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Can we use Positivstellensatz (P-satz) below for a non-polynomial term?

P-satz: Let $R$ be real closed field. Let $f,g,h$ be finite families of polynomials in $R[X_{1} ,...,X_{n}]$. Denote by P the cone generated by $f$, $M$ the multiplicative monoid generated by $g$ and $I$ the ideal generated by $h$. Then the following properties are equivalent:

(i) The set $\{x\in R^n| f\geq 0, g\neq 0 , h=0\}$ is empty (ii) There exist $f \in P, g\in M, h\in I$ such that $f+g^2 +h=0$.

Reference: Bochnak et al. 1999

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  • $\begingroup$ For which Ring you would like to apply it? $\endgroup$ Mar 7, 2014 at 6:45
  • $\begingroup$ If you do not want to apply it to polynomials, then what else do you have in mind? $\endgroup$
    – jmc
    Mar 7, 2014 at 7:56
  • $\begingroup$ Yes, that is exactly what I wanted to say. $\endgroup$ Mar 7, 2014 at 8:31
  • $\begingroup$ In my case, $f$ and $g$ have some non-linear term, e.g. $f(x)=\sin x (x^2+x^3)$. So there are some non-linear terms in the function (not only polynomials). (Now, I am working on some robotics system that represented in some non-linear terms and some polynomial terms. I am just wondering: Whether can I apply the P-satz to my system or not to simplify the analysis. Thank you before) $\endgroup$ Mar 7, 2014 at 8:42
  • $\begingroup$ So you want to replace the ring of polynomials with the ring of continuous functions? Or of all functions? $\endgroup$ Mar 7, 2014 at 10:12

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