# Mathematica and Finite Geometry

## Authors

Dr Neil Gordon N.A.Gordon@hull.ac.uk
Senior Lecturer

## Contributors

V. Keranen
Editor

P. Mitic
Editor

### Abstract

In this paper we aim to show the use of Mathematica in the investigation of various aspects of finite geometry, in particular the exploration of the interaction between algebra and geometry. We will look at two examples of this use. For more details on these particular examples see  and . The first example we will look at is in finding the single polynomial function that corresponds with a set of polynomial functions, the simultaneous zeros of this set of polynomial functions determining a certain subset of elements of a set. In a vector space, that is an-dimensional vector space over a finite field, and given a set of polynomial functions,, the simultaneous zeros of which characterize a set, then it is known that there is a single function,Fsay, the zeros of which correspond with those of--- for more details of this see . It is of interest to ask, what is the degree and the structure of the single polynomial functionFcorresponding to the setin the case of some interesting and important point-sets. In , Shaw and Gordon examined the case of the set of polynomial functions that characterize the set of decomposable bivectors,. Here it was shown that the set of 5 conditions corresponded to a single poylnomial of degree 5. This case was done ``by hand'', and the result confirmed on a ad-hoc computer program. The coding for the program was made much easier since we were dealing only with the field consisting of. For the ``next'' cases,, andwe made use of Mathematica. We will look briefly at one way of modeling such finite fields (prime order only) on Mathematica, and then give some details of looking at the structure of the polynomials themselves --- that is by considering the terms as corresponding to graphs. Again, Mathematica has a role to play here, as using the graphs package (see ), it is possible to get Mathematica to determine the distinct classes represented. For another example of using Mathematica in finite geometry, we will look at the case of trying to find the orbit structure of, in particular the structure of those figures generated by line complements. The case of was possible ``by hand'', but in the higher dimensional case the complexity increased sufficiently to make it impractical. We will describe the role played by Mathematica in this work. To facilitate this, we define the signature of a figure, and show how, by using these signatures, Mathematica can be utilised to study the structure of this space. Moreover, by showing the links between the algebra and the geometry, we will show how Mathematica can calculate the polynomials corresponding to a given point set, and equally can calculate the point set corresponding to a given polynomial, and so we can cross from one viewpoint to another. Finally, we will describe a canonical form for these polynomials, and describe the different ways of producing a list of these canonical forms. Given some other knowledge of the structure of these spaces, in particular their Tower structure and TGL orbits, (see ) we will describe how we hope to look at more complex cases, in particular.

Publication Date Jul 1, 1995 Computational Mechanics Publications 159 - 166 Mathematics with Vision, Proceedings of the First International Mathematica Symposium 1 85312 386 2 Gordon, N. (1995). Mathematica and Finite Geometry. In V. Keranen, & P. Mitic (Eds.), Mathematics with Vision, Proceedings of the First International Mathematica Symposium, 159 - 166 Applied mathematics