Wikipedia on "Handelman's theorem: If $K$ is a compact polytope in Euclidean $d$-space, defined by linear inequalities $g_i ≥ 0$, and if $f$ is a polynomial in $d$ variables that is positive on $K$, then $f$ can be expressed as a linear combination with non-negative coefficients of products of members of ${g_i}$." where the context is the positive polynomials and Farkas' lemma is the Positivstellensatz but often not called by that name so I am curious which theorems fall under Nichtnegativstellensatzen. I am particularly interested in compact polytopes.
Related on this question here on using Handelman's theorem to check nonnegativivity of a polynomial in a compact polynomial but now focus on terminology and jargon.
Are Handelman's theorem and Nichtnegativstellensatz the same thing?
How are Handelman's theorem and Nichtnegativstellensatz related for compact polytopes?
Which Nichtnegativstellensatz is relevant to determine non-negativity in compact polytopes? What about with Handelman's theorem?
What are the difference between different Nichtnegativstellensatzen? The Satz by Stengle
$$f\geq0 \text{ on } K \Leftrightarrow hf=f^{2s}+p$$ for some integer $s$and polynomials $h,p\in P(g)$. Moreover bounds for $s$ and degrees of $h,p$ exist.
I cannot answer the question but I am trying to gather potentially relevant material to answer this question, work in progress until trying to target the question.
I could find the threads and material sometimes relevant for positive polynomials but also focus on nonnegative polynomials
Nonnegativity condition for a polynomial in two variables
Lecture slides on positive polynomials introduce you to non-negative polynomials.
Stengle's Nichtnegativstellensatz