On Some Computations on Sparse Polynomials
In arithmetic circuit complexity the standard operations are +,x. Yet, in some scenarios exponentiation gates are considered as well. In this paper we study the question of efficiently evaluating a polynomial given an oracle access to its power. Among applications, we show that:
* A reconstruction algorithm for a circuit class c can be extended to handle f^e for f in C.
* There exists an efficient deterministic algorithm for factoring sparse multiquadratic polynomials.
* There is a deterministic algorithm for testing a factorization of sparse polynomials, with constant individual degrees, into sparse irreducible factors. That is, testing if f = g_1 x ... x g_m when f has constant individual degrees and g_i-s are irreducible.
* There is a deterministic reconstruction algorithm for multilinear depth-4 circuits with two multiplication gates.
* There exists an efficient deterministic algorithm for testing whether two powers of sparse polynomials are equal. That is, f^d = g^e when f and g are sparse.
Derandomization
Arithmetic Circuits
Reconstruction
48:1-48:21
Regular Paper
Ilya
Volkovich
Ilya Volkovich
10.4230/LIPIcs.APPROX-RANDOM.2017.48
M. Agrawal, C. Saha, R. Saptharishi, and N. Saxena. Jacobian hits circuits: Hitting-sets, lower bounds for depth-d occur-k formulas & depth-3 transcendence degree-k circuits. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC), pages 599-614, 2012.
M. Agrawal and V. Vinay. Arithmetic circuits: A chasm at depth four. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 67-75, 2008.
D. Angluin, L. Hellerstein, and M. Karpinski. Learning read-once formulas with queries. J. ACM, 40(1):185-210, 1993.
M. Beecken, J. Mittmann, and N. Saxena. Algebraic independence and blackbox identity testing. Information &Computation, 222:2-19, 2013. URL: http://dx.doi.org/10.1016/j.ic.2012.10.004.
http://dx.doi.org/10.1016/j.ic.2012.10.004
M. Ben-Or and P. Tiwari. A deterministic algorithm for sparse multivariate polynominal interpolation. In Proceedings of the 20th Annual ACM Symposium on Theory of Computing (STOC), pages 301-309, 1988.
D. Bshouty and N. H. Bshouty. On interpolating arithmetic read-once formulas with exponentiation. JCSS, 56(1):112-124, 1998.
N. H. Bshouty and R. Cleve. Interpolating arithmetic read-once formulas in parallel. SIAM J. on Computing, 27(2):401-413, 1998.
N. H. Bshouty, T. R. Hancock, and L. Hellerstein. Learning boolean read-once formulas with arbitrary symmetric and constant fan-in gates. JCSS, 50:521-542, 1995.
D. Coppersmith and J. Davenport. Polynomials whose powers are sparse. Acta Arith., 58:79-87, 1991.
D. A. Cox, J. Little, and D. O'Shea. Ideals, varieties, and algorithms - an introduction to computational algebraic geometry and commutative algebra (4. ed.). Undergraduate texts in mathematics. Springer, 2015.
Z. Dvir and R. Mendes de Oliveira. Factors of sparse polynomials are sparse. CoRR, abs/1404.4834, 2014.
Z. Dvir, A. Shpilka, and A. Yehudayoff. Hardness-randomness tradeoffs for bounded depth arithmetic circuits. SIAM J. on Computing, 39(4):1279-1293, 2009.
P. Erdös. On the number of terms of the square of a polynomial. Nieuw Arch. Wisk, 23:63-65, 1949.
S. Gao, E. Kaltofen, and A. G. B. Lauder. Deterministic distinct-degree factorization of polynomials over finite fields. J. Symb. Comput., 38(6):1461-1470, 2004.
K. O. Geddes, S. R. Czapor, and G. Labahn. Algorithms for computer algebra. Kluwer, 1992.
A. Gupta, N. Kayal, and S. V. Lokam. Reconstruction of depth-4 multilinear circuits with top fanin 2. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC), pages 625-642, 2012. Full version at http://eccc.hpi-web.de/report 153.
V. Guruswami and M. Sudan. Improved decoding of reed-solomon codes and algebraic-geometry codes. IEEE Transactions on Information Theory, 45(6):1757-1767, 1999.
T. R. Hancock and L. Hellerstein. Learning read-once formulas over fields and extended bases. In Proceedings of the 4th Annual Workshop on Computational Learning Theory (COLT), pages 326-336, 1991.
V. Kabanets and R. Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity, 13(1-2):1-46, 2004.
E. Kaltofen. Single-factor hensel lifting and its application to the straight-line complexity of certain polynomials. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC), pages 443-452, 1987. URL: http://dx.doi.org/10.1145/28395.28443.
http://dx.doi.org/10.1145/28395.28443
E. Kaltofen. Factorization of polynomials given by straight-line programs. In S. Micali, editor, Randomness in Computation, volume 5 of Advances in Computing Research, pages 375-412. JAI Press Inc., Greenwhich, Connecticut, 1989.
