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