Abstract
Given a multivariate polynomial P(X 1,…,X n ) over a finite field \(\ensuremath {\mathbb {F}_{q}}\) , let N(P) denote the number of roots over \(\ensuremath {\mathbb {F}_{q}}^{n}\) . The modular root counting problem is given a modulus r, to determine N r (P)=N(P)mod r. We study the complexity of computing N r (P), when the polynomial is given as a sum of monomials. We give an efficient algorithm to compute N r (P) when the modulus r is a power of the characteristic of the field. We show that for all other moduli, the problem of computing N r (P) is \({\rm NP}\) -hard. We present some hardness results which imply that our algorithm is essentially optimal for prime fields. We show an equivalence between maximum-likelihood decoding for Reed-Solomon codes and a root-finding problem for symmetric polynomials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adleman, L., Huang, M.-D.: Counting rational points on curves and Abelian varieties over finite fields. In: Proceedings of the 1996 Algorithmic Number Theory Symposium. LNCS, vol. 1122, pp. 1–16. Springer, Berlin (1996)
Artin, M.: Algebra. Prentice Hall, New York (1991)
Beigel, R., Tarui, J.: On ACC. Comput. Complex. 4, 350–366 (1994)
Ehrenfeucht, A., Karpinski, M.: The computational complexity of (xor, and)-counting problems. Tech. Report 8543-CS, ICSI, Berkeley (1990)
Grigoriev, D., Karpinski, M.: An approximation algorithm for the number of zeroes of arbitrary polynomials over GF[q]. In: Proc. 32nd IEEE Symposium on Foundations of Computer Science (FOCS’91), pp. 662–669 (1991)
Guruswami, V., Vardy, A.: Maximum-likelihood decoding of Reed-Solomon codes is NP-hard. In: Proceedings of the ACM-SIAM symposium on Discrete Algorithms (SODA’05), pp. 470–478 (2005)
Huang, M.-D., Ierardi, D.: Counting rational points on curves over finite fields. In: Proceedings of the 34th IEEE Symposium on Foundations of Computer Science (FOCS’93), pp. 616–625 (1993)
Huang, M.-D., Wong, Y.: Solvability of systems of polynomial congruences modulo a large prime. J. Comput. Complex. 8, 227–257 (1999)
Karpinski, M., Luby, M.: Approximating the number of zeroes of a GF[2] polynomial. J. Algorithms 14, 280–287 (1993)
Kedlaya, K.: Computing Zeta functions via p-adic cohomology. In: Algorithmic Number Theory Symposium (ANTS), pp. 1–17 (2004)
Lauder, A., Wan, D.: Computing zeta functions of Artin-Schreier curves over finite fields ii. J. Complex. 20(2–3), 331–349 (2004)
Lauder, A., Wan, D.: Counting points on varieties over finite fields of small characteristic. In: Buhler, J.P., Stevenhagen, P. (eds.) Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography (Mathematical Sciences Research Institute Publications). Cambridge University Press (2007, to appear)
Lidl, R., Niederreiter, H.: Finite Fields, Encylopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (1997)
Luby, M., Velicković, B., Wigderson, A.: Deterministic approximate counting of depth-2 circuits. In: Israel Symposium on Theory of Computing Systems, pp. 18–24 (1993)
Mason, R.C.: Diophantine Equations Over Function Fields. Cambridge University Press, Cambridge (1984)
Moreno, O., Moreno, C.J.: Improvements of the Chevalley-Warning and the Ax-Katz theorems. Am. J. Math. 117(1), 241–244 (1995)
Papadimitriou, C.: Computational Complexity. Addison–Wesley, Reading (1994)
Pila, J.: Frobenius maps of Abelian varieties and finding roots of unity in finite fields. Math. Comput. 55, 745–763 (1990)
Schoof, R.: Counting points on elliptic curves over finite fields. J. Th’eor. Nr. Bord. 7, 219–254 (1995)
Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM J. Comput. 20(5), 865–877 (1991)
Valiant, L.G.: The complexity of computing the permanent. Theor. Comput. Sci. 189–201 (1979)
Valiant, L.G.: Completeness for parity problems. In: Proc. 11th International Computing and Combinatorics Conference (COCOON’05), pp. 1–8 (2005)
Valiant, L.G.: Accidental algorithms. In: Proc. 47th IEEE Symposium on Foundations of Computer Science (FOCS’06), pp. 509–517 (2006)
Valiant, L.G., Vazirani, V.V.: NP is as easy as detecting unique solutions. Theor. Comput. Sci. 47, 85–93 (1986)
von zur Gathen, J., Karpinski, M., Shparlinski, I.: Counting curves and their projections. Comput. Complex. 6, 64–99 (1996)
Wan, D.: A Chevalley-Warning approach to p-adic estimate of character sums. Proc. Am. Math. Soc. 123, 45–54 (1995)
Wan, D.: Computing Zeta functions over finite fields. Contemp. Math. 225, 135–141 (1999)
Wan, D.: Modular counting of rational points on sparse equations over finite fields. Found. Comput. Math. (2006, to appear)
Wan, D.: Personal communication (April 2006)
Yao, A.C.: On ACC and threshold circuits. In: 31st IEEE Symposium on Foundations of Computer Science (FOCS’90), pp. 619–627 (1990)
Author information
Authors and Affiliations
Corresponding author
Additional information
P. Gopalan’s and R.J Lipton’s research was supported by NSF grant CCR-3606B64.
V. Guruswami’s research was supported in part by NSF grant CCF-0343672 and a Sloan Research Fellowship.
Rights and permissions
About this article
Cite this article
Gopalan, P., Guruswami, V. & Lipton, R.J. Algorithms for Modular Counting of Roots of Multivariate Polynomials. Algorithmica 50, 479–496 (2008). https://doi.org/10.1007/s00453-007-9097-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00453-007-9097-3