Advertisement

The Ramanujan Journal

, Volume 47, Issue 1, pp 67–84 | Cite as

Counting polynomial subset sums

  • Jiyou Li
  • Daqing Wan
Article
  • 65 Downloads

Abstract

Let D be a subset of a finite commutative ring R with identity. Let \(f(x)\in R[x]\) be a polynomial of degree d. For a nonnegative integer k, we study the number \(N_f(D,k,b)\) of k-subsets S in D such that
$$\begin{aligned} \sum _{x\in S} f(x)=b. \end{aligned}$$
In this paper, we establish several bounds for the difference between \(N_f(D,k, b)\) and the expected main term \(\frac{1}{|R|}{|D|\atopwithdelims ()k}\), depending on the nature of the finite ring R and f. For \(R=\mathbb {Z}_n\), let \(p=p(n)\) be the smallest prime divisor of n, \(|D|=n-c \ge C_dn p^{-\frac{1}{d}}\,+\,c\) and \(f(x)=a_dx^d +\cdots +a_0\in \mathbb {Z}[x]\) with \((a_d, \ldots , a_1, n)=1\). Then
$$\begin{aligned} \left| N_f(D, k, b)-\frac{1}{n}{n-c \atopwithdelims ()k}\right| \le {\delta (n)(n-c)+(1-\delta (n))\left( C_dnp^{-\frac{1}{d}}+c\right) +k-1\atopwithdelims ()k}, \end{aligned}$$
answering an open question raised by Stanley (Enumerative combinatorics, 1997) in a general setting, where \(\delta (n)=\sum _{i\mid n, \mu (i)=-1}\frac{1}{i}\) and \(C_d=e^{1.85d}\). Furthermore, if n is a prime power, then \(\delta (n) =1/p\) and one can take \(C_d=4.41\). Similar and stronger bounds are given for two more cases. The first one is when \(R=\mathbb {F}_q\), a q-element finite field of characteristic p and f(x) is general. The second one is essentially the well-known subset sum problem over an arbitrary finite abelian group. These bounds extend several previous results.

Keywords

Polynomial subset sums Inclusion–exclusion Character sums Subset sum problem Counting problems 

Mathematics Subject Classification

11T06 11T24 05A15 05A16 

Notes

Acknowledgements

The authors wish to thank Professor Richard Stanley for his helpful suggestions.

