Let $P\_{d, n}$ be the space of polynomial maps $\mathbb{R}^n\to \mathbb{R}$ of degree at most $d$.


Is the subset $S\subset P\_{d, n}$ of nowhere negative polynomials semialgebraic?



Nowhere negative polynomials form a semialgebraic set
As I said in the comments this is very well known: $S$ is the complement of the projection of the semialgebraic set $\{(f,a)\in P\_{d,n}\times\mathbb{R}^n:f(a)<0\}$, hence semialgebraic by the Tarski-Seidenberg theorem.


