be the corresponding lengths of the line segments projected onto the x-axis (i.e. if we have a line segment stretching between the points P i {\displaystyle P_{i}} and P j {\displaystyle P_{j}} then the length is j − i {\displaystyle j-i} ). Then for each integer 1 ≤ κ ≤ r {\displaystyle 1\leq \kappa \leq r} , f ( x ) {\displaystyle f(x)} has exactly λ κ {\displaystyle \lambda _{\kappa }} roots with valuation − μ κ {\displaystyle -\mu _{\kappa }} .