Interpolation in affine and projective space over a finite field
type: Item
instance of: scholarly article
title: Interpolation in affine and projective space over a finite field
zbMATH Open document ID: 1346.14063
DOI: 10.1216/JCA-2015-7-2-207
author: Michael Hellus
author: R. Waldi
published in: Journal of Commutative Algebra
publication date: 2015-08-25T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jca/1436909532
review text: 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.
reviewed by: Guillermo Matera
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 14G05
zbMATH DE Number: 6475295
zbMATH Keywords: projective space
zbMATH Keywords: affine space
zbMATH Keywords: finite fields
zbMATH Keywords: Castelnuovo-Mumford regularity
zbMATH Keywords: rational points
zbMATH Keywords: Hilbert function
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6475295