References

  1. 1.
    Andrews, G.: On a conjecture of Peter Borwein. J. Symb. Comput. 20, 487–501 (1995)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bourgain, J.: Sum-product theorems and applications. In: Additive Number Theory: Festschrift. In Honor of the Sixtieth Birthday of Melvyn B, Nathanson (2010)Google Scholar
  3. 3.
    Bourgain, J., Konyagin, S.: Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order. C. R. Math. Acad. Sci. Paris 337, 75–80 (2003)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bourgain, J., Glibichuk, A., Konyagin, S.: Estimates for the number of sums and products and for exponential sums in fields of prime order. J. Lond. Math. Soc. (2) 73, 380–398 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cheng, Q.: Hard problems of algebraic geometry codes. IEEE Trans. Inf. Theory 54, 402–406 (2008)CrossRefGoogle Scholar
  6. 6.
    Cheng, Q., Hill, J., Wan, D.: Counting value sets: algorithm and complexity. In: Proceedings of the Tenth Algorithmic Number Theory Symposium. Open Book Series 1, pp. 235–248. Mathematical Science Publishers, Berkeley (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cochrane, T., Zheng, Z.: On upper bounds of Chalk and Hua for exponential sums. Proc. Am. Math. Soc. 129, 2505–2516 (2001)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cochrane, T., Zheng, Z.: A survey on pure and mixed exponential sums modulo prime powers. In: Number Theory for the Millennium, I, pp. 273–300. AK Peters, Natick (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ding, P., Qi, M.: Further estimate of complete trigonometric sums. J. Tsinghua Univ. 29, 74–85 (1989)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Erdős, P., Heilbronn, H.: On the addition of residue classes mod p. Acta Arith. 9, 149–159 (1964)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Galil, Z., Margalit, O.: An almost linear-time algorithm for the dense subset-sum problem. SIAM J. Comput. 20, 1157–1189 (1991)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Heath-Brown, D.R., Konyagin, S.V.: New bounds for Gauss sums derived from \(k\)th powers, and for Heilbronn’s exponential sum. Q. J. Math. 51, 221–235 (2000)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Heilbronn, H.: Lecture Notes on Additive Number Theory mod \(p\). California Institute of Technology, Pasadena (1964)Google Scholar
  14. 14.
    Hua, L.K.: On an exponential sum. J. Chin. Math. Soc. 2, 301–312 (1940)zbMATHGoogle Scholar
  15. 15.
    Hua, L.K.: On exponential sums. Sci. Rec. (N.S.) 1, 1–4 (1957)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Hua, L.K.: Additive Primzahltheorie (German). B. G. Teubner Verlagsgesellschaft, Leipzig (1959)Google Scholar
  17. 17.
    Konyagin, S.: Estimates for Gaussian sums and Waring’s problem modulo a prime (Russian). Trudy Mat. Inst. Steklov. 198, 111–124 (1992); translation in Proc. Steklov Inst. Math. 198, 105–117 (1994)Google Scholar
  18. 18.
    Konyagin, S., Shparlinski, I.E.: Character Sums with Exponential Functions and Their Applications. Cambridge Tracts in Mathematics, vol. 136. Cambridge University Press, Cambridge (1999)Google Scholar
  19. 19.
    Kosters, M.: The subset sum problem for finite abelian groups. J. Combin. Theory Ser. A 120, 527–530 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kitchloo, N., Pachter, L.: An interesting result about subset sums. MIT Unpublished Notes (1994)Google Scholar
  21. 21.
    Li, J.: On the Odlyzko–Stanley enumeration problem and Warings problem over finite fields. J. Number Theory 133, 2267–2276 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Li, J.: A note on the Borwein conjecture (2017). arXiv:1512.01191
  23. 23.
    Li, J., Wan, D.: On the subset sum problem over finite fields. Finite Fields Appl. 14, 911–929 (2008)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Li, J., Wan, D.: A new sieve for distinct coordinate counting. Sci. China Ser. A 53, 2351–2362 (2010)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Li, J., Wan, D.: Counting subsets of finite abelian groups. J. Combin. Theory Ser. A 19, 170–182 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Li, J., Wan, D., Zhang, J.: On the minimum distance of elliptic curve codes. In: Proceedings of the IEEE International Symposium on Information Theory (ISIT), pp. 2391–2395 (2015)Google Scholar
  27. 27.
    Lu, M.: Estimate of a complete trigonometric sum. Sci. Sin. Ser. A 28, 561–578 (1985)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Odlyzko, A.M., Stanley, R.P.: Enumeration of power sums modulo a prime. J. Number Theory 10, 263–272 (1978)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Ramanathan, K.G.: Some applications of Ramanujan’s trigonometrical sum \(C_m(n)\). Proc. Indian Acad. Sci. 20, 62–69 (1945)zbMATHGoogle Scholar
  30. 30.
    Stanley, R.P.: Enumerative Combinatorics, vol. 1, Second edn. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  31. 31.
    Stanley, R.P., Yoder, M.F.: A study of Varshamov codes for asymmetric channels. JPL Technical Report 32-1526, DSM, vol. XIV, pp. 117–123 (1973)Google Scholar
  32. 32.
    Stečkin, S.B.: Estimate of a complete rational trigonometric sum, Proc. Inst. Steklov. 143, 188–220 (1977) (English translation, A.M.S. Issue 1, pp. 201–220, 1980)Google Scholar
  33. 33.
    Wan, D.: Generators and irriducible polyniomials over finite fields. Math. Comput. 66, 1195–1212 (1997)CrossRefGoogle Scholar
  34. 34.
    Zhang, J., Fu, F., Wan, D.: Stopping sets of algebraic geometry codes. IEEE Trans. Inf. Theory 60, 1488–1495 (2014)CrossRefGoogle Scholar
  35. 35.
    Zhu, G., Wan, D.: An asymptotic formula for counting subset sums over subgroups of finite fields. Finite Fields Appl. 18, 192–209 (2012)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Zhu, G., Wan, D.: Computing the error distance of Reed–Solomon codes. In: Agrawal, M., Cooper, S.B., Li, A. (eds.) TAMC 2012. LNCS, vol. 7287, pp. 214–224. Springer, Berlin (2012)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  3. 3.Department of MathematicsUniversity of CaliforniaIrvineUSA

Personalised recommendations