Timeline for Notions of positivity for q-polynomials
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 29, 2019 at 18:04 | comment | added | Valerio_xula | A stronger notion of positivity for a polynomial $p(x)$ is that $|p(z)|\leq p(|z|)$ for all complex $z$. Assuming $p(0)\neq 0$ and the $\gcd$ of the exponents appearing in $p(x)$ to be $1$, an even stronger version is that $|p(z)|< p(|z|)$ for all complex $z$ not on the positive real axis. An example of such a polynomial with a negative coefficients is $p(x) = 2+2x-x^2+2x^3+2x^4$, while the polynomial $1-x+x^2$ does not satisfy the weaker version. | |
| Dec 28, 2014 at 3:31 | comment | added | James Propp | 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. | |
| Dec 26, 2014 at 16:53 | comment | added | Richard Stanley | 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 $\Phi_1(q)=q-1$. | |
| Dec 26, 2014 at 0:41 | comment | added | James Propp | Can one give an intrinsic characterization of such polynomials? | |
| Dec 25, 2014 at 19:58 | comment | added | darij grinberg | There is also the concept of being the result of a subtraction-free rational expression (e.g., arXiv:1307.8425v4). | |
| Dec 25, 2014 at 19:52 | history | edited | James Propp | CC BY-SA 3.0 |
added 5 characters in body
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| Dec 25, 2014 at 4:53 | comment | added | Michael | Another natural notion of positivity would be a positivity of the classical limit at $q\to 1$. | |
| Dec 25, 2014 at 3:54 | history | edited | James Propp | CC BY-SA 3.0 |
Clarified that coefficients need not be positive
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| Dec 25, 2014 at 3:45 | history | asked | James Propp | CC BY-SA 3.0 |