E. Kaltofen. Polynomial factorization: a success story. In ISSAC, pages 3-4, 2003.
E. Kaltofen and B. M. Trager. Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators. J. of Symbolic Computation, 9(3):301-320, 1990.
I. Kaplansky. An Introduction to Differential Algebra. Hermann, Paris, 1957.
M. Karchmer, N. Linial, I. Newman, M. E. Saks, and A. Wigderson. Combinatorial characterization of read-once formulae. Discrete Mathematics, 114(1-3):275-282, 1993.
Z. S. Karnin, P. Mukhopadhyay, A. Shpilka, and I. Volkovich. Deterministic identity testing of depth 4 multilinear circuits with bounded top fan-in. SIAM J. on Computing, 42(6):2114-2131, 2013.
N. Kayal. Derandomizing some number-theoretic and algebraic algorithms. PhD thesis, Indian Institute of Technology, Kanpur, India, 2007.
N. Kayal. An exponential lower bound for the sum of powers of bounded degree polynomials. Electronic Colloquium on Computational Complexity (ECCC), 19:81, 2012. URL: https://eccc.weizmann.ac.il/report/2012/081/.
https://eccc.weizmann.ac.il/report/2012/081/
A. Klivans and D. Spielman. Randomness efficient identity testing of multivariate polynomials. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC), pages 216-223, 2001.
S. Kopparty, S. Saraf, and A. Shpilka. Equivalence of polynomial identity testing and deterministic multivariate polynomial factorization. In Proceedings of the 29th Annual IEEE Conference on Computational Complexity (CCC), pages 169-180, 2014. URL: http://dx.doi.org/10.1109/CCC.2014.25.
http://dx.doi.org/10.1109/CCC.2014.25
A. K. Lenstra, H. W. Lenstr, and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen,, 261(4):515-534, 1982.
R. J. Lipton and N. K. Vishnoi. Deterministic identity testing for multivariate polynomials. In Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 756-760, 2003.
D. Minahan and I. Volkovich. Complete derandomization of identity testing and reconstruction of read-once formulas. Manuscript, 2016. (submitted).
C. Saha, R. Saptharishi, and N. Saxena. A case of depth-3 identity testing, sparse factorization and duality. Computational Complexity, 22(1):39-69, 2013. URL: http://dx.doi.org/10.1007/s00037-012-0054-4.
http://dx.doi.org/10.1007/s00037-012-0054-4
S. Saraf and I. Volkovich. Blackbox identity testing for depth-4 multilinear circuits. Combinatorica, 2016. (accepted).
V. Shoup. A fast deterministic algorithm for factoring polynomials over finite fields of small characteristic. In ISSAC, pages 14-21, 1991.
A. Shpilka and I. Volkovich. On the relation between polynomial identity testing and finding variable disjoint factors. In Automata, Languages and Programming, 37th International Colloquium (ICALP), pages 408-419, 2010. Full version at http://eccc.hpi-web.de/report 036.
A. Shpilka and I. Volkovich. On reconstruction and testing of read-once formulas. Theory of Computing, 10:465-514, 2014.
A. Shpilka and I. Volkovich. Read-once polynomial identity testing. Computational Complexity, 24(3):477-532, 2015.
A. Shpilka and A. Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science, 5(3-4):207-388, 2010.
M. Sudan. Decoding of reed solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180-193, 1997.
M. Sudan, L. Trevisan, and S. P. Vadhan. Pseudorandom generators without the XOR lemma. J. Comput. Syst. Sci., 62(2):236-266, 2001. URL: http://dx.doi.org/10.1006/jcss.2000.1730.
http://dx.doi.org/10.1006/jcss.2000.1730
L. G. Valiant. Negation can be exponentially powerful. Theoretical Computer Science, 12(3):303-314, 1980.
I. Volkovich. Deterministically factoring sparse polynomials into multilinear factors and sums of univariate polynomials. In APPROX-RANDOM, pages 943-958, 2015.
I. Volkovich. Characterizing arithmetic read-once formulae. ACM Transactions on Computation Theory (ToCT), 8(1):2, 2016. URL: http://dx.doi.org/10.1145/2858783.
http://dx.doi.org/10.1145/2858783
I. Volkovich. A guide to learning arithmetic circuits. In Proceedings of the 29th Conference on Learning Theory, (COLT), pages 1540-1561, 2016. URL: http://jmlr.org/proceedings/papers/v49/volkovich16.html.
http://jmlr.org/proceedings/papers/v49/volkovich16.html
J. von zur Gathen. Who was who in polynomial factorization. In ISSAC, page 2, 2006.
J. von zur Gathen and J. Gerhard. Modern computer algebra. Cambridge University Press, 1999.
J. von zur Gathen and E. Kaltofen. Factoring sparse multivariate polynomials. Journal of Computer and System Sciences, 31(2):265-287, 1985. URL: http://dx.doi.org/10.1016/0022-0000(85)90044-3.
http://dx.doi.org/10.1016/0022-0000(85)90044-3
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode