2 problems

Let $d\geq 1$ and let $\gamma(n)=(P_1(n),\ldots,P_d(n))$ be a polynomial curve in $\mathbb{Z}^d$, where $P_1,\ldots,P_d\in\mathbb{Z}[X]$ have separated degrees, meaning…

Let $d\geq 2$ be an integer, let $p(x)$ be any polynomial of degree $d$ mapping the integers to the integers, and let $N\geq 1$. Define … For the interval $I=[0,N^d]$, let $f$ be s…