On the nuclei of semifields and Cofman's many subplane problem
(p,m,n) is a Cofman or C-triple, if an affine translation plane of order \(q^ n\), where \(q=p^ m\), contains more subaffine planes of order q than exist in the Desarguesian plane of the same size, viz., \(q^ n\). \textit{J. W. Freeman} [Geom. Dedicata 9, 267-280 (1980; Zbl 0436.51005)] showed that (p,2M,2) is always a C-triple, thus answering a question of \textit{J. Cofman} [Proc. Internat. Conf. Projective Planes, Washington State Univ. 1973, 59-70 (1973; Zbl 0281.50008)]. In the article being discussed it is conjectured that (p,m,n) is a C-triple iff both m, \(n>1\). The conjecture is proved for even n, and also for many odd values of n: e.g. when \(\gcd (m,n)\) has a square divisor \(>1\). These results are obtained by analysing the conditions under which every semifield, coordinatising a given semifield plane \(\pi\), has GF(q) in the intersection subfield of two of its three seminuclei.
scientific article; zbMATH DE number 3996505