Abstract:Let $$\varvec{F}_q$$
F
q
be the finite field of q elements, where $$q=p^r$$
q
=
p
r
is a power of the prime p, and $$\left( \beta _1, \beta _2, \dots , \beta _r \right) $$
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“…In the case of function fields, several significant results have also been obtained, see [6–9, 14, 17, 21, 28, 33, 34] and references therein. However, in general, this direction falls behind its counterpart over .…”
We estimate mixed character sums of polynomial values over elements of a finite field with sparse representations in a fixed ordered basis over the subfield . First we use a combination of the inclusion–exclusion principle with bounds on character sums over linear subspaces to get nontrivial bounds for large . Then we focus on the particular case , which is more intricate. The bounds depend on certain natural restrictions. We also provide families of examples for which the conditions of our bounds are fulfilled. In particular, we completely classify all monomials as argument of the additive character for which our bound is applicable. Moreover, we also show that it is applicable for a large family of rational functions, which includes all reciprocal monomials.
“…In the case of function fields, several significant results have also been obtained, see [6–9, 14, 17, 21, 28, 33, 34] and references therein. However, in general, this direction falls behind its counterpart over .…”
We estimate mixed character sums of polynomial values over elements of a finite field with sparse representations in a fixed ordered basis over the subfield . First we use a combination of the inclusion–exclusion principle with bounds on character sums over linear subspaces to get nontrivial bounds for large . Then we focus on the particular case , which is more intricate. The bounds depend on certain natural restrictions. We also provide families of examples for which the conditions of our bounds are fulfilled. In particular, we completely classify all monomials as argument of the additive character for which our bound is applicable. Moreover, we also show that it is applicable for a large family of rational functions, which includes all reciprocal monomials.
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