We determine the relationship between the multiplicities of the zeros of certain rational functions defined on the${\rm SL}_2(\mathbb C)$ and ${\rm PSL}_2(\mathbb C)$ character varieties of knot exteriors.This leads to a formula for the Culler--Shalen seminorms associated to small Seifert Dehn fillings of such manifolds. 2000 Mathematics Subject Classification: 57M05, 57M27, 57R65.