5,053 research outputs found
Square-full polynomials in short intervals and in arithmetic progressions
We study the variance of sums of the indicator function of square-full
polynomials in both arithmetic progressions and short intervals. Our work is in
the context of the ring of polynomials over a finite field
of elements, in the limit . We use a recent
equidistribution result due to N. Katz to express these variances in terms of
triple matrix integrals over the unitary group, and evaluate them
On vanishing sums of th roots of unity in finite fields
In an earlier work, the authors have determined all possible weights for
which there exists a vanishing sum of th roots
of unity in characteristic 0. In this paper, the same problem is
studied in finite fields of characteristic . For given and , results
are obtained on integers such that all integers are in the
``weight set'' . The main result in this paper guarantees,
under suitable conditions, the existence of solutions of
with all coordinates not equal to zero over a finite field
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