Progress on Polynomial Identity Testing - II

We survey the area of algebraic complexity theory; with the focus being on the problem of polynomial identity testing (PIT). We discuss the key ideas that have gone into the results of the last few years.

[1]  Adi Shamir,et al.  IP = PSPACE , 1992, JACM.

[2]  Avi Wigderson,et al.  Partial Derivatives in Arithmetic Complexity and Beyond , 2011, Found. Trends Theor. Comput. Sci..

[3]  Manindra Agrawal,et al.  Classifying polynomials and identity testing , 2009 .

[4]  Nitin Saxena,et al.  Blackbox identity testing for bounded top fanin depth-3 circuits: the field doesn't matter , 2010, STOC '11.

[5]  Leslie G. Valiant,et al.  Fast Parallel Computation of Polynomials Using Few Processors , 1983, SIAM J. Comput..

[6]  Manindra Agrawal,et al.  PRIMES is in P , 2004 .

[7]  Ketan Mulmuley,et al.  On P vs. NP and geometric complexity theory: Dedicated to Sri Ramakrishna , 2011, JACM.

[8]  Marek Karpinski,et al.  An exponential lower bound for depth 3 arithmetic circuits , 1998, STOC '98.

[9]  Meena Mahajan,et al.  Non-Commutative Arithmetic Circuits: Depth Reduction and Size Lower Bounds , 1998, Theor. Comput. Sci..

[10]  Vikraman Arvind,et al.  New Results on Noncommutative and Commutative Polynomial Identity Testing , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[11]  Nitin Saxena,et al.  Diagonal Circuit Identity Testing and Lower Bounds , 2008, ICALP.

[12]  Nitin Saxena,et al.  From Sylvester-Gallai Configurations to Rank Bounds: Improved Black-Box Identity Test for Depth-3 Circuits , 2010, FOCS.

[13]  Pavel Pudlák,et al.  Pseudorandom generators for group products: extended abstract , 2011, STOC '11.

[14]  Joos Heintz,et al.  Testing polynomials which are easy to compute (Extended Abstract) , 1980, STOC '80.

[15]  Nitin Saxena,et al.  An Almost Optimal Rank Bound for Depth-3 Identities , 2008, 2009 24th Annual IEEE Conference on Computational Complexity.

[16]  Michael Ben-Or,et al.  Computing algebraic formulas with a constant number of registers , 1988, STOC '88.

[17]  Joachim von zur Gathen,et al.  Factoring Sparse Multivariate Polynomials , 1983, J. Comput. Syst. Sci..

[18]  Peter B. Borwein The Desmic Conjecture , 1983, J. Comb. Theory, Ser. A.

[19]  Ketan Mulmuley,et al.  The GCT program toward the P vs. NP problem , 2012, Commun. ACM.

[20]  Michael Ben-Or,et al.  Computing Algebraic Formulas Using a Constant Number of Registers , 1992, SIAM J. Comput..

[21]  Russell Impagliazzo,et al.  Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds , 2003, STOC '03.

[22]  Ketan Mulmuley,et al.  Geometric Complexity Theory V: Equivalence between Blackbox Derandomization of Polynomial Identity Testing and Derandomization of Noether's Normalization Lemma , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[23]  Manindra Agrawal,et al.  The Polynomially Bounded Perfect Matching Problem Is in NC 2 , 2007, STACS.

[24]  Ramprasad Saptharishi,et al.  Hitting sets for multilinear read-once algebraic branching programs, in any order , 2014, STOC.

[25]  Manindra Agrawal,et al.  Proving Lower Bounds Via Pseudo-random Generators , 2005, FSTTCS.

[26]  Manindra Agrawal Determinant versus permanent , 2006 .

[27]  Amir Shpilka,et al.  Explicit Noether Normalization for Simultaneous Conjugation via Polynomial Identity Testing , 2013, APPROX-RANDOM.

[28]  Amir Yehudayoff,et al.  Arithmetic Circuits: A survey of recent results and open questions , 2010, Found. Trends Theor. Comput. Sci..

[29]  C. Jacobi De Determinantibus functionalibus. , 1841 .

[30]  Pascal Koiran,et al.  Arithmetic circuits: The chasm at depth four gets wider , 2010, Theor. Comput. Sci..

[31]  Nitin Saxena,et al.  Algebraic Independence in Positive Characteristic -- A p-Adic Calculus , 2012, Electron. Colloquium Comput. Complex..

[32]  Manindra Agrawal,et al.  Primality and identity testing via Chinese remaindering , 2003, JACM.

[33]  Zhi-Zhong Chen,et al.  Reducing Randomness via Irrational Numbers , 2000, SIAM J. Comput..

[34]  Zeev Dvir,et al.  Locally Decodable Codes with Two Queries and Polynomial Identity Testing for Depth 3 Circuits , 2007, SIAM J. Comput..

[35]  Neeraj Kayal,et al.  A super-polynomial lower bound for regular arithmetic formulas , 2014, STOC.

[36]  Ran Raz,et al.  Lower Bounds and Separations for Constant Depth Multilinear Circuits , 2008, Computational Complexity Conference.

[37]  Neeraj Kayal,et al.  Polynomial Identity Testing for Depth 3 Circuits , 2006, 21st Annual IEEE Conference on Computational Complexity (CCC'06).

[38]  A. Wigderson,et al.  Partial Derivatives in Arithmetic Complexity and Beyond (Foundations and Trends(R) in Theoretical Computer Science) , 2011 .

