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Yankl
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Generalization of stationary points to stationary lines (valleys)?
To add onto my previous comment, it seems that the condition which you're looking for is that the curve is tangent to the gradient of $f$.
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Generalization of stationary points to stationary lines (valleys)?
A question to clarify what you're looking for: Why do you not see/wish to include the x-axis as a stationary curve for the function you used as an example?
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Polynomial entropy of topological dynamical systems
It might help to have some more context on what you're looking for exactly.These slides of Paris-Romaskevich may be useful for you: people.math.osu.edu/gogolyev.1/DynScreen/ParisR_dys.pdf
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