8 problems

Let $K[p(t),q(t)]=K[u(t),v(t)]$ for one-variable polynomials $p(t),q(t),u(t),v(t)$. Write … Assume that $\max(m,n)>\max(k,l)$. Degree-divisibility conjecture. Either $m$ divides…

Let $p_1,p_2,p_3,p_4$ be the four polynomials constructed in the paper, evaluated at $x,y,z\in\mathbb{N}^*$. The relevant generated parameter is a natural number $q\geq1$. Four-pol…

Four-polynomial coverage conjecture. The union of the images of these four families covers every integer $a\equiv1\pmod{4}$. The claim is supported by computational verification up…

Let $f\in\mathbb{R}[x_1,x_2]$ define the curve being approximated, and let $\mathbf{d}=(d_1,d_2)$ be the bidegree of the parametrizations, with $d_1=d_2$. A real $\mathbf{d}$-inter…

Let $\mathbf{SL}_n(\mathbb{Z})$ denote the group of $n\times n$ integer matrices with determinant $1$. A subset $\mathcal{X}$ of $\mathfrak{R}^h$ is a polynomial family over a comm…

The authors consider polynomial parametrizations of rational knots by Chebyshev polynomials, including representations of the form $x=T_a(t)$, $y=T_b(t)$, $z=C(t)$, where $C$ is ch…

Let $T_n$ denote the degree-$n$ Chebyshev polynomial of the first kind. Let $i,j$ be relatively prime positive integers, and let $k,\ell$ belong to the semigroup denoted by…

Let $T_n$ denote the degree-$n$ Chebyshev polynomial of the first kind, and let $(i,j,k)$ be a reduced triple as in the source. Primality conjecture. Every knot of the form … for t…