[39]  Ilya Volkovich,et al.  Improved Polynomial Identity Testing for Read-Once Formulas , 2009, APPROX-RANDOM.

[40]  Manindra Agrawal,et al.  On derandomizing tests for certain polynomial identities , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[41]  Moni Naor,et al.  Small-Bias Probability Spaces: Efficient Constructions and Applications , 1993, SIAM J. Comput..

[42]  Amir Shpilka,et al.  Black box polynomial identity testing of generalized depth-3 arithmetic circuits with bounded top fan-in , 2011, Comb..

[43]  Ramprasad Saptharishi,et al.  Pseudorandomness for Multilinear Read-Once Algebraic Branching Programs, in any Order , 2013, Electron. Colloquium Comput. Complex..

[44]  Valentine Kabanets,et al.  2 Better Tradeoffs 2 . 1 Hardness amplification via error-correcting codes , 2007 .

[45]  H. Coxeter A Problem of Collinear Points , 1948 .

[46]  K. Forsman Constructive Commutative Algebra in Nonlinear Control Theory , 1991 .

[47]  V. Vinay,et al.  Arithmetic Circuits: A Chasm at Depth Four , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[48]  Neeraj Kayal,et al.  Arithmetic Circuits: A Chasm at Depth Three , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[49]  Noam D. Elkies,et al.  Sylvester–Gallai Theorems for Complex Numbers and Quaternions , 2004, Discret. Comput. Geom..

[50]  Amir Shpilka,et al.  Black box polynomial identity testing of generalized depth-3 arithmetic circuits with bounded top fan-in , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[51]  Amir Shpilka,et al.  Quasipolynomial-Time Identity Testing of Non-commutative and Read-Once Oblivious Algebraic Branching Programs , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[52]  Nitin Saxena,et al.  Progress on Polynomial Identity Testing , 2009, Bull. EATCS.

[53]  Nitin Saxena,et al.  Jacobian hits circuits: hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits , 2011, STOC '12.

[54]  Noga Alon,et al.  Simple construction of almost k-wise independent random variables , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[55]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[56]  Nitin Saxena,et al.  The Power of Depth 2 Circuits over Algebras , 2009, FSTTCS.

[57]  Nitin Saxena,et al.  Algebraic independence and blackbox identity testing , 2011, Inf. Comput..

[58]  Leslie G. Valiant,et al.  Completeness classes in algebra , 1979, STOC.

[59]  B. M. Fulk MATH , 1992 .

[60]  Amir Shpilka,et al.  Reconstruction of Generalized Depth-3 Arithmetic Circuits with Bounded Top Fan-in , 2009, 2009 24th Annual IEEE Conference on Computational Complexity.

[61]  Nitin Saxena,et al.  From Sylvester-Gallai Configurations to Rank Bounds: Improved Black-Box Identity Test for Depth-3 Circuits , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[62]  M. Ben-Or,et al.  A Deterministic Algorithm for Sparse Multivariate Polynominal Interpolation (Extended Abstract) , 1988, Symposium on the Theory of Computing.

[63]  I. Fischer Sums of like powers of multivariate linear forms , 1994 .

[64]  Nitin Saxena,et al.  Quasi-polynomial hitting-set for set-depth-Δ formulas , 2012, STOC '13.

[65]  Ran Raz,et al.  Deterministic extractors for affine sources over large fields , 2008, Comb..

[66]  Nitin Saxena,et al.  Blackbox Identity Testing for Bounded Top-Fanin Depth-3 Circuits: The Field Doesn't Matter , 2012, SIAM J. Comput..

[67]  Ran Raz,et al.  Deterministic polynomial identity testing in non-commutative models , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..

[68]  Noam Nisan,et al.  Pseudorandom generators for space-bounded computation , 1992, Comb..

[69]  Nitin Saxena,et al.  Hitting-sets for low-distance multilinear depth-3 , 2013, Electron. Colloquium Comput. Complex..

[70]  Vikraman Arvind,et al.  The Monomial Ideal Membership Problem and Polynomial Identity Testing , 2007, ISAAC.

[71]  Meena Mahajan,et al.  Determinant: Combinatorics, Algorithms, and Complexity , 1997, Chic. J. Theor. Comput. Sci..

[72]  Amir Shpilka Interpolation of Depth-3 Arithmetic Circuits with Two Multiplication Gates , 2009, SIAM J. Comput..

[73]  Michael J. Fischer,et al.  Separating Nondeterministic Time Complexity Classes , 1978, JACM.

[74]  Nisheeth K. Vishnoi,et al.  Deterministically testing sparse polynomial identities of unbounded degree , 2009, Inf. Process. Lett..

[75]  Russell Impagliazzo,et al.  Pseudorandomness from Shrinkage , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[76]  Noam Nisan,et al.  Hardness vs Randomness , 1994, J. Comput. Syst. Sci..

[77]  Sébastien Tavenas,et al.  Improved bounds for reduction to depth 4 and depth 3 , 2013, Inf. Comput..

[78]  J. von zur Gathen Factoring sparse multivariate polynomials , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[79]  W. T. Tutte The Factorization of Linear Graphs , 1947 .

[80]  Sten Hansen A Generalization of a Theorem of Sylvester on the Lines Determined by a Finite Point Set. , 1965 .

[81]  Shubhangi Saraf,et al.  Blackbox Polynomial Identity Testing for Depth 3 Circuits , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[82]  Amir Shpilka,et al.  On identity testing of tensors, low-rank recovery and compressed sensing , 2011, STOC '12.