9
votes

### Does there exist a study of entire functions which satisfy $|F(x+iy)| \leq a e^{-bx^2}e^{cy^2}$?

The inequality in the post can be rewritten as
$$\left|F(z)e^{\frac{b+c}{2}z^2}\right|\leq ae^{\frac{c-b}{2}|z|^2},\qquad z\in\mathbb{C}.$$
For $c<b$ it follows that $F=0$, and for $c=b$ it follows ...

- 87k

8
votes

Accepted

### Geometry of critical points of holomorphic maps in projective space

For $n=2$, the locus $J$ is smooth and irreducible for a general $f$; i.e., these $f$ form a Zariski dense subset of the parameter space of such $f$. For $n\ge3$ and for general $f$, the locus $J$ ...

- 42.7k

6
votes

### Geometry of critical points of holomorphic maps in projective space

For n=1, every point in the critical divisor has degree $≤d-1$, where $d$ is the degree of the map, and the total degree of the critical divisor is $2d−2$, and any such divisor can occur.
To state it ...

- 79.5k

2
votes

Accepted

### Singularity on the boundary of domain of holomorphy

Write $\phi=\phi_1+i\phi_2$. A counterexample is given by
$$
\phi_2(x)=\begin{cases} 0 & x<0 \\ x & 0\le x\le 1 \end{cases} .
$$
We also give $\phi_2$ compact support and keep it smooth ...

Only top scored, non community-wiki answers of a minimum length are eligible

#### Related Tags

holomorphic-maps × 3cv.complex-variables × 1

differential-equations × 1

projective-varieties × 1

critical-point-theory × 1