For which class of polynomials over $\mathbb{C}^{n}$ does the following growth inequality hold? For any multi-index $\alpha$, there are positive constants $A, B, C, D < \infty$ such that \begin{align} |\partial^{\alpha} f | \leqslant A |\partial f|^{2} + B |f | + C |z|^{2} + D \end{align} holds for sufficiently large $|z|$, where $| \cdot |$ denotes the standard Euclidean norm. It seems that polynomials which violate this inequality do not vanish at the origin and have non-isolated critical points there. Does the inequality hold for polynomials which vanish at the origin and have an isolated critical point there?
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$\begingroup$ Pointwise inequality? Of course not. Take $\alpha = (2,0)$ and consider $f(z)=Mz^2$ at the origin. The inequality is $|2M| \le D$, which is pretty hard to satisfy for all $M$. $\endgroup$– user53153Mar 2, 2013 at 18:13
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$\begingroup$ Thanks, 5pm. Your example illustrates a condition that is necessary (and one that I forgot to include). I need only the inequality to hold for $|z|>0$. $\endgroup$– user02138Mar 2, 2013 at 18:21
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$\begingroup$ That does not help. Since both sides of the inequality are continuous in $z$, if it were true for all $z\ne 0$, it would also be true for $z=0$. $\endgroup$– user53153Mar 2, 2013 at 18:25
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1$\begingroup$ You should define "large": for all $z$ such that $|z|\ge M$, where $M$ depends on ? And please do this by editing your question, so that other readers will not have to read a chain of comments to find what the question actually is. $\endgroup$– user53153Mar 2, 2013 at 18:39
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$\begingroup$ For $n>1$ the answer to the edited question is no, no matter how we interpret "large". Take $f(z)=Mz_1^2 z_2^3$, and $\alpha=(2,0)$ as before. At the points where $z_1=0$, the stated inequality becomes $|2Mz_2^3|\le C|z_2|^2 +D $ which is false. $\endgroup$– user53153Mar 2, 2013 at 18:54
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Of course, there is nothing special about the origin: we can shift the variables so that the origin is not a critical point at all, or is an isolated critical point. The former is achieved with $f(z)=M (z_1+1)^2 (z_2+1)^3$, the latter with $f(z)=M (z_1+1)^2 (z_2+1)^3-M(2z_1+3z_2)$. In both cases, the stated inequality fails for $\alpha=(2,0)$ when $z_1=-1$ and $|z_2|$ is large.
Edit: if you want $f$ to vanish at the origin (in addition to having an isolated critical point there, take $$f(z)=M (z_1+1)^2 (z_2+1)^3-M(1+2z_1+3z_2)$$
I think you should focus on why any such inequality should be true (and what you need it for), instead of trying to escape known counterexamples with minor modifications.
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$\begingroup$ Thank you +1. However, I disagree with your remark about the origin. I believe it to be a key component with this growth inequality. Polynomials which vanish at the origin and have isolated critical points there seem to satisfy it. $\endgroup$ Mar 6, 2013 at 3:02
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$\begingroup$ @user02138 I just gave a counterexample to that. $\endgroup$– user53153Mar 6, 2013 at 3:07
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