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Small generators of function fields. (English. French summary) Zbl 1233.11120
It is well known that a finite extension of global fields can be generated by one element. In the paper under review the upper bound on the size of the minimal generator is established in the function field case. More precisely, let \(k\subset K\) be two finite separable extension of \(\mathbb F_q(t).\) Then it is proven that there exists an element \(\alpha\) and a constant \(C=C(k, [K:k])\) depending only on \(k\) and \([K:k]\) such that \(K=k(\alpha)\) and \(h(1,\alpha)<\frac{g_K}{d(K/k)}+C,\) where \(d(K/k)=[K:k]/[K_0:k_0],\) \(K_0\) and \(k_0\) being the corresponding fields of constants and \(h(1, \alpha)\) is the logarithmic projective Weil height on \(\mathbb P^1.\) The main ingredient of the proof is an application of Weil bounds for the number of places of given degree on curves over finite fields.

11R58 Arithmetic theory of algebraic function fields
11G50 Heights
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