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What are some existing notions of positivity (or nonnegativity) for $\mathbf{R}[q]$ such that all $q$-integers $[n]_q = 1+q+...+q^{n-1}$ are positive, as well as all polynomials that can be expressed in the form $[m_1]_q [m_2]_q \dots / [n_1]_q [n_2]_q \dots$, such as the algebraically natural $q$-Catalan numbers $[2n]_q / [n]_q [n+1]_q$?

Of course, one such property is that the polynomial takes on only nonnegative values when $q \geq 0$, but I suspect that there are others.

Note however that the coefficients need not be nonnegative; e.g., the cyclotomic polynomial $1-q+q^2$ can be written as $[6]_q / [2]_q [3]_q$.

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    $\begingroup$ Another natural notion of positivity would be a positivity of the classical limit at $q\to 1$. $\endgroup$
    – Michael
    Dec 25, 2014 at 4:53
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    $\begingroup$ There is also the concept of being the result of a subtraction-free rational expression (e.g., arXiv:1307.8425v4). $\endgroup$
    – darij grinberg
    Dec 25, 2014 at 19:58
  • $\begingroup$ Can one give an intrinsic characterization of such polynomials? $\endgroup$
    – James Propp
    Dec 26, 2014 at 0:41
  • $\begingroup$ If you want the set of all "positive" polynomials to form a cone, then the smallest such cone is genererated by products of cyclotomic polynomials $\Phi_n(q)$ excluding $\Phi_1(q)=q-1$. $\endgroup$
    – Richard Stanley
    Dec 26, 2014 at 16:53
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    $\begingroup$ Apropos of Darij Grinberg's comment and my query pertaining to that comment, Sergey Fomin has explained to me that a polynomial in $q$ can be written as a ratio of polynomials with non-negative coefficients if and only if the polynomial is positive on the positive ray. $\endgroup$
    – James Propp
    Dec 28, 2014 at 3:31

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