Polynomial Identity Testing via Evaluation of Rational Functions

by Ivan Hu, Dieter van Melkebeek, and Andrew Morgan

Theory of Computing, Volume 20(1), pp. 1-70, 2024

Bibliography with links to cited articles

[1]   William Adams and Philippe Loustaunau: An Introduction to Gröbner Bases. Amer. Math. Soc., 1994. [doi:10.1090/gsm/003]

[2]   Manindra Agrawal: Proving lower bounds via pseudo-random generators. In Proc. 25th Found. Softw. Techn. Theoret. Comp. Sci. Conf. (FSTTCS’05), pp. 92–105. Springer, 2005. [doi:10.1007/11590156_6]

[3]   Manindra Agrawal, Rohit Gurjar, Arpita Korwar, and Nitin Saxena: Hitting-sets for ROABP and sum of set-multilinear circuits. SIAM J. Comput., 44(3):669–697, 2015. [doi:10.1137/140975103]

[4]   Manindra Agrawal, Chandan Saha, and Nitin Saxena: Quasi-polynomial hitting-set for set-depth-Δ formulas. In Proc. 45th STOC, pp. 321–330. ACM Press, 2013. [doi:10.1145/2488608.2488649]

[5]   Ravindra Ahuja, Thomas Magnanti, and James Orlin: Network Flows: Theory, Algorithms, and Applications. Pearson, 1993. [doi:10.5555/137406]

[6]   Matthew Anderson, Michael A. Forbes, Ramprasad Saptharishi, Amir Shpilka, and Ben Lee Volk: Identity testing and lower bounds for read-k oblivious algebraic branching programs. ACM Trans. Comput. Theory, 10(1/3):1–30, 2018. [doi:10.1145/3170709]

[7]   Matthew Anderson, Dieter van Melkebeek, and Ilya Volkovich: Deterministic polynomial identity tests for multilinear bounded-read formulae. Comput. Complexity, 24(4):695–776, 2015. [doi:10.1007/s00037-015-0097-4]

[8]   Robert Andrews and Michael A. Forbes: Ideals, determinants, and straightening: proving and using lower bounds for polynomial ideals. In Proc. 54th STOC, pp. 389–402. ACM Press, 2022. [doi:10.1145/3519935.3520025]

[9]   Vikraman Arvind, Pushkar S. Joglekar, Partha Mukhopadhyay, and S. Raja: Randomized polynomial-time identity testing for noncommutative circuits. Theory of Computing, 15(7):1–36, 2019. [doi:10.4086/toc.2019.v015a007]

[10]   Vishwas Bhargava and Sumanta Ghosh: Improved hitting set for orbit of ROABPs. Comput. Complexity, 31(15), 2022. [doi:10.1007/s00037-022-00230-9]

[11]   Prerona Chatterjee and Anamay Tengse: On annihilators of explicit polynomial maps, 2023. [arXiv:2309.07612]

[12]   David A. Cox, John Little, and Donal O’Shea: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer, 2013. [doi:10.1007/978-3-319-16721-3]

[13]   Richard A. Demillo and Richard J. Lipton: A probabilistic remark on algebraic program testing. Inform. Process. Lett., 7(4):193–195, 1978. [doi:10.1016/0020-0190(78)90067-4]

[14]   Michael A. Forbes: Deterministic divisibility testing via shifted partial derivatives. In Proc. 56th FOCS, pp. 451–465. IEEE Comp. Soc., 2015. [doi:10.1109/FOCS.2015.35]

[15]   Michael A. Forbes, Ramprasad Saptharishi, and Amir Shpilka: Hitting sets for multilinear read-once algebraic branching programs, in any order. In Proc. 46th STOC, pp. 867–875. ACM Press, 2014. Full version arXiv:1309.5668. [doi:10.1145/2591796.2591816]

[16]   Michael A. Forbes and Amir Shpilka: On identity testing of tensors, low-rank recovery and compressed sensing. In Proc. 44th STOC, p. 163–172. ACM Press, 2012. [doi:10.1145/2213977.2213995]

