## Journal of Commutative Algebra

### Interpolation in affine and projective space over a finite field

#### Abstract

Let $s(n,q)$ be the smallest number $s$ such that any $n$-fold $\mathbb{ F}_q$-valued interpolation problem in $\mathbb{P}^k_{\mathbb{F}_q}$ has a solution of degree~$s$, that is: for any pairwise different $\mathbb{F}_q$-rational points $P_1,\ldots ,P_n$, there exists a hypersurface $H$ of degree~$s$ defined over $\mathbb{F}_q$ such that $P_1,\ldots ,P_{n-1}\in H$ and $P_n\notin H$. This function $s(n,q)$ was studied by Kunz and the second author in \cite{KuW} and completely determined for $q=2$ and $q=3$. For $q\geq 4$, we improve the results from \cite{KuW}.

The affine analogue to $s(n,q)$ is the smallest number $s=s_a(n,q)$ such that any $n$-fold $\mathbb{F}_q$-valued interpolation problem in $\mathbb{A}^k(\mathbb{F}_q)$, $k\in \mathbb{N}_{>0}$ has a polynomial solution of degree $\leq s$. We exactly determine this number.

#### Article information

Source
J. Commut. Algebra, Volume 7, Number 2 (2015), 207-219.

Dates
First available in Project Euclid: 14 July 2015

https://projecteuclid.org/euclid.jca/1436909532

Digital Object Identifier
doi:10.1216/JCA-2015-7-2-207

Mathematical Reviews number (MathSciNet)
MR3370484

Zentralblatt MATH identifier
1346.14063

Subjects
Primary: 14G15: Finite ground fields

#### Citation

Hellus, Michael; Waldi, Rolf. Interpolation in affine and projective space over a finite field. J. Commut. Algebra 7 (2015), no. 2, 207--219. doi:10.1216/JCA-2015-7-2-207. https://projecteuclid.org/euclid.jca/1436909532

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