On Lie’s functional equation of translation type

Summary.

The functional equation of translation type, as originally introduced by Sophus Lie, will be solved for the paraboloid under various regularity assumptions and in different algebraic situations. Real paraboloids are characterized as surfaces permitting two special decompositions as sum of two curves. Moreover, we show that such a characterization does not generally hold true for paraboloids in Galois spaces. Finally, some problems related to a generalization of Lie’s functional equation are presented.