Proceedings of the Japan Academy, Series A, Mathematical Sciences

On complex-tangential curves on the unit sphere on $\mathbf {C}^2$ and homogeneous polynomials

Hong Oh Kim

Full-text not supplied by publisher

Abstract

We show that a closed complex-tangential $C^2$-curve $\gamma$ of constannt curvature on the unit sphere $\partial B_2$ of $\mathbf{C}^2$ is unitarily equivalent to \[ \gamma_{l,m}(t) = \left( \sqrt{l/d} e^{it\sqrt{m/l}}, \sqrt{m/d} e^{-it\sqrt{l/m}} \right) \] where $d = l + m$, $l, m \geq 1$ integers. As an application, we propose a conjecture that if a homogeneous polynomial $\pi$ on $\mathbf{C}^2$ admits a complex-tangential analytic curve on $\partial B_2$ with $\pi(\gamma(t)) = 1$ then $\pi$ is unitarily equivalent to a monomial \[ \pi_{l,m}(z,w) = \sqrt{\frac{d^d}{l^l m^m}} z^l w^m \] where $l, m \geq 1$ integers and show that the conjecture is true for homogeneous polynomial of degree $\leq 5$. A relevant conjecture and partial answer on the maximum modulus set of a homogeneous polynomial $\pi$ on $\mathbf{C}^2$ is also given.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 76, Number 3 (2000), 39-43.

Dates
First available in Project Euclid: 23 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.pja/1148393559

Digital Object Identifier
doi:10.3792/pjaa.76.39

Mathematical Reviews number (MathSciNet)
MR1762069

Zentralblatt MATH identifier
0968.32003

Subjects
Primary: 32C05: Real-analytic manifolds, real-analytic spaces [See also 14Pxx, 58A07] 14P05: Real algebraic sets [See also 12D15, 13J30]

Keywords
Complex tangential curve homogeneous polynomial maximum mudulus set

Citation

Kim, Hong Oh. On complex-tangential curves on the unit sphere on $\mathbf {C}^2$ and homogeneous polynomials. Proc. Japan Acad. Ser. A Math. Sci. 76 (2000), no. 3, 39--43. doi:10.3792/pjaa.76.39. https://projecteuclid.org/euclid.pja/1148393559


Export citation

References

  • Choa, J. S., and Kim, H. O.: Homogeneous polynomials satisfying Cauchy integral equalities. J. Korean Math. Soc., 27, 193–202 (1990).
  • Duchamp, T., and Stout, E. L.: Maximum modulus sets. Ann. Inst. Fourier (Grenoble), 31, 37–69 (1981).
  • Nagel, A. and Rosay, J.-P.: Maximum modulus sets and reflection sets. Ann. Inst. Fourier (Grenoble), 41, 431–466 (1991).
  • Rudin, W.: Functionn Theory in the Unit Ball of $\C^n$. Springer-Verlag, Berlin-Heidelberg-New York (1980).