4 problems

Let $G$ be a graph, and let $D(G,x)$ denote its domination polynomial. An integer root is an integer number $r$ such that $D(G,r)=0$. Integer-root conjecture. Every integer root of…

Let $n,m\in\mathbb{N}$ satisfy $n>2$, $\lfloor\log_2(n)\rfloor+1\geq m>1$, and suppose there is no $k\in\mathbb{N}$ such that $k^m=n$. Then Shunia's conjecture. … The formula gives…

Let $a,n\in\mathbb{Z}^+$ satisfy $a>2$, $\left\lfloor\log_2(a)\right\rfloor+1\geq n>1$, and suppose there is no $k\in\mathbb{Z}^+$ such that $k^n=a$. Integer-part root conjecture.…

Let $f$ be a univariate polynomial whose arithmetic circuit complexity is bounded by $s$. Shub–Smale's $\tau$-conjecture. The number of integer zeros of $f$ is bounded by a polynom…