# Put positive polynomial in finite intersection of half-spaces

This is a cross-posting of a MSE question (which did not attract any attention there so far).

Denote by $V={\mathcal P}_{n,d}$ the space of polynomials in $n$ variables with degree at most $d$, with rational coefficients. Thus ${\mathcal P}_{n,d}$ has dimension $\binom{n+d}{n}$ over $\mathbb Q$ .

We can consider arbitrary intersection of closed half-spaces in $V$ : $N_{\Phi}=\lbrace p \in V | \forall \phi \in \Phi, \ \phi(p) \geq 0 \rbrace$, where $\Phi$ is a subset of $V^{*}$. By analogy with nef polygones, I call $N_{\Phi}$ a nef, and I say that it is of finite or infinite type, according to whether $\Phi$ is finite or infinite (perhaps there is a better terminology for this, that I’m unaware of).

Let $W$ denote the polynomials in $V$ that are nonnegative on ${\mathbb N}^n$. Then $W$ is clearly a nef of infinite type. I ask if for any $p\in W$, one can find a nef $W'$ of finite type, with nonempty interior, such that $p \in W' \subseteq W$.

This is possible when $n=1$ : if $p_0$ is a polynomial of degree $d$, nonnegative on $\mathbb N$, then there is an integer $k_0$ such that all the derivatives of $p_0$ are nonnegative at $k_0$. Then, we can take

$$W'=\bigg\lbrace p \in V \ \bigg| \ p(k) \geq 0 \ {\rm for} \ k \leq k_0, p^{(j)}(k_0) \geq 0 \ {\rm for} \ 1\leq j \leq d \bigg\rbrace$$