[17]   Michael A. Forbes and Amir Shpilka: Quasipolynomial-time identity testing of non-commutative and read-once oblivious algebraic branching programs. In Proc. 54th FOCS, pp. 243–252. IEEE Comp. Soc., 2013. [doi:10.1109/FOCS.2013.34]

[18]   Michael A. Forbes, Amir Shpilka, Iddo Tzameret, and Avi Wigderson: Proof complexity lower bounds from algebraic circuit complexity. Theory of Computing, 17(10):1–88, 2021. [doi:10.4086/toc.2021.v017a010]

[19]   Michael A. Forbes, Amir Shpilka, and Ben Lee Volk: Succinct hitting sets and barriers to proving lower bounds for algebraic circuits. Theory of Computing, 14(18):1–45, 2018. Preliminary version in STOC’17. [doi:10.4086/toc.2018.v014a018]

[20]   Hervé Fournier and Arpita Korwar: Limitations of the Shpilka–Volkovich generator. In Workshop on Algebraic Complexity Theory (WACT), Paris, 2018. CONF and SLIDES.

[21]   Ariel Gabizon and Ran Raz: Deterministic extractors for affine sources over large fields. Combinatorica, 28(4):415–440, 2008. [doi:10.1007/s00493-008-2259-3]

[22]   Hermann Grassmann: Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik: dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert. Otto Wigand, Leipzig, 1844. Available at HathiTrust.

[23]   Hermann Grassmann: Extension theory. Amer. Math. Soc., 2000. Translation of [22], translated by Lloyd C. Kannenberg.

[24]   Zeyu Guo and Rohit Gurjar: Improved explicit hitting-sets for ROABPs. In Proc. 24th Internat. Conf. on Randomization and Computation (RANDOM’20), pp. 4:1–16. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2020. [doi:10.4230/LIPIcs.APPROX/RANDOM.2020.4]

[25]   Rohit Gurjar, Arpita Korwar, and Nitin Saxena: Identity testing for constant-width, and any-order, read-once oblivious arithmetic branching programs. Theory of Computing, 13(2):1–21, 2017. [doi:10.4086/toc.2017.v013a002]

[26]   Rohit Gurjar, Arpita Korwar, Nitin Saxena, and Thomas Thierauf: Deterministic identity testing for sum of read-once oblivious arithmetic branching programs. Comput. Complexity, 26(4):835–880, 2017. [doi:10.1007/s00037-016-0141-z]

[27]   Joos Heintz and Claus-Peter Schnorr: Testing polynomials which are easy to compute. In Proc. 12th STOC, pp. 262–272. ACM Press, 1980. [doi:10.1145/800141.804674]

[28]   Russell Impagliazzo and Avi Wigderson: P=BPP if E requires exponential circuits: Derandomizing the XOR lemma. In Proc. 29th STOC, pp. 220–229. ACM Press, 1997. [doi:10.1145/258533.258590]

[29]   Maurice Jansen, Youming Qiao, and Jayalal Sarma M. N.: Deterministic identity testing of read-once algebraic branching programs, 2009. [arXiv:0912.2565]

[30]   Maurice Jansen, Youming Qiao, and Jayalal Sarma M. N.: Deterministic black-box identity testing π-ordered algebraic branching programs. In Proc. 30th Found. Softw. Techn. Theoret. Comp. Sci. Conf. (FSTTCS’10), pp. 296–307. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2010. [doi:10.4230/LIPIcs.FSTTCS.2010.296]

[31]    Valentine Kabanets and Russell Impagliazzo: Derandomizing polynomial identity tests means proving circuit lower bounds. Comput. Complexity, 13(1–2):1–46, 2004. [doi:10.1007/s00037-004-0182-6]

[32]   Zohar S. Karnin, Partha Mukhopadhyay, Amir Shpilka, and Ilya Volkovich: Deterministic identity testing of depth-4 multilinear circuits with bounded top fan-in. SIAM J. Comput., 42(6):2114–2131, 2013. [doi:10.1137/110824516]

[33]   Zohar S. Karnin and Amir Shpilka: Black box polynomial identity testing of generalized depth-3 arithmetic circuits with bounded top fan-in. Combinatorica, 31(3):333–364, 2011. [doi:10.1007/s00493-011-2537-3]

[34]   Adam R. Klivans and Daniel Spielman: Randomness efficient identity testing of multivariate polynomials. In Proc. 33rd STOC, p. 216–223. ACM Press, 2001. [doi:10.1145/380752.380801]

[35]   Arpita Korwar: Personal communication, 2021.

