## 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

#### 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

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

#### 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

#### References

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