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].

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].