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Counting polynomials with zeros of given multiplicities in finite fields. (English) Zbl 1024.11076

Summary: We consider the set of polynomials in \(r\) indeterminates over a finite field and with bounded degree. We give here a way to count the number of elements of some of its subsets, namely those sets defined by the multiplicities of their elements at some points of \(\mathbb{F}^r_q\). The number of polynomials having at least one zero in a given finite field is computed as a particular applications.

MSC:

11T06 Polynomials over finite fields
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