Near-rings of Polynomials and Polynomial Functions
Sprache der Bezeichnung:
Our world is not linear. If a car drives twice as fast, it needs much more than the double distance to come to a full stop. Many processes and dependencies are so highly non-linear that a precise mathematical formulation is impossible or so complicated that a mathematical treatment is too difficult even for the best mathematicians.
The usual reaction is to ?linearize? the model, but this is dangerous. If one replaces a curve by a straight line, one can make terrible mistakes. A very intelligent compromise is to use ?polynomial? models. Polynomial functions are complicated enough to capture most of the essential features of non-linear models, but they are at the same time tame enough to enable a proper mathematical treatment.
The algebraic properties of polynomials and polynomial functions are the main object of this project. These objects can ?live? on fields, rings, groups, or on other algebraic structures and usually form so-called ?near-rings?. One of the big open questions ? unanswered since about 40 years ? is to determine the bijective polynomial functions on groups. They describe dependencies such that every input has a unique output, and every output comes from a unique input.