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Q&A

Is there a two variable quartic polynomial with two strict local minima and no other critical point?

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Does there exist a degree 4 polynomial $p:\Bbb R^2 \to \Bbb R$ that has two strict local minima and no other critical point?

This is the same as this question rock-star, except for that I'm only interested in quartic polynomials instead of general functions $f:\Bbb R^2 \to \Bbb R, C^\infty$.

One option would be to consider polynomials in the form $p(x,y)=u(x,y)^2 + v(x,y)^2$. Since we need $p$ to be degree 4 polynomial, $u$ and $v$ must be quadratic polynomials. We need there to be a set $P\subset \Bbb R^2$ containing two points such that:

  1. $(x,y)\in P \ \ \Longleftrightarrow \ \ u(x,y)=v(x,y)=0$;
  2. $(x,y)\in P \ \ \Longleftrightarrow \ \ u_x u + v_x v = u_y u + v_y v =0$ when evaluated at $(x,y)$.

Do there exist quadratic polynomials $u,v:\Bbb R^2 \to \Bbb R$ and a two-point set $P$ satisfying conditions 1 and 2?

Disclaimer. I asked this question in a more general form ($P$ consisting of 2-4 points) on MSE.

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