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Smith normal form of augmented degree matrix and its applications. (English) Zbl 1184.15012
Let \(\mathbb{F}_{q}\) be a finite field of \(q\) elements and \(p\) its characteristic. Let \(f\left( x_{1},x_{2},\dots,x_{n}\right) \) be a nonzero polynomial in \(n\) variables over \(\mathbb{F}_{q}\). For each positive integer \(r\), let \(N_{q}(f)\) denote the number of \(\mathbb{F}_{q^{r}}\)-rational points on the affine hypersurface \(f=0\), i.e.
\[ N_{q}(f)=\#\left\{ \left( x_{1},x_{2},\dots,x_{n}\right) \in \mathbb{A} ^{n}\left( \mathbb{F}_{q^{r}}\right) \left| f\left( x_{1},x_{2},\dots,x_{n}\right) =0\right. \right\} \] The author uses the Smith normal forms of an augmented degree matrix to evaluate the number \(N_{q}(f).\) This work is a generalization of the author’s previous results. As he mentions, one of the benefit of this result is that there are algorithms for finding the Smith normal form of a matrix, and these are programmed into common Computer Algebra packages such as Maple and Mupad. He gives, at the end of the paper an example to compare this result with the results previously obtained.

MSC:
15A21 Canonical forms, reductions, classification
11T24 Other character sums and Gauss sums
11C20 Matrices, determinants in number theory
15B33 Matrices over special rings (quaternions, finite fields, etc.)
Software:
EDIM; GAP; Maple
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References:
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