I'm working on a problem and I'd like to compute a couple examples. For one example I'd like to create a function from $\mathbb{R}^2 \rightarrow \mathbb{R}$ with the following properties:

  1. It has a saddle at zero
  2. It goes to positive infinity on the x AND y axis (so maybe cross sections along these axes are upward opening parabolas)
  3. It goes to negative infinity along the line $x = y$ AND $x = -y$ (so maybe cross sections along these axes are downward opening parabolas).

What is an example of a function with such properties?


  • $\begingroup$ A degree $2$ polynomial obviously won't do. Have you tried a degree $4$ polynomial? $\endgroup$ Dec 20, 2020 at 20:11
  • $\begingroup$ Haha, nope. I'm totally lost on what should I guess. $\endgroup$
    – yoshi
    Dec 20, 2020 at 20:14
  • 1
    $\begingroup$ Here's a hint: Have you played with a monkey saddle? It only has three directions of interest, not four, but understanding this might give you an idea. $\endgroup$ Dec 20, 2020 at 20:19

1 Answer 1


Following the commentor, I searched Monkey Saddles. I found more complicated saddles: I guess these types of surfaces are called n-th order saddles.

Formulas for them can be found here: https://rivix.com/download/Docs/Peckham_2011_Saddles_Final.pdf

For my particular problem: $f(x,y) = x^4 - 6x^2y^2 +y^4$


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