# Coefficients of polynomials forming an unimodal sequence

Let $f(x)=\sum_{i=0}^{n}a_{i}x^{i}$ be a polynomial with rational coefficients. I am interested in an efficient way which allows checking whether the finite sequence $A=(a_{i})_{i\in\{0,\ldots,n\}}$ is unimodal. Let us recall that the sequence is unimodal if there is $k\in\{0,\ldots,n\}$ such that $a_{0}\leq \ldots\leq a_{k-1}\leq a_{k}\geq a_{k+1}\geq \ldots \geq a_{n}$.

• So you want to use the software Mathematica for this? – J. M. is away Sep 26 '17 at 6:08
• @J. M. Yes, I want to use Mathematica for this computation for several values of $n$. – Maciej Ulas Sep 26 '17 at 11:04
• Would Differences[Sign[Differences[CoefficientList[poly, x]]]] be useful to you? – J. M. is away Sep 26 '17 at 12:07
• The best answer for me would be the information whether the sequence of coefficients is unimodal. Proposed approach gives another sequence of numbers and it is not clear to me how you want to extract the information concerning unimodality. – Maciej Ulas Sep 27 '17 at 7:48