Timeline for Does there exist a homogeneous polynomial $F$ whose partial derivatives satisfy the following inequalities?

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Dec 3, 2017 at 18:42	comment	added	Rodrigo de Azevedo		Are $n$ and $d$ inputs? Or can one choose them?
S Dec 3, 2017 at 18:37	history	suggested	Rodrigo de Azevedo	CC BY-SA 3.0	Minor improvements
Dec 3, 2017 at 18:05	review	Suggested edits
S Dec 3, 2017 at 18:37
S Dec 3, 2017 at 18:02	history	suggested	Rodrigo de Azevedo	CC BY-SA 3.0	Minor improvements
Dec 3, 2017 at 18:00	review	Suggested edits
S Dec 3, 2017 at 18:02
Dec 3, 2017 at 17:17	comment	added	Johnny T.		@JasonStarr True. Thanks very much!
Dec 3, 2017 at 16:34	comment	added	Jason Starr		There are still plenty of examples. For instance, fix an integer $d$ with $2\leq d \leq n$, and consider $F(x_1,x_2,\dots,x_n) = \sum_{1\leq i_1<i_2<\dots<i_d \leq n} a_{i_1,i_2,\dots,i_d}x_{i_1}x_{i_2}\cdots x_{i_d},$ i.e., a general linear combination of square-free monomials of total degree $d$. The partial derivatives $\partial^2 F/\partial x_i^2$ are all zero, but the singular set does not equal all of $W$.
Dec 3, 2017 at 16:20	history	edited	Johnny T.	CC BY-SA 3.0	added 33 characters in body
Dec 3, 2017 at 16:19	comment	added	Johnny T.		@JasonStarr Let me fixed this...
Dec 3, 2017 at 15:59	comment	added	Jason Starr		@gcousin. More generally, if $W$ is contained in the singular locus of $F$, e.g., for $F=G^2$. Perhaps the OP would like to specify that $W$ is not contained in the singular locus of $F$.
Dec 3, 2017 at 15:52	comment	added	gcousin		F=0 does the job ;-)
Dec 3, 2017 at 15:36	history	asked	Johnny T.	CC BY-SA 3.0