Closed formulas for exponential sums of symmetric polynomials over Galois fields
Exponential sums have applications to a variety of scientific fields, including, but not limited to, cryptography, coding theory and information theory. Closed formulas for exponential sums of symmetric Boolean functions were found by Cai, Green and Thierauf in the late 1990s. Their closed formulas imply that these exponential sums are linearly recursive. The linear recursivity of these sums has been exploited in numerous papers and has been used to compute the asymptotic behavior of such sequences. In this article, we extend the result of Cai, Green and Thierauf, that is, we find closed formulas for exponential sums of symmetric polynomials over any Galois fields. Our result also implies that the recursive nature of these sequences is not unique to the binary field, as they are also linearly recursive over any finite field. In fact, we provide explicit linear recurrences with integer coefficients for such sequences. As a byproduct of our results, we discover a link between exponential sums of symmetric polynomials over Galois fields and a problem for multinomial coefficients which is similar to the problem of bisecting binomial coefficients.
KeywordsExponential sums Symmetric functions Linear recurrences
Mathematics Subject Classification05E05 11T23 11B37
The authors would like to thank Oscar E. González for reading a previous version of this article. The authors also appreciate the comments and suggestions to the article made by the referees. Their comments and suggestions improve the presentation of this work.
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