Few inflection points of holomorphic curves

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $n\ge1$ and let $f:\mathbb C\to\mathbb P^n$ have a reduced entire representation $f=[f_0:\cdots:f_n]$, where $f_0,\ldots,f_n$ have no common zero and are linearly independent over $\mathbb C$. Define \[ T(r,f)=\frac{1}{2\pi}\int_0^{2\pi}\log\left(\sum_{j=0}^{n}|f_j(re^{i\theta})|^2\right)^{1/2}\,d\theta -\log\left(\sum_{j=0}^{n}|f_j(0)|^2\right)^{1/2}. \] Let $W_f=\det(f_j^{(i)})_{0\le i,j\le n}$ be its Wronskian. Write $m_0$ for the zero multiplicity of $W_f$ at $0$, and let $n_1(t)$ count its zeros in $|z|\le t$, with multiplicity. For $r\ge1$, put \[ N_1(r)=m_0\log r+\int_0^r\frac{n_1(t)-m_0}{t}\,dt, \qquad\lambda=\liminf_{r\to\infty}\frac{\log T(r,f)}{\log r}. \] The assumption of linear independence makes $W_f\not\equiv0$. Changing the reduced representation multiplies it by a nonvanishing factor, so the zero count is intrinsic; $T$ changes by at most a harmless normalization constant.

Statement recorded in the supplied source.

Assume $\lambda<\infty$ and $N_1(r)=o(T(r,f))$ as $r\to\infty$. Must $\lambda$ be rational, and must the limit \[ \lim_{r\to\infty}\frac{\log T(r,f)}{\log r} \] exist (and thus equal $\lambda$)? The original survey §4 proposes the stronger regularly varying form $T(r,f)=r^\lambda\ell(r)$, with $\ell(cr)/\ell(r)\to1$ uniformly for $1\le c\le2$. The registered question asks for rationality and existence of the growth exponent. [S2]

Short English statement

For a linearly nondegenerate entire projective curve of finite lower order with negligible Wronskian zero count, prove that its growth exponent exists and is rational.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3700043 (AMR-036-0043). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Alexandre Eremenko, Value Distribution and Potential Theory (2003), §§3–4, pp.685–688; §4 Conjecture. https://arxiv.org/abs/math/0304332.