[36]   Klaus Kühnle and Ernst W. Mayr: Exponential space computation of Gröbner bases. In Proc. 21st Internat. Symp. Symbolic and Algebraic Computation (ISSAC’96), pp. 63–71. ACM Press, 1996. [doi:10.1145/236869.236900]

[37]   Mrinal Kumar and Shubhangi Saraf: Arithmetic circuits with locally low algebraic rank. Theory of Computing, 13(6):1–33, 2017. [doi:10.4086/toc.2017.v013a006]

[38]   Ernst W. Mayr: Some complexity results for polynomial ideals. J. Complexity, 13(3):303–325, 1997. [doi:10.1006/jcom.1997.0447]

[39]   Dori Medini and Amir Shpilka: Hitting sets and reconstruction for dense orbits in VPe and ΣΠΣ circuits. In Proc. 36th Comput. Complexity Conf. (CCC’21), pp. 19:1–27. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2021. [doi:10.4230/LIPIcs.CCC.2021.19]

[40]   Dieter van Melkebeek and Andrew Morgan: Polynomial identity testing via evaluation of rational functions. In Proc. 13th Innovations in Theoret. Comp. Sci. Conf. (ITCS’22), pp. 119:1–24. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2022. [doi:10.4230/LIPIcs.ITCS.2022.119]

[41]   Daniel Minahan and Ilya Volkovich: Complete derandomization of identity testing and reconstruction of read-once formulas. ACM Trans. Comput. Theory, 10(3/10):1–11, 2018. [doi:10.1145/3196836]

[42]   Noam Nisan: Lower bounds for non-commutative computation. In Proc. 23rd STOC, pp. 410–418. ACM Press, 1991. [doi:10.1145/103418.103462]

[43]   Noam Nisan and Avi Wigderson: Hardness vs randomness. J. Comput. System Sci., 49(2):149–167, 1994. Preliminary version in FOCS’88. [doi:10.1016/S0022-0000(05)80043-1]

[44]   Øystein Ore: Über höhere Kongruenzen. Norsk Mat. Forenings Skrifter, Ser. I, 1(7):1–15, 1922.

[45]   Ran Raz and Amir Shpilka: Deterministic polynomial identity testing in non-commutative models. Comput. Complexity, 14(1):1–19, 2005. [doi:10.1007/s00037-005-0188-8]

[46]   Chandan Saha and Bhargav Thankey: Hitting sets for orbits of circuit classes and polynomial families. In Proc. 25th Internat. Conf. on Randomization and Computation (RANDOM’21), pp. 50:1–26. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2021. [doi:10.4230/LIPIcs.APPROX/RANDOM.2021.50, ECCC:TR21–015]

[47]   Jacob T. Schwartz: Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4):701–717, 1980. [doi:10.1145/322217.322225]

[48]    Amir Shpilka and Ilya Volkovich: Read-once polynomial identity testing. Comput. Complexity, 24(3):477–532, 2015. [doi:10.1007/s00037-015-0105-8]

[49]   Amir Shpilka and Amir Yehudayoff: Arithmetic circuits: A survey of recent results and open questions. Found. Trends Theor. Comp. Sci., 5(3–4), 2010. [doi:10.1561/0400000039]

[50]   Richard Zippel: Probabilistic algorithms for sparse polynomials. In Proc. Internat. Symp. Symbolic and Algebraic Computation (EUROSAM’79), pp. 216–226. ACM Press, 1979. [doi:10.1007/3-540-09519-5_73]