7 problems

Suppose $z_1,\ldots,z_n\in\mathbb{D}_p\subset\mathbb{C}_p$ are $\mathbb{Q}$-linearly independent, where $\mathbb{D}_p$ is the domain of the $p$-adic exponential function. Define…

Let $x_1,x_2,y_1,y_2$ be complex numbers such that $x_1,x_2$ are linearly independent over the rational numbers and $y_1,y_2$ are linearly independent over the rational numbers. Fo…

Let $Sk$ be the smallest set of functions $f:\mathbb R^{>0}\to\mathbb R^{>0}$ containing the constant function $1$ and the identity function $\mathbf{x}$, and such that if…

Let $p(x,y_1,\ldots,y_k)$ be a nonzero irreducible polynomial in $\mathbb C[x,y_1,\ldots,y_k]$ that depends on $x$ and the last variable $y_k$. A complex number $a$ is generic for…

The exponential addition-law conjecture. The relationship

For each positive real number $a$, consider the curve through the points $(n,a^n)$ with integer coordinates and its tangent line at $(0,1)$. The tangent-slope conjecture. The slope…

Let $f$ be the sign function considered in the two-frequency root mean square approximation problem, and let $a9e(a9e_1,a9e_2)$ denote the resulting approximation error for two dis…