1. The Problems
The following problems raised/discussed during the workshop:-
Problem 1.05.
[Apoorva Khare, Petter Branden, ...]- (i) Let $\mathcal{P}_d := \left\{f(z)= \sum\limits_{j=0}^{d}a_j z^j \ | \ a_j \geq 0~ \forall j, ~~f(z)=0 \implies z \in \mathbb{R}\right\}$. Which functions $F: \mathbb{R}_+ \longrightarrow \mathbb{R}_+$ such that $F(0)=0$ map $\mathcal{P}_d$ into itself, where $F[f](z) := \sum\limits_{j=0}^{d}F(a_j)z^j$? Equivalently, let \[ T_{\bf a} := {\scriptsize \begin{pmatrix} a_0 & 0 & 0 & \cdots\\ a_1 & a_0 & 0 & \cdots\\ a_2 & a_1 & a_0 & \cdots\\ \vdots & \vdots & \vdots & \ddots \end{pmatrix}} \in TN, \] where ${\bf a}= (a_0,\ldots,a_d)\in (0,\infty)^{d+1}$. When is $F[T_{\bf a}]\in TN$ for all such $\bf a$?
- (ii) The same question for fixed $d$.
- (iii) One can replace $\mathcal{P}_d$ with $\tilde{\mathcal{P}}_d:=\left\{f(z)= \sum\limits_{j=0}^{d}\frac{a_jz^j}{j !}~|~a_j\geq 0~ \forall j, ~~f(z)=0 \implies z \in \mathbb{R}\right\}$ or replace $F[f](z)$ with $\tilde{F}[f](z):=\sum\limits_{j=0}^{d}\frac{F(a_j)z^j}{j !}$ and consider the above questions (four possible cases).
- (iv) What about preservers of the form $F_g[f](z)=\sum\limits_{j=0}^{d}g(j)F(a_j)z^j$? Such $g$ are called multiplier sequences and have been classified (Pólya, Laguerre and Schur).
- (v) Let $h: \mathbb{R}^3 \longrightarrow \mathbb{R}$ and let $F_h[\sum\limits_{j=0}^{d}a_jz^j]=\sum\limits_{j=0}^{d}h(a_{j-1},a_j,a_{j+1})z^j$, where $a_{-1}=a_{d+1}=0$. E.g., if $h(x,y,z)=y^2-xz$, then $F_h[-]$ preserves real rooted polynomials.
What other $h$ work?
What about higher order determinants?
-
Problem 1.1.
[Petter Branden] Given positive scalars $a_0, \dots, a_d$ such that $\frac{a_k^2}{a_{k-1}a_{k+1}}\geq 4$ for $0 < k < d$, Hutchinson’s theorem says that the generating polynomial $p(x) := \sum\limits_{j=0}^{d} a_jx^j$ is real-rooted. One can homogenize this to $p(x,y)$.
Now consider the analogue for homogeneous polynomials of three variables. It is known that for each degree $d \geq 1$, there exists $\lambda_d > 0$ such that if $\lambda \geq \lambda_d$, then the “$\lambda$-rhombus log-concavity” of the Maclaurin coefficients of a homogeneous polynomial $p(x,y,z)$ implies that $p$ is real-stable.
Can one remove the dependence of $\lambda_d$ on $d$? If yes, then what about for $n > 3$ variables? -
Problem 1.15.
[Alan Sokal] The Aissen–Edrei–Schoenberg–Whitney (AESW) theorem states that a real sequence $(a_0,a_1,a_2,\ldots)$ with $a_0 \neq 0$ is a one-sided PF sequence $\Leftrightarrow F(z)=a_0 e^{\delta z} \prod\limits_{j=0}^{\infty}\frac{1+\alpha_j z}{1-\beta_jz}$ for $\alpha_j, \beta_j, \delta, a_0 \geq 0$ and $\sum_j (\alpha_j + \beta_j) < \infty$. Find a proof that does not use Nevanlinna theory: in other words, the fact that if $F(z)$ is entire of genus at most one, and has no zeros, then it is an exponential function $ae^{\delta z}$ for some $\delta \geq 0$. -
Problem 1.2.
[Pavlo Pylyavskyy] Let \[ M_{\bf A} := {\scriptsize \begin{pmatrix} A_0 & 0 & 0 & \cdots\\ A_1 & A_0 & 0 & \cdots\\ A_2 & A_1 & A_0 & \cdots\\ \vdots & \vdots & \vdots & \ddots \end{pmatrix}} \in TN, \] and $F(t)=A_0+tA_1+t^2A_2+\cdots \in M_2[[t]],$ where $A_0$ is invertible. Is there an AESW-type factorization for $F(t)$? -
An algorithmic process is known to identify inequalities that compare the products of two minors for all $TN$ matrices.
Problem 1.25.
[Prateek Kumar Vishwakarma] Classify real linear combinations of products of minors that are nonnegative for all $TN$ matrices. More concretely, given a polynomial $p( \{ x_{I \times J} \ | \ I,J \subset [n], |I| = |J| \} )$, when can one certify, via some constructive algorithmic process, that evaluating $p$ at $x_{I \times J} = \det A_{I \times J}$ yields a non-negative output for all $TN$ matrices $A$. -
Problem 1.3.
