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Polynomial values in affine subspaces of finite fields

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Abstract

We introduce a new approach and obtain new results for the problem of studying polynomial images of affine subspaces of finite fields. We improve and generalise several previously known results, and also extend the range of such results to polynomials of degrees higher than the characteristic of the field. Our approach is based on estimates for a certain new type of exponential sums. The results we obtain have a wide scope of applications similar to those associated with their counterparts studying consecutive intervals over prime fields instead of affine subspaces. Here we give only two immediate consequences: to bounding the size of the intersection of orbits of polynomial dynamical systems with affine subspaces and to the Waring problem over affine subspaces.

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Acknowledgements

The author would like to thank Igor Shparlinski for suggesting some extensions of initial results, and for his important comments on earlier versions of the paper. The author is also grateful to the Max Planck Institute for Mathematics for hosting the author for two months during the program “Dynamics and Numbers” when important progress on this paper was made. Finally, the author is grateful to the anonymous referee for helpful comments which improved the exposition.

During the preparation of this paper the author was supported by the UNSW Vice Chancellor’s Fellowship.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    Alina Ostafe

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  1. Alina Ostafe
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Correspondence to Alina Ostafe.

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Ostafe, A. Polynomial values in affine subspaces of finite fields. JAMA 138, 49–81 (2019). https://doi.org/10.1007/s11854-019-0021-y

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  • DOI: https://doi.org/10.1007/s11854-019-0021-y

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