Surfaces containing two circles through each point
Abstract

the set \(\{p+q:p\in \alpha ,q\in \beta \}\), where \(\alpha ,\beta \) are two circles in \(\mathbb {R}^3\);

the set \(\{2\frac{p \times q^{}}{p+q^2}:p\in \alpha ,q\in \beta ,p+q\ne 0\}\), where \(\alpha ,\beta \) are circles in the unit sphere \({S}^2\);

the set \(\{(x,y,z): Q(x,y,z,x^2+y^2+z^2)=0\}\), where \(Q\in \mathbb {R}[x,y,z,t]\) has degree 2 or 1.
The proof uses a new factorization technique for quaternionic polynomials.
Mathematics Subject Classification
51B10 13F15 16H05Notes
Acknowledgements
This results have been presented at Moscow Mathematical Society seminar, SFB “Discretization in geometry and dynamics” colloquim in Berlin, Ya. Sinai—G. Margulis conference, and the conference “Perspectives in real geometry” in Luminy. The authors are especially grateful to A. Pakharev for joint numerical experiments, related studies [26], and for pointing out that numerous surfaces we tried to invent all have form (4). The authors are grateful to N. Lubbes and L. Shi for parts of Fig. 1, to A. Gaifullin and S. Ivanov for finding gaps in earlier versions of the proofs, to A. Bobenko, J. Capco, S. Galkin, A. Kanunnikov, O. Karpenkov, A. Klyachko, J. Kollár, W. Kühnel, A. Kuznetsov, N. Moshchevitin, S. Orevkov, F. Petrov, R. Pignatelli, F. Polizzi, H. Pottmann, G. Robinson, I. Sabitov, J. Schicho, K. Shramov, S. Tikhomirov, V. Timorin, M. Verbitsky, E. Vinberg, J. Zahl, S. Zubė for useful discussions. The first author is grateful to King Abdullah University of Science and Technology for hosting him during the start of the work over the paper.
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