Fermat curves over finite fields and cyclic subsets in high-dimensional projective spaces
The authors prove the following theorem: Let \(q\) be a power of the prime \(p\), \(N=(q^m-1)/(q-1), n\) a divisor of \(N\) and \(d=\gcd(N/n,t+1),\) where \(t\equiv q\bmod 0 N/n\), \(0<t<N/n.\) Assuming that \[ p> \left({2\over\root t+1 \of{\sin(n\pi/(2N))}}+1\right)^{(t-1)(N/n-d)}, \] the number of \(\text{GF}(q^m)\)-rational points of the Fermat curve \(X^n+Y^n+Z^n=0\) is \[ 3n+n^2(q-2)+(d-1)(d-2). \] This interesting theorem is proved using methods and results from finite geometry and combinatorics. More specific: The authors study a set \(\Omega_n\) inside the projective space \(\text{PG}(m-1,q)\) (\(\Omega_n\) is a Singer cycle of size \(N/n\)). The proof is based on counting the number of intersection points a line can have with \(\Omega_n\). From this the above formula is derived by looking at the case where only the numbers \(0,1,2,d\) occur. The authors show that the general approach can also be used to find explicit formulas in cases not covered by the theorem.
scientific article; zbMATH DE number 1295221
scientific article