1. Preservers
These are questions about maps preserving properties of interest for a polynomial, such as stability, Lorentzianity, real rootedness, etc.-
Projections and Lorentzian Polynomials
Problem 1.1.
If $X \subseteq \mathbb{R}^n$ is an $M$-convex set, and we take a linear projection $L : R^n → R$ that preserves integer points (i.e. $L(x) = \langle v,x \rangle$ with $v \in \mathbb{Z}^n$,and then let $s_i =|\{x\in X:L(x)=i\}|$. When is this a log-concave sequence? -
Preserving real roots on a subspace
Problem 1.2.
Let $S \subseteq \R[x]_d$ be a subspace of univariate polynomials, characterize linear maps sending real rooted polynomials in $S$ to real rooted polynomials in $S$. As an example, if $S$ is the space of polynomials with roots summing to 0, then it is conjectured that a diagonal linear map $T$ preserves real rootedness on $S$if and only if $T((x-1)^{n-1}(x+n-1))$ has real roots so that either $n-1$ of these roots are nonnegative or $n-1$ of them are nonpositive. -
Preservers of Real Rootedness with Nonnegative Roots
Problem 1.3.
What are linear maps $T:\mathbb{R}[t] \rightarrow \mathbb{R}[t]$ so that for any real rooted polynomial $f$ with nonnegative real coefficients, $T(f)$ is real rooted.
Cite this as: AimPL: The geometry of polynomials in combinatorics and sampling, available at http://aimpl.org/geompolysampling.