Hellus, Michael; Waldi, Rolf Interpolation in affine and projective space over a finite field. (English) Zbl 1346.14063 J. Commut. Algebra 7, No. 2, 207-219 (2015). 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 [E. Kunz and 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. Reviewer: Guillermo Matera (Buenos Aires) Cited in 1 Document MSC: 14G15 Finite ground fields in algebraic geometry 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 14G05 Rational points Keywords:projective space; affine space; finite fields; Castelnuovo-Mumford regularity; rational points; Hilbert function Citations:Zbl 1282.14043 PDFBibTeX XMLCite \textit{M. Hellus} and \textit{R. Waldi}, J. Commut. Algebra 7, No. 2, 207--219 (2015; Zbl 1346.14063) Full Text: DOI Euclid References: [1] W. Bruns and J. Herzog, Cohen-Macaulay rings , Cambr. Stud. Adv. Math. 39 , Cambridge University Press, Cambridge, 1993. · Zbl 0788.13005 [2] G.F. Clements and B. Lindström, A generalization of a combinatorial theorem of Macaulay , J. Combin. Theor. 7 (1969), 230-238. · Zbl 0186.01704 · doi:10.1016/S0021-9800(69)80016-5 [3] D. Eisenbud, The geometry of syzygies , Grad. Texts Math. 229 , Springer, New York, 2005. [4] D. Eisenbud, M. Green and J. Harris, Cayley-Bacharach theorems and conjectures , Bull. Amer. Math. Soc. 33 (1996), 295-324. · Zbl 0871.14024 · doi:10.1090/S0273-0979-96-00666-0 [5] A.V. Geramita and M. Kreuzer, On the uniformity of zero-dimensional complete intersections , J. Algebra 391 (2013), 82-92. · Zbl 1291.14075 · doi:10.1016/j.jalgebra.2013.05.027 [6] A.V. Geramita, P. Maroscia and L.G. Roberts, The Hilbert function of a reduced \(k\)-algebra , J. Lond. Math. Soc. 28 (1983), 443-452. · Zbl 0535.13012 · doi:10.1112/jlms/s2-28.3.443 [7] M. Kreuzer and R. Waldi, On the Castelnuovo-Mumford regularity of a projective system , Comm. Alg. 25 (1997), 2919-2929. · Zbl 0883.13014 · doi:10.1080/00927879708826031 [8] E. Kunz and R. Waldi, On the regularity of configurations of \(\mathbb{F}_q\)-rational points in projective space , J. Comm. Alg. 5 (2013), 269-280. · Zbl 1282.14043 · doi:10.1216/JCA-2013-5-2-269 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.