On the number of rational points on codimension 1 algebraic sets in \(\mathbb{P}^ n (\mathbb{F}_ q)\)
As an example of the type of results obtained by the author, we have the following result. Let \(X \subset \mathbb{P}^ n (\mathbb{F}_ q)\) be a pure \((n - 1)\)-dimensional algebraic set which is defined over \(\mathbb{F}_ q\). Suppose \(X\) has degree \(d\) and let \(| X |_{\mathbb{F}_ q}\) denote the number of points of \(X\) rational over \(\mathbb{F}_ q\). Theorem: We have \(| X |_{\mathbb{F}_ q} \leq dq^{n-1} + {q^{n- 1} - 1 \over q - 1}\). These results are then generalized through the use of coding theory, and applications to coding theory are given.
scientific article; zbMATH DE number 716466