[Melissa Sherman-Bennett] Consider $TN$ Toeplitz matrices. What are the allowed patterns of zero and nonzero minors (i.e., which positroid cells contain Toeplitz matrices)? How does this relate to Pólya frequency sequences? (Check work by Rietsch for the lower triangular Toeplitz case.) Is it enough to check all the corner minors to identify $TN$ Toeplitz matrices? -
Problem 1.35.
[Alan Sokal, ...] Consider semi-infinite lower triangular $TN$ Toeplitz matrices. Under what conditions can a non-trivial $k\times k$ minor be zero? -
Problem 1.4.
[Robert Angarone, Alan Sokal] There are many classical theorems about operations that preserve negative-real-rootedness of polynomials: for instance, $d/dx + a$ or $x \cdot d/dx + a$ with $a \ge 0$; or Hadamard product; or Hadamard product with an extra $n!$ (sometimes known as Schur composition); or Brändén’s (2011) log-concavity operation $\ldots$. These can be reinterpreted as statements about pointwise total positivity for certain Toeplitz matrices involving elementary symmetric functions (e.g. $(n+a) e_n$). Can these results be upgraded to total positivity in the monomial basis? Computational tests suggest that the answer is yes. But we have been unable, thus far, to come up with a plausible strategy for proving any of these conjectures. -
Problem 1.45.
[Apoorva Khare, Alan Sokal] It was recently shown that if $\lambda, \mu$ are $N$-tuples of positive integers, then the ratio of Schur polynomials $s_\lambda / s_\mu$, when evaluated at $N$ real variables with values in $(0,1]^N$ (the “log-negative orthant”), is minimized at $(1,\dots,1)$ if and only if $-\lambda$ weakly majorizes $-\mu$.
The analogous statement, in which the minimum over the entire positive orthant $(0,\infty)^N$ is achieved at $(1,\dots,1)$, is equivalent to $\lambda$ majorizing $\mu$. What can one say these domains are replaced by $I^N$ for other intervals $I \subset (0,\infty)$? -
Problem 1.5.
[Pavlo Pylyavskyy] Consider a bipartite graph $G$ on an annulus, with a fixed “base” perfect matching $\mu$. Given any other $\mu'$, superimpose it on $\mu$; this gives a bunch of cycles. Count the number of non-contractible cycles $=f(\mu')$. Running over all $\mu' \neq \mu$, we get $$ \sum_{k \geq 0} \ \sum_{\mu'\neq \mu \ | \ f(\mu')=k} z^{k} = \sum_{k\geq 0} a_{k}z^{k}. $$ (This is known to be independent of the base matching $\mu$.) Is it true that $\sum_{k\geq 0} a_{k}z^{k}$ generates a Pólya frequency sequence? If one attaches weights to the edges, does one get a coefficientwise-$TN$ infinite triangular $TN$ Toeplitz matrix? (It is known that this Toeplitz matrix is coefficientwise ${TN}_2.$) -
This has now been answered affirmatively.
Problem 1.55.
[Apoorva Khare] Suppose $n,r \geq 1$ are integers, and $(\lambda_1, \dots, \lambda_r)$ and $(\mu_1, \dots, \mu_r)$ are (non-increasing) partitions of $n$. If $\mu$ majorizes $\lambda$, then for all $TN$ matrices $A$ we have: \[ \lambda_1! \cdots \lambda_r! \sum_I \prod_{k=1}^r \det A_{I_k \times I_k} \geq \mu_1! \cdots \mu_r! \sum_J \prod_{k=1}^r \det A_{J_k \times J_k}, \] for all partitions $I = (I_1, \dots, I_r)$ and $J = (J_1, \dots, J_r)$ of $[n]$ with $|I_k| = \lambda_k, |J_k| = \mu_k$ for all $k$.
Is the converse true? -
Problem 1.6.
[Petter Branden] Is Fisk’s conjecture true for $3 \times 3$ minors / $n \times n$ minors? Namely, if $\sum_{k=0}^d a_k z^k$ is a real-rooted polynomial with all $a_k > 0$, then Brändén showed the “$2 \times 2$ analogue”: \[ \sum_{k \geq 0} \left| \begin{matrix} a_k & a_{k-1} \\ a_{k+1} & a_k \end{matrix} \right| z^k \] is also real-rooted. The conjecture by Fisk asks if the $3 \times 3$ analogue (or $n \times n$ for higher $n$) is true: is \[ \sum_{k \geq 0} \left| \begin{matrix} a_k & a_{k-1} & a_{k-2} \\ a_{k+1} & a_k & a_{k-1} \\ a_{k+2} & a_{k+1} & a_k \end{matrix} \right| z^k \] also real-rooted? -
A quantitative condition in $\lambda$ has been formulated and shown to be sufficient; it is not clear if the value of $\lambda$ is tight.
Problem 1.65.
[Charles Johnson, Steven Karp] Suppose $\lambda \geq 4$. Given a (strictly) ${TP}_1$ square matrix $A$, it is known that if $ad - \lambda bc > 0$ for every $2 \times 2$ submatrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ of $A$, then $A$ is (strictly) TP.
The question is if this fact can be extended to higher orders of total positivity. Namely, can the above result be extended to finding a quantitative (in $\lambda > 0$) sufficient condition, ideally involving $k \times k$ or smaller minors of $A$, such that every (strictly) ${TP}_{k-1}$ matrix satisfying this condition is automatically (strictly) $TP$? If yes, what is the smallest value of $\lambda$ for each such $k$?
Cite this as: AimPL: Theory and applications of total positivity, available at http://aimpl.org/totalpos.