Shape-regular polygons in finite planes (Q2563485)

The authors define the points of the affine plane \(AG(2,q)\) over \(GF(q)\) as elements \(a+b i\in GF(q^2)\) where \(i\) is a root of an irreducible polynomial of degree two over \(GF(q)\). With this, the triangle \((uvw)\) has shape \(S(uvw)= (u-w) (u-v)^{-1}\), an element of \(GF(q^2)\) but not of \(GF(q)\) whenever it is non-degenerate. A shape-regular \(n\)-polygon is formed by \(n\) triangles of equal shape (the ``centre'' being the common point of these triangles). The authors show that shape-regular \(n\)-polygons in \(AG(2,q)\) with \(q\) odd exist if and only if is a divisor of \(q^2-1\) and not of \(q-1\). On the other hand, an \(n\)-polygon of \(AG (2,q)\) is called affine-regular, if (for suitable mapping of the vertices) parallelism and non-parallelism of diagonals are the same as in euclidean space. The authors show that a shape regular \(n\)-polygon is also affine-regular, if \(s+s^{-1} \in GF(q)\), where \(s=S (u_{k-1} u_k u_{k+2})\). Remarks: 1. The authors allow common divisors of \(n\) and \(q-1\) (other than 2), so it may happen that subpolygons become degenerate. 2. For Corollary 3.4 it is sufficient to assume that \(n\) divides \(q+1\), because in this case \(s+s^{-1} =s+ s^q\in GF(q)\).