Monomial graphs and generalized quadrangles (Q439070)

In this paper, the author proves the nonexistence of certain generalized quadrangles of order \((s,s)\), with \(s\) an odd prime power. This is achieved by considering the graph of points and lines far away from a flag of such a quadrangle, identifying the point and line sets with 3-dimensional vector space over \(\mathrm{GF}(s)\), coordinatizing, and expressing the incidence relation by means of two equations. In the classical case, the two equations have a linear part and a quadratic/cubic part. The latter is a monomial, and the authors would like to know whether other monomials would work to give new quadrangles. They prove that in a number of cases, the classical monomials are the unique ones leading to generalized quadrangles.NEWLINENEWLINEThe method employed in the paper, and stemming from \textit{V. Dmytrenko, F. Lazebnik} and \textit{J. Williford} [Finite Fields Appl. 13, No. 4, 828--842 (2007; Zbl 1135.05032)] boils down to the coordinatization of generalized quadrangles as introduced by \textit{G. Hanssens} and the reviewer [Ann. Discrete Math. 37, 195--207 (1988; Zbl 0643.51011)] in the eighties. These authors have also tried to find new quadrangles by changing the non-linear part of the same equations, without success in the odd case. In the even case, they found back examples related to hyperovals, and in the case of other parameters, they found back the examples of Kantor using Knuth semifields. In any case, it should be clear that this is not the way anymore to find new finite generalized quadrangles.