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Additive decompositions induced by multiplicative characters over finite fields. (English) Zbl 1298.11112
Lavrauw, Michel (ed.) et al., Theory and applications of finite fields. The 10th international conference on finite fields and their applications, Ghent, Belgium, July 11–15, 2011. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-5298-9/pbk; 978-0-8218-9157-5/ebook). Contemporary Mathematics 579, 179-186 (2012).
Summary: In 1952, O. Perron [Math. Z. 56, 122–130 (1952; Zbl 0048.03002)] showed that quadratic residues in a field of prime order satisfy certain additive properties. This result has been generalized in different directions, and our contribution is to provide a further generalization concerning multiplicative quadratic and cubic characters over any finite field. In particular, recalling that a character partitions the multiplicative group of the field into cosets with respect to its kernel, we derive the number of representations of an element as a sum of two elements belonging to two given cosets. These numbers are then related to the equations satisfied by the polynomial characteristic functions of the cosets.
Further, we show a connection, a quasi-duality, with the problem of determining how many elements can be added to each element of a subset of a coset in such a way as to obtain elements still belonging to a subset of a coset.
For the entire collection see [Zbl 1253.00023].
11T30 Structure theory for finite fields and commutative rings (number-theoretic aspects)
12E20 Finite fields (field-theoretic aspects)
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