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A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its highest homogeneous part does not vanish on $\mathbb{R}^2\setminus\{0\}$.

Inspired by the first part of the Hilbert 16th problem we ask that:

Is there an elliptic polynomial $P(x,y)\in \mathbb{R}[x,y]$ of degree $n$ which has a level set $P^{-1}(c)$ with more than $n$ connected components? Can elliptic polynomials produce $M$-curves?

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    $\begingroup$ what do you mean by "last homogeneous part"? $\endgroup$ – user347489 May 20 at 20:53
  • $\begingroup$ @user347489 for example $x^2+y^2$ is the last homogenous part of $x^2+y^2+ax+by+c$. $\endgroup$ – Ali Taghavi May 21 at 10:51
  • $\begingroup$ Every polynomial is a sum if homogenous polynomials.the last homogenous part is the homogenous part with highest degree. $\endgroup$ – Ali Taghavi May 21 at 10:54
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    $\begingroup$ @user347489 I revised the word "last" to "highest" $\endgroup$ – Ali Taghavi 2 days ago
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    $\begingroup$ What have you tried? Is it true for $n=2$? $n=4$? Also, possibly a more intrinsic way to describe your condition is in terms of projective space. Let $\overline P(x,y,z)\in\mathbb R[x,y,z]$ be the homogenization of your $P$. Then your condition is that $\overline P=0$ has no points on the line $z=0$ "at infinity" in $\mathbb P^2(\mathbb R)$. $\endgroup$ – Joe Silverman 2 days ago

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