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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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