Let $f:\mathbb R^2\rightarrow\mathbb R$ be a smooth function such that for every point $(x,y)\in\mathbb R^2$ the system $$f_{11}+2tf_{12}+t^2f_{22}=0$$ $$f_{111}+3tf_{112}+3t^2f_{122}+t^3f_{222}=0$$ has a solution in $t$. In other words, $f$ satisfies a certain complicated partial differential equation (think of the resultant: http://en.wikipedia.org/wiki/Resultant)

(Here $f_{ij}$ and $f_{ijk}$ are second and third partial derivatives computed at $(x,y)$.)

Is there any chance that the above conditions imply the following statement:

"For a given constant $c$, the set of points $(x,y)$ that give a solution $t=c$, if nonempty, is a line, or at least must contain a line."

Or, if it doesn't hold, do you see a way to add some natural generically satisfied condition on $f$ so that it holds?

This is not my specialty, but I would be surprised if it's true.

fancyasconvoluted. – Noam D. Elkies Nov 9 '12 at 4:50