Additive solutions of polynomial equations on fields and decompositions of positive real numbers into sets that are closed under addition and multiplication

In my talk I will present two problems addressed in the title. Both of these problems lead to derivations defined on fields but for basically different reasons. The solutions of the corresponding equations via the concept of polynomial exponential functions on the multiplicative group of a field naturally implies solutions that are (higher order) derivations. On the other hand, it is a challenging task to find the precize bound for the order of such derivations. These results are joint work with Eszter Gselmann. The decomposition problem can be solved in a relatively simple way using derivations. On the other hand, the characterization of finite decompositions is highly non-trivial. This second part is based on our results with Gábor Somlai and Tamás Terpai.