Results

SUCCESS!

The execution of the program finished at the end of december 2005, thanks to the great number of computers connected to the grid. The aim was to get a complete list of polynomials of degree 11 with constant term 2 and leading coefficient 1. Among these one can find all generalized binary number system bases.

The output of the program

The program generated 550 polynomials as output. At the writing of the program we had to allow it to produce a larger output than the actual set of expansive polynomials. This is due to an optimization of speed. A total of 339 polynomials turned out to be expansive. (The rest can be written as the product of expansive and cyclotomic polynomials. These polynomials are interesting as well, that’s why we do not include additional code to get rid of them.)

Number system polynomials

We used a specific  algorithm to determine which expansive polynomials are number system polynomials. This procedure is performed at the Eotvos Lorand University, Budapest. As we expected, only a few polynomials are number system polynomials. Their exact number is 11. We give the now complete list containing the number of expansive and number system polynomials in dimensions at most 11.

 Degree 2 3 4 5 6 7 8 9 10 11 Expansive 5 7 29 29 105 95 309 192 623 339 Number system 4 4 12 7 12 7 32 12 42 11

The numbers in the table and the speed of growth in terms of the degree are interesting issues and subject to our mathematical investigation.

Further projects

The obtained number systems are being tested in applications related to data compression etc. Besides, mathematical analysis is in progress.

Further work in this area includes two immediate generalizations. On one hand, dimensions higher than 12 could be attacked by the help of a currently unproved mathematical conjecture. This conjecture could reduce the size of the parameter space at hand so that a list of polynomials can be obtained. This list will not be 100% sure to be complete but from a practical point of view, a possibly uncomplete list is still valuable.  This program is now executed on the grid for degree 12.

A generalization of the problem deals with polynomials with constant term greater than 2. In case of constant term 3, we speak of genaralized ternary number systems. Currently a program for ternary number systems is in the development phase. We estimate that we can generate a complete list of canonical ternary number systems up to degrees 9 or 10. This program will be started paralelly on the grid.