Near-Rings of Polynomial Functions and their Applications

Our world is not linear. Many phenomena, however, are often "linearized" because only then a reasonably well- working mathematical machinery can describe the phenomena and produce meaningful forecasts. Between the two extremes of "brutal" linearization and inacceptable mathematical complication, the compromise of "polynomialization" turns out to be the best one. Polynomial functions are highly non-linear, but are still "tame" enough to allow a mathematical treatment which produces excellent results in otherwise hopeless situations. In the area of algebra, the use of polynomials and polynomial functions has a much shorter tradition than in analysis, and much less is known about the value of these concepts for algebra itself and for its applications. So the goal of this project is to investigate polynomials and polynomial functions on algebraic structures more closely and to apply the results to areas outside of algebra, namely to - the construction of Balanced Incomplete Block Designs (BIB-designs) in Combinatorics, - the optimal design of statistical experiments, - the characterization of synchronizing words in automata theory, and to possible other areas.

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. So we collected the best researchers in this area from all parts of the world. Together we succeeded in solving this problem. The solution turned out to be really very difficult, and it is doubtful that any single researcher would ever have cracked the problem. All this sounds very theoretical, mainly because it is very theoretical. Surprisingly enough, some considerations in the area of near-rings are very practical. Suppose one wants to find the best combination of ingredients to a toothpaste, or the best mix of fertilizers for the field of a farmer. It is very time-consuming and sometimes extremely costly to conduct a large number of tests. Remember that a test in agriculture usually takes (at least) one year. So it is essential to find very few "representative" tests from which one can compute the very best mixture of ingredients. Using certain types of near-rings, we could show that one can construct the most economical designs for these tests which are theoretically possible. A cooperation with farmers etc. was established. It is really very unusual that a very abstract parts of mathematics produces so concrete applications in daily life!