Two-flag degenerations and real circles tangent to three conics
Abstract
Three general plane conics admit $184$ complex tangent circles, and it had been conjectured that at most $136$ of them could be real. We construct an explicit strongly general triple of smooth conics over ℚ with exactly $160$ real tangent circles; the same count therefore occurs on a nonempty Euclidean chamber. The construction combines a fourfold splitting theorem for two flagged double-line degenerations with exact Sturm--Tarski, elimination, and interval certificates. We also show that the Grothendieck--Witt-valued count is 92ℍ and express its real local signs, up to a fixed orientation convention, in terms of curvature differences, residual intersection divisors, and contact normals. The exact-arithmetic code and certificate data are archived in the accompanying repository.
Related
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- Quartic polynomial differential systems with three disjoint invariant circles
- Positivity of simplicial volume and uniqueness for the twelve-vertex flag three-spheres
- Enumerative geometry of conics and double covers on projective complete intersections
- Khayyam's Choices of Conics
BibTeX
@misc{liu2026twoflag,
title = {{Two-flag degenerations and real circles tangent to three conics}},
author = {Haoyang Liu and Tianle Liu and Guorui Xu and Fan Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26285},
eprint = {2609.26285},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26285},
}
RIS
TY - UNPB
TI - Two-flag degenerations and real circles tangent to three conics
AU - Haoyang Liu
AU - Tianle Liu
AU - Guorui Xu
AU - Fan Yang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.26285
UR - https://arxiv.org/abs/2609.26285
AB - Three general plane conics admit $184$ complex tangent circles, and it had been conjectured that at most $136$ of them could be real. We construct an explicit strongly general triple of smooth conics over ℚ with exactly $160$ real tangent circles; the same count therefore occurs on a nonempty Euclidean chamber. The construction combines a fourfold splitting theorem for two flagged double-line degenerations with exact Sturm--Tarski, elimination, and interval certificates. We also show that the Grothendieck--Witt-valued count is 92ℍ and express its real local signs, up to a fixed orientation convention, in terms of curvature differences, residual intersection divisors, and contact normals. The exact-arithmetic code and certificate data are archived in the accompanying repository.
ER -