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3. Even Delta Matroids, Jump Systems and Fractional Log-Concavity

    1.     A jump-system is given by a subset $J \subseteq \mathbb{Z}^n$ with the following property: define a step from $a$ to $b$ to be a vector $c \in \Z_{\ge 0}^n$ so that $\|a-c\|_1 = 1$ (i.e. $c$ differs from $a$ either by a coordinate vector or the negative of a coordinate vector) and $|b-c| < |b-a|$. $J$ is a jump system if for all steps $c$ from $a$ to $b$, there exists a step $c$ to $b$ in $J$.

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      Problem 3.1.

      Let $J \subseteq \mathbb{Z}^n$ be a jump system so that either every element of $\{\sum_{i=0}^n a_i : a \in J\}$ is even, or every element of it is odd. For every $k \in \mathbb{N}$, let $c_k$ be the number of points in $J$ with the sum of the coordinates equal to $2k$. Is $c_0, \dots, $ ultra-log concave?

        •     A posinomial is a polynomial whose exponents may be fractional,i.e. \[ f(x) = \sum_{\alpha \in \mathbb{R}_{\ge 0}^n, |\alpha| = d} c_{\alpha}x^{\alpha}\text{ with }c_{\alpha} \ge 0. \] for some $d \in \mathbb{Q}_{\ge 0}$.

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          Problem 3.2.

          Suppose that we know that some posinomial $f(x)$ does not vanish whenever it is evaluated when $x$ is in the right half-plane, does that imply that $f(x)$ is log concave on the positive orthant?

              Cite this as: AimPL: The geometry of polynomials in combinatorics and sampling, available at http://aimpl.org/geompolysampling.