36 problems

Let $R=\mathbb{Z}[A^{\pm1},w,x,y,z]$. For $n\geq1$, let $D_n^{\mathit{Mb}}(d,w,x,y,z)$ be the Gram determinant of type $\mathit{Mb}$. Let $T_k$ and $S_k$ denote the Chebyshev polyn…

Let $R=\mathbb{Z}[A^{\pm1},w,x,y,z]$. For $n\geq 1$, let $D_n^{(Mb)_1}$ denote the Gram determinant of type $(Mb)_1$, where the Gram matrix is formed from crossingless-connection d…

Möbius–Chebyshev conjecture. For all integers $i\leq j$, $\mu(a^i,c^j)$ is the coefficient of $x^{j-i}$ in $T_{i+j}(x)$.

Simion's non-crossing matching conjecture. For every positive integer $n$,

Simion's refined determinant conjecture. For every positive integer $n$,

Simion's type-B determinant conjecture. For every positive integer $n$,

Let $d\ge 3$, and let $p(x)$ range over polynomials of total degree less than $2d$. For the monomial $(x_1\cdots x_d)^2$, consider polynomials of the form $(x_1\cdots x_d)^2-p(x)$…

Let $T_m(x)$ denote the Chebyshev polynomial of index $m$, and let $\ell$ be a non-zero multiple of $4$. Consider … A continued fraction is semi-specializable if it has the partial…

Let $h$ be a function admitting a Fourier-like expansion in the Chebyshev basis, represented by an infinite vector $\vec{h}$, and let $D_{\infty}$ denote the diagonal matrix arisin…

For a prime $p$, let $R_p$ denote the relevant Chebyshev residue structure, and let $\omega_a$ be the element associated with $a\in R_p$. A Chebyshev primitive root is an element…

Let $f\in \mathbb{C}[x]$ be a complex polynomial, and let $G_\infty(\mathbb{C},f,t)$ denote its geometric iterated Galois group over a transcendental parameter $t$. A polynomial is…

Let $Mb_{n,1}$ be the set of crossingless-connection diagrams between $2n$ marked boundary points of the Möbius band, with exactly one curve intersecting the crosscap. Let…

Prime-divisor density conjecture. The result that the set of prime divisors of the orbit has density zero remains valid for rational initial conditions and the resulting almost int…

Let $0<\alpha<1$ and let $0\leq\beta\leq 1-\alpha$. Write $\tau_{\alpha,\beta}$ for the asymptotic degree ratio associated with approximating a Chebyshev polynomial after the affin…

Merino's irreducibility conjecture. This polynomial is irreducible over $\mathbb{R}$ if and only if $m$ and $n$ are coprime. The claim gives a precise arithmetic criterion for irre…

Merino's conjecture. This curve is the union of a finite number of Lissajous curves. The conjecture concerns the decomposition of the Chebyshev-defined curve into parametrized Liss…

Unit-primitive matrix conjecture. The characteristic polynomial of $A_n$ is

Let $g_E(E_1,E/E_1)$ be the function used in the paper's threshold equations. Energy-one-half threshold conjecture. For $\kappa=8$, the energy $E=0.5$ is a threshold; the observed…

Let $\kappa\geq2$ be even and fix $1\leq j\leq\lfloor\kappa/2\rfloor$. Let $T_\kappa$ be the Chebyshev polynomial used in the paper, and choose real numbers satisfying … with…

Let ${\mathtt{p}}=(p_1,\ldots,p_L)$ be the probability vector under consideration, and let $H_1({\mathtt{p}})$ and $H_s({\mathtt{p}})$ denote its Shannon entropy and Tsallis entrop…

Let $(r_k(n))_{k\geq 0}$ be the companion sequence associated with the positive integer parameter $n$, and let $\mathcal{T}_p(j)$ denote the relevant dilated Chebyshev value. Prime…

Let $(s_k(n))_{k\geq 0}$ be the sequence associated with the positive integer $n$. For a fixed $n$, define $(k_N)_{N\geq 1}$ by requiring that $s_{k_N}(n)$ is the $N$th prime term…

Let $(s_k(n))_{k\geq 0}$ be the sequence associated with the positive integer parameter $n$, and let $\mathcal{T}_p(j)$ denote the dilated Chebyshev polynomial of the first kind ev…

Periodicity conjecture. The difficulty of the degree decision problem over a ring of modulo $2^w$ is related to the periodicity of Chebyshev polynomials over the ring.

Let $f(z)=z^2+1/4$ and let $J(f)$ denote its Julia set. For a non-polar infinite compact set $K\subset\mathbb{C}$, write … where $P_n$ is the Chebyshev polynomial of degree $n$ on…