  A non-negativity certificate (NNC) is a way to write a polynomial so that its
non-negativity on a semialgebraic set becomes evident. Positivstellens\"atze
(Ps\"atze) guarantee the existence of NNCs. Both, NNCs and Ps\"atze underlie
powerful algorithmic techniques for optimization. This paper proposes a
universal approach to derive new Ps\"atze for general semialgebraic sets from
ones developed for simpler sets, such as a box, a simplex, or the non-negative
orthant. We provide several results illustrating the approach. First, by
considering Handelman's Positivstellensatz (Psatz) over a box, we construct
non-SOS Schm\"{u}dgen-type Ps\"atze over any compact semialgebraic set. That
is, a family of Ps\"atze that follow the structure of the fundamental
Schm\"{u}dgen's Psatz, but where instead of SOS polynomials, any class of
polynomials containing the non-negative constants can be used, such as SONC,
DSOS/SDSOS, hyperbolic or sums of AM/GM polynomials. Secondly, by considering
the simplex as the simple set, we derive a sparse Psatz over general compact
sets, which does not require any structural assumptions of the set. Finally, by
considering P\'olya's Psatz over the non-negative orthant, we derive a new
non-SOS Psatz over unbounded sets which satisfy some generic conditions. All
these results contribute to the literature regarding the use of non-SOS
polynomials and sparse NNCs to derive Ps\"atze over compact and unbounded sets.
Throughout the article, we illustrate our results with relevant examples and
numerical experiments.
