Set $x=(x_{1}, \dots, x_{n}).$ Consider the set $\mathbb{R}[x]_{d}$ of polynomials with coef. in $\mathbb{R}$ in $n$ variables up to degree $d.$ This set can be seen as a finite-dimensional vector space with basis formed by all the monomials up to degree $d$ in $n$ variables and thus endowed accordingly with a(n Euclidean) topology coming naturally from this vector space structure. Now consider particularly the subset $\mathbb{R}0[x]_{d}$ of polynomials in $\mathbb{R}[x]_{d}$ that, when restricted to a real line, have only real zeros (see Section 2.1 here where RZ polynomials are defined). Finally, take the subset $b\mathbb{R}0[x]_{d}\subsetneq\mathbb{R}0[x]_{d}$ of polynomials defining bounded real algebraic varieties. Does $b\mathbb{R}0[x]_{d}$ has non-empty interior in $\mathbb{R}[x]_{d}$ with respect to the Euclidean topology previously mentioned?
I really intuit that this question has to be easily answerable and that $b\mathbb{R}0[x]_{d}$ has in fact non-empty interior in $\mathbb{R}[x]_{d}$ but I cannot see how to prove it right now. Thanks for any insight!
