Weierstrass points on cyclic covers of the projective line

We are interested in cyclic covers of the projective line which are totally ramified at all of their branch points. We begin with curves given by an equation of the form $y^{n}=f(x)$, where $f$ is a polynomial of degree $d$. Under a mild hypothesis, it is easy to see that all of the branch points must be Weierstrass points. Our main problem is to find the total Weierstrass weight of these points, $BW$. We obtain a lower bound for $BW$, which we show is exact if $n$ and $d$ are relatively prime. As a fraction of the total Weierstrass weight of all points on the curve, we get the following particularly nice asymptotic formula (as well as an interesting exact formula):

where $g$ is the genus of the curve. In the case that $n=3$ (cyclic trigonal curves), we are able to show in most cases that for sufficiently large primes $p$, the branch points and the non-branch Weierstrass points remain distinct modulo $p$.