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Sums of reciprocals of polynomials over finite fields. (English) Zbl 1266.11129
Let \({\mathcal P}_q(n)\) be the set of all monic polynomials of degree \(n\) over the finite field \(\mathbb{F}_q\). The authors study the sum of the \(k\)th powers of the polynomial reciprocals, and find a surprising result for \(k\leq q\): \[ \sum_{f\in{\mathcal P}_q(n)} \frac{1}{f^k}=\bigg(\frac{1}{\prod_{i=1}^n (x-x^{q^i})}\bigg)^k. \]

MSC:
11T55 Arithmetic theory of polynomial rings over finite fields
11T06 Polynomials over finite fields
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