  Let $S$ be a polynomial ring in $n$ variables over a field. Let $I$ be a
homogeneous ideal in $S$ generated by forms of degree at most $d$ with
$\text{dim}(S/I)=r$. In the first part of this paper, we show how to derive
from a result of Hoa an upper bound for the regularity of $\sqrt{I}$. More
specifically we show that $\text{reg}(\sqrt{I})\leq d^{(n-1)2^{r-1}}$. In the
second part, we show that the $r$-th arithmetic degree of $I$ is bounded above
by $2\cdot d^{2^{n-r-1}}$. This is done by proving upper bounds for arithmetic
degrees of strongly stable ideals and ideals of Borel type.
