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Towards Deterministic Algorithms for Constant-Depth Factors of Constant-Depth Circuits (2403.01965v1)

Published 4 Mar 2024 in cs.CC and cs.DS

Abstract: We design a deterministic subexponential time algorithm that takes as input a multivariate polynomial $f$ computed by a constant-depth circuit over rational numbers, and outputs a list $L$ of circuits (of unbounded depth and possibly with division gates) that contains all irreