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On complex-tangential curves on the unit sphere on \(\mathbb{C}^2\) and homogeneous polynomials. (English) Zbl 0968.32003

Let \(\partial B_2\) denote the unit sphere of \(\mathbb{C}^2\). A curve \(\gamma: (a,b)\to \partial B_2\) is called complex-tangential if \(\langle \gamma' (t),\gamma(t) \rangle=0\) for all \(t\in(a,b)\).
In the paper it is proved that a closed complex-tangential \(\mathbb{C}^2\)-curve \(\gamma\) of constant curvature on \(\partial B_2\) is unitarily equivalent to \[ \gamma_{l,m} (t)=\bigl(\sqrt {l/ d} e^{it \sqrt{m/l}}, \sqrt{m/l} e^{-it \sqrt{l/m}}\bigr) \] where \(d=l+m\), \(l,m \geq 1\) integers. The author proposes and gives a partial answer to the following conjecture: if a homogeneous polynomial \(\pi\) on \(\mathbb{C}^2\) admits a closed complex-tangential analytic curve \(\gamma\) on \(\partial B_2\) with \(\pi( \gamma(t)) =1\) then \(\pi\) reduces to a monomial \(\pi_{l,m}\) with \(l,m\geq 1\) integers by a unitary change of variables.

MSC:

32C05 Real-analytic manifolds, real-analytic spaces
14P15 Real-analytic and semi-analytic sets
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