files / problems / hilbert16-limit-cycles / compute / q1 / e-bezout-bautin / bezout_samples.json
{
"lemma": "A real polynomial map \u03a6=(p,q) of degree \u2264 m has at most m^2 regular real preimages of any point (B\u00e9zout + regularity).",
"remark": "Answers arXiv:2604.12883v1 Remark 4 at the one-step sheet count: non-separable maps cannot beat m^2 regular sheets.",
"seed": 1,
"violations": 0,
"max_regular_by_m": {
"2": 4,
"3": 9,
"4": 16
},
"n_samples": 44,
"n_shared_component": 0,
"samples": [
{
"kind": "chebyshev",
"m": 2,
"target": [
0.5,
0.3
],
"status": "ok",
"count": 4,
"ceiling": 4,
"ok": true
},
{
"kind": "chebyshev",
"m": 3,
"target": [
0.4,
-0.2
],
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"count": 9,
"ceiling": 9,
"ok": true
},
{
"kind": "chebyshev",
"m": 4,
"target": [
0.2,
0.6
],
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"count": 16,
"ceiling": 16,
"ok": true
},
{
"kind": "two_quadrics",
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2,
0
],
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"ceiling": 4,
"ok": true,
"exact_count": 4
},
{
"kind": "complex_square",
"m": 2,
"target": [
1,
0
],
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"count": 2,
"ceiling": 4,
"ok": true,
"exact_count": 2
},
{
"kind": "nonseparable_cubic",
"m": 3,
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1,
-2
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"ok": true
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{
"kind": "random",
"m": 2,
"target": [
-1,
2
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"status": "ok",
"count": 4,
"ceiling": 4,
"ok": true,
"index": 0,
"p": "-2*u**2 - 3*u*v + 3*u + 2*v**2",
"q": "-u**2 + 3*u*v - u - 2*v**2 + 3*v + 2",
"deg_p": 2,
"deg_q": 2
},
{
"kind": "random",
"m": 2,
"target": [
0,
1
],
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"ceiling": 4,
"ok": true,
"index": 1,
"p": "-3*u**2 - 3*u*v + v**2 - v - 2",
"q": "-u**2 + 2*u*v + 2*v**2 + 2*v + 3",
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},
{
"kind": "random",
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-2,
1
],
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"ok": true,
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"q": "-2*u**2 - 2*u + 3*v**2 - v - 1",
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},
{
"kind": "random",
"m": 2,
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2,
-1
],
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"p": "3*u**2 - 3*u*v - 2*v - 3",
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{
"kind": "random",
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-1,
1
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"ok": true,
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"q": "2*u**2 + u - v**2 - 3*v - 1",
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},
{
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1,
2
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"p": "2*u*v - 3*u - v**2 + 3*v + 2",
"q": "u**2 - 3*u - v**2 + 3",
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{
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-2,
-2
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"p": "2*u**2 - u*v - 2*u + 2*v**2 + v - 2",
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{
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0,
-2
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{
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2,
1
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"p": "3*u**2 + 2*u*v + 3*u + v**2 + 2*v - 2",
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{
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2,
2
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{
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0,
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"q": "-3*u**2 + 3*u*v + u + v**2 - v + 2",
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{
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"p": "3*u**2 + 2*u + 3*v**2 + 3*v",
"q": "2*u**2 + u*v - 2*u - 2*v**2 + v - 3",
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{
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{
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{
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{
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{
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"q": "-3*u**4 - 3*u**3 + 2*u**2*v**2 - 2*u**2*v + 3*u**2 + u*v**3 + u*v - 3*u + 3*v**4 + 2*v**2 + 3*v + 2",
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{
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{
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}
]
}