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 ...
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 $J$ ...
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 ...
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 ...

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