On the internal nuclei of sets in \(PG(n,q)\), \(q\) is odd (Q5947344)

The main result of the article generalizes some former results of \textit{F. Wettl} [J. Geom. 30, 157-163 (1987; Zbl 0629.51017)] and \textit{T. Szönyi} [Combinatorics '88, Vol. 2: Proceedings of the international conference on incidence geometries and combinatorial structures, Ravello, Italy (Mediterranean Press, Research and Lecture Notes in Mathematics) 449-458 (1991; Zbl 0945.51521)], who described the set \(IN(K)\) of internal nuclei of a set \(K \in PG(n,q)\) for \(n=2\), \(q\) odd and \(k=|K|=q+1\). The author proves, under some appropriate restrictions, that for general \(n\), \(IN(K)\) is contained in a normal rational curve of order \(n\) if \(i=|IN(K)|\leq \frac{q+1}{2}\) and is projectively equivalent to the set of points of an explicit curve, if \(i=\frac{q+1}{2}\). The proof is based on a property in the plane (a condition in terms of \(k\) and \(i\) under which \(IN(K)\) must be a part of a conic) and successive reductions to lower dimension. An application in cryptography is presented to the so-called threshold schemes.