  This paper studies the problem of which sequences of non-negative integers
arise as the functions $\operatorname{reg} I^{n-1}/I^n$, $\operatorname{reg}
R/I^n$, $\operatorname{reg} I^n$ for an ideal $I$ generated by forms of degree
$d$ in a standard graded algebra $R$. These functions are asymptotically linear
with slope $d$. If $\dim R/I = 0$, we give a complete characterization of all
numerical functions which arise as the functions $\operatorname{reg}
I^{n-1}/I^n$, $\operatorname{reg} R/I^n$ and show that $\operatorname{reg} I^n$
can be any numerical function $f(n) \ge dn$ that weakly decreases until it
becomes a linear function with slope $d$. The latter result gives a negative
answer to a question of Eisenbud and Ulrich. If $\dim R/I \ge 1$, we show that
$\operatorname{reg} I^{n-1}/I^n$ can be any numerical asymptotically linear
function $f(n) \ge dn-1$ with slope $d$ and $\operatorname{reg} R/I^n$ can be
any numerical asymptotically linear function $f(n) \ge dn-1$ with slope $d$
that is weakly increasing. Inspired of a recent work of Ein, Ha and Lazarsfeld
on non-singular complex projective schemes, we also prove that the function of
the saturation degree of $I^n$ is asymptotically linear for an arbitrary graded
ideal $I$ and study the behavior of this function.
