On complex-tangential curves and homogeneous polynomials on the unit sphere on C~2

摘要

For d = l + m with I, m ≥ 1 integers, the monomial πl,m(z,ω)= d~d/l~lm~m~(1/2)z~lω~m (1.1) has maximum modulus one on the unit ball B_2 of C~2. The solution of πl,m (z,ω) = 1 on partial derivB_2 is easily seen to be a closed curve on the unit sphere partial derivB_2 given by γl, m(t) = (l/d~(1/2) e~(it) m/1~(1/2),m/d~(1/2) e~(-it)l/m~(1/2)). (1.2) The curve γl,m(t) is complex-tangential in the sense that < γi,m, γl,m ＞ = 0. See [4] for more on the complex-tangential curves on the unit sphere of C~n. In this short paper, we propose and give a partial answer to Conjecture A If a homogeneous polynomial π on C~2 admits a closed complex-tangential analytic curve γ on partial derivB_2 with π(γ(t)) = 1 then π reduces to a monomial πl,m with l,m ≥ 1 integers by a unitary change of variables.