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Some function field estimates with applications. (English) Zbl 0930.11041

Yıldırım, Cem Y. (ed.) et al., Number theory and its applications. Proceedings of a summer school at Bilkent University, Ankara, Turkey. New York, NY: Marcel Dekker (ISBN 978-0-8247-1969-2/pbk; 978-1-138-40407-6/hbk; 978-0-429-33267-8/ebook). Lect. Notes Pure Appl. Math. 204, 23-45 (1999).
Let \(f_1(x)\) and \(f_2(x)\) be two monic and coprime polynomials in \(\mathbb{F}_q[x]\) and suppose that \(m=\deg f_1>\deg f_2\). The author considers the function field extension \(L\mid\mathbb{F}_q(y)\), where \(L\) is a splitting field over \(\mathbb{F}_q(y)\) for the polynomial \(F(x,y)\): \[ F(x,y)=f_1(x)-yf_2(x). \] The following bound on the genus \(g\) of \(L\) is obtained: \[ 2g\leq(m-3)\cdot\bigl[L:\mathbb{F}_q(y)\bigr]+2. \] Quite related to the number of rational places of \(L\) is the number \(N\) of elements \(a\in\mathbb{F}_q\) such that the polynomial \(f_1(x)-af_2(x)\) is a product of distinct linear factors. A bound on \(N\) is then deduced from the Hasse-Weil bound; clearly, one assumes that \(\mathbb{F}_q\) is the full constant field of \(L\) and two interesting criteria for this assumption to hold are given.
Applications to graph theory and coding theory are given (Steiner systems, covering and packing radius of codes,…).
For the entire collection see [Zbl 0905.00051].

MSC:

11G20 Curves over finite and local fields
14H05 Algebraic functions and function fields in algebraic geometry
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
05C20 Directed graphs (digraphs), tournaments
94B75 Applications of the theory of convex sets and geometry of numbers (covering radius, etc.) to coding theory
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