# The first part of the Hilbert sixteenth problem for elliptic polynomials

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?

• what do you mean by "last homogeneous part"? – user347489 May 20 at 20:53
• @user347489 for example $x^2+y^2$ is the last homogenous part of $x^2+y^2+ax+by+c$. – Ali Taghavi May 21 at 10:51
• Every polynomial is a sum if homogenous polynomials.the last homogenous part is the homogenous part with highest degree. – Ali Taghavi May 21 at 10:54
• @user347489 I revised the word "last" to "highest" – Ali Taghavi 2 days ago
• 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 $\mathbb P^2(\mathbb R)$. – Joe Silverman 2 days ago