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2. General Polynomials and Real Rootedness


    1. Add remark

      Problem 2.1.

      Given a palindromic polynomial $\sum_{i=0}^Nc_ix^i$, consider the following conjecture:

      There exists a cone $C$ such that $(c_0,c_1,\cdots,c_N)\in C$ if and only if the roots of the polynomial are on the unit circle.
          One such family of polynomials where this seems to occurs is $\sum_{j=0}^N {\rm gcd}(j,N)x^j$. Techniques for proving this may include the discrete Fourier transform and common interlacing.


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          Problem 2.2.

          Given a polynomial $ \sum_{n=0}^N a_nx^n$ where $a_n$ is hypergeometric (i.e $\frac{a_{n+1}}{a_n}$ is a rational function in $n$), determine real rootedness algorithmically.


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              Problem 2.3.

              Let $\sum_{i=0}^n b_ix^i$ be a real rooted polynomial with $b_i\geq 0$ for all $i$ and let $A_i(x)$ denote the $i$th Eulerian polynomial. Is the polynomial $\sum_{i=0}^nb_iA_i(x)$ real rooted? Determine this in general or for particular families of real-rooted polynomials.

                •     $\eta$-quotients:

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                  Problem 2.4.

                  Let $s(q):=\prod_{j=1}^\infty\frac{(1-q^{2j})^2}{1-q^j}$. Suppose that $s(q)=\sum_{n\geq 0}s_n q^n$, is $s_n\geq 0$ for all $n$?
                    1. Remark. As mentioned during the Workshop, the series $s(q)=\sum_{n\geq 0} q^{\binom{n}2}$. So, the coefficients $s_n$ are indeed non-negative. This is due to Euler.


                        • Add remark

                          Problem 2.5.

                          Let $p(n)$ be a polynomial with positive coefficients. $p(n)=a(n)+nb(n)$ uniquely where $a(n)$ and $b(n)$ are palidromic. Do $a(n)$ and $b(n)$ have nonnegative coefficients?

                              Cite this as: AimPL: Polyhedral geometry and partition theory, available at http://aimpl.org/polypartition.