-1,1)^2": true,
"jacobian_det": "T3'(u) T3'(v)",
"jacobian_nonzero": true,
"circle_in_[-1/2,1/2]^2": true,
"circle_compactly_in_(-1,1)^2": true,
"one_compact_oval": true,
"oval_strictly_inside_open_rectangle": true
}
],
"nine_ovals": true,
"proof": "T3'=3(2t-1)(2t+1) has no zero in any open I_k; T3(I_k)=(-1,1) by endpoint values \u00b11 and strict monotonicity; \u03a6 is a product diffeomorphism on each I_i\u00d7I_j; the circle of radius 1/2 is compactly in (-1,1)^2; each preimage is therefore one compact oval strictly inside its rectangle. The degree-6 curve does not split into nine factors over Q; the ovals are real components."
}