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Value Sets of Sparse Polynomials

  • Igor E. Shparlinski (a1) and José Felipe Voloch (a2)

Abstract

We obtain a new lower bound on the size of the value set $\mathscr{V}(f)=f(\mathbb{F}_{p})$ of a sparse polynomial $f\in \mathbb{F}_{p}[X]$ over a finite field of $p$ elements when $p$ is prime. This bound is uniform with respect to the degree and depends on some natural arithmetic properties of the degrees of the monomial terms of $f$ and the number of these terms. Our result is stronger than those that can be extracted from the bounds on multiplicities of individual values in $\mathscr{V}(f)$ .

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Author I. E. S. was supported by ARC Grants DP170100786 and DP180100201.

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[BCR16] Bi, J., Cheng, Q., and Rojas, J. M., Sub-linear root detection, and new hardness results, for sparse polynomials over finite fields . SIAM J. Comput. 45(2016), 14331447. https://doi.org/10.1137/140990401
[BS-D59] Birch, B. J. and Swinnerton-Dyer, H. P. F., Note on a problem of Chowla . Acta Arith. 5(1959), 417423. https://doi.org/10.4064/aa-5-4-417-423
[CFKLLS] Canetti, R., Friedlander, J. B., Konyagin, S. V., Larsen, M., Lieman, D., and Shparlinski, I. E., On the statistical properties of Diffie-Hellman distributions . Israel J. Math. 120(2000), 2346. https://doi.org/10.1007/s11856-000-1270-1
[CLMS61] Carlitz, L., Lewis, D. J., Mills, W. H., and Strauss, E. G., Polynomials over finite fields with minimal value sets . Mathematika 8(1961), 121130. https://doi.org/10.1112/S0025579300002230
[CGRW17] Cheng, Q., Gao, S., Rojas, J. M., and Wan, D., Sparse univariate polynomials with many roots over finite fields . Finite Fields Appl. 46(2017), 235246. https://doi.org/10.1016/j.ffa.2017.03.006
[Kel16] Kelley, A., Roots of sparse polynomials over a finite field . LMS J. Comput. Math. 19(2016), suppl. A, 196204. https://doi.org/10.1112/S1461157016000334
[Kur09] Kurlberg, P., Poisson spacing statistics for value sets of polynomials . Int. J. Number Theory 5(2009), 489513. https://doi.org/10.1142/S1793042109002237
[Lor96] Lorenzini, D., An invitation to arithmetic geometry . Graduate Studies in Mathematics, 9, American Mathematical Society, Providence, RI, 1996. https://doi.org/10.1090/gsm/009
[Mil64] Mills, W. H., Polynomials with minimal value sets . Pacific J. Math. 14(1964), 225241.
[MZ13] Mullen, G. and Zieve, M., Value sets of polynomials . In: Handbook of finite fields , CRC Press, Boca Raton, FL, 2013, pp. 232235. https://doi.org/10.1201/b15006
[St09] Stichtenoth, H., Algebraic function fields and codes . Graduate Texts in Mathematics, 254, Springer-Verlag, Berlin, 2009.
[SV86] Stöhr, K. O. and Voloch, J. F., Weierstrass points and curves over finite fields . Proc. London Math. Soc. 52(1986), 119. https://doi.org/10.1112/plms/s3-52.1.1
[Vol85] Voloch, J. F., Diagonal equations over function fields . Bol. Soc. Brasil. Mat. 16(1985), 2939. https://doi.org/10.1007/BF02584799
[Vol89] Voloch, J. F., On the number of values taken by a polynomial over a finite field . Acta Arith. 52(1989), 197201. https://doi.org/10.4064/aa-52-2-197-201
[WSC93] Wan, D., Shiue, P. J.-S., and Chen, C. S., Value sets of polynomials over finite fields . Proc. Amer. Math. Soc. 119(1993), 711717. https://doi.org/10.2307/2160504
[Zan07] Zannier, U., On the number of terms of a composite polynomial . Acta Arith. 127(2007), 157168. https://doi.org/10.4064/aa127-2-5
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Value Sets of Sparse Polynomials

  • Igor E. Shparlinski (a1) and José Felipe Voloch (a2)

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