Interpolation in affine and projective space over a finite field
Let \(\mathbb{F}_q\) be the finite field of \(q\) elements and let \(\mathbb{P}^k(\mathbb{F}_q)\) and \(\mathbb{A}^k(\mathbb{F}_q)\) be the projective and affine \(k\)-dimensional spaces over \(\mathbb{F}_q\) respectively. Let \(s(n,q)\) be the smallest number such that the following interpolation problem has a solution of degree \(s\): given pairwise distinct points \(P_1,\dots,P_n\in \mathbb{P}^k(\mathbb{F}_q)\), there exists a hypersurface \(H\) of degree \(s\) defined over \(\mathbb{F}_q\) such that \(P_1,\dots,P_{n-1}\in H\) and \(P_n\notin H\). The number \(s(n,q)\) can also be defined as the maximal Castelnuovo-Mumford regularity \(r_\mathcal{X}\) of a subset \(\mathcal{X}\subset\mathbb{P}^k(\mathbb{F}_q)\) with \(|\mathcal{X}|=n\). Further, \(s_a(n,q)\) denotes the corresponding affine analogous of \(s(n,q)\). In [\textit{E. Kunz} and \textit{R. Waldi}, J. Commut. Algebra 5, No. 2, 269--280 (2013; Zbl 1282.14043)] the function \(s(n,q)\) was analyzed and completely determined for \(q=2\) and \(q=3\). Since the inequality \(s(n,q)\leq s(n+1,q)\leq s(n,q)+1\) holds for any \(n\in\mathbb{Z}_{>0}\), \(s(n,q)\) is determined by its initial value \(s(1,q)=0\) and its ``jump discontinuities'', namely those \(n\) where \(s(n+1,q)=s(n,q)+1\). In [loc. cit.] the first \(q-1\) jump discontinuities, and the jump discontinuities of order \((m-1)(q-1)+1\) and \(m(q-1)\) are determined for every \(m\geq 2\) and every \(q\geq 4\). The main result of the paper under review is the determination of the jump discontinuities of order \(q-1+r\) with \(2\leq r\leq q-2\) for every \(q\geq 4\). Further, the function \(s_a(n,q)\) is completely determined.
scientific article; zbMATH DE number 6475295