18 problems

Let $n\ge 3$, let $\mathbb{H}_n=\{(0,n,0,0),(0,n-2,0,2),(2,n-2,0,0)\}$, and let an $n\times n$ sign pattern ${\cal A}$ require $\mathbb{H}_n$ when … A sign pattern is irreducible i…

Baruah–Sarma sign conjecture. For all integers $n\geq 0$,

Let $s\geq 1$ and let $(\epsilon_1,\ldots,\epsilon_s)\in\{-1,1\}^s$ be a sign pattern. Chowla's sign-pattern conjecture. The set … has natural density $2^{-s}$. This is the sign-pa…

Let … The sign patterns are coefficient patterns in the displayed order. Schlosser and Zhou's conjecture. For $\delta=1$, the coefficients of $Q_{10}(q)^{\delta}$ exhibit…

Schlosser's conjecture. If $\delta$ is real and

Conjectured sign pattern. For all integers $n\geq0$,

Let $\Sigma \in \operatorname{PR}_5$ be a real symmetric matrix whose principal-minor sign pattern $\operatorname{signs}_\Sigma$ agrees with $s_*$ on all subsets of sizes $1$ and…

Modulus-seven Borwein conjecture. For all $m$ and $n$, $d_{7m}(n)\ge0$ and $d_{7m+1}(n),d_{7m+3}(n),d_{7m+4}(n),d_{7m+6}(n)\le0$. Further, $d_{7m+5}(n)\ge0$ when…

Ismail–Kim–Stanton conjecture. If $m\equiv \pm(2l+1)a\pmod K$ for some $l$ with $0\le l<K/2$, then $b_m\le0$; otherwise $b_m\ge0$. Thus the sign of $b_m$ is determined by $m$ modul…

Third Borwein conjecture. The sign pattern of the coefficients in the expansion of $S_n(q)$ is $+----+----+----\cdots$, with the convention that zero coefficients may be regarded a…

Cubic Borwein conjecture. The sign pattern of the coefficients in the expansion of $P_n(q)^3$ is $+--+--+--\cdots$, with the same convention concerning zero coefficients as above.…

The Second Borwein conjecture. The sign pattern of the coefficients in the expansion of $P_n(q)^2$ is $+--+--+--\cdots$, with the same convention that a zero coefficient is conside…

Let $c_t^{(m)}(n)$ denote the coefficient of $q^n$ in $(q;q)_\infty^m(q^t;q^t)_\infty^{-m}$, and let a sequence be a UPS sequence when its signs are periodic with a least sign peri…

Converse characterization conjecture. The following are equivalent:

A sign pattern is a matrix whose entries are $0$, $+$, or $-$. It is spectrally arbitrary if every monic real polynomial of degree $n$ is the characteristic polynomial of an…

Realizability conjecture. For an arbitrary sign pattern ${\bar \sigma}$, the only type of pairs $(pos,\neg)$ which can be non-realizable has either $pos$ or $\neg$ vanishing. Equival…

Let ${\cal A}$ be a sign pattern, and let $mr({\cal A})$ denote its minimum rank over the reals while $mr_{\mathbb Q}({\cal A})$ denotes its minimum rank over the rationals. Ration…

Let $N_1,N_2,\ldots,N_k$ be positive integers, all at least $2$, and let ${\bf z}_{\bf N}(q)$ be the corresponding mirror map. Sign-pattern conjecture. 1. If $\Phi_{\mathbf N}\le3$…