Suppose $f$ is a bivariate polynomial that defines an irreducible plane curve of degree $d$ over some algebraically closed field $k$. We know that by Bézout's that summing over points on the curve: \begin{align*} \sum_P m_P(f)(m_P(f)-1) \leq d(d-1). \end{align*}
Is there any bound on $\sum_P m_P(f)$? Just like for the univariate polynomial that $\sum_P m_P(f) = d$.
I’m interested in bounding the sum of multiplicities of singular points on an irreducible curve over the finite field. Any literature recommendation along that direction is also appreciated.