real analysis – Set of smooth hyperbolic polynomials of fixed degree $d$ and number of variables $n$ nonvanishing in any point at infinity.


This question is about the topology of the set of smooth hyperbolic polynomials of fixed degree $d$ and number of variables $n$ nonvanishing in any point at infinity inside its corresponding set of homogeneous polynomials of fixed degree $d$ and number of variables $n$.

We know that the set of smooth hyperbolic polynomials has nonempty interior in the space of homogeneous polynomials of same degree and number of variables. See discussions after Theorem 1.1 here and Corollary 5.1 here. Do you know if the set of smooth hyperbolic polynomials nonvanishing in any point at the infinity has also nonempty interior in the space of homogeneous polynomials of same degree and number of variables?

My guess is that this has to be true (that is, the set of all smooth hyperbolic polynomials nonvanishing in any point at the infinity has also nonempty interior in the (vector) space of all homogeneous polynomials of same degree and number of variables) but I am not sure. Observe, in particular, that this question is not absolutely trivial as imposing some conditions at infinity might result in a drastical reduction of the number of polynomials that when imposed in finite points are much more innocuous. E.g., the set of all polynomials in one variable that are not zero at the origin is open in the vector space of all polynomials (true?) while the set of all polynomials in one variable that are not infinite (in modulus) at infinity is formed only by the (finite) constants and thus has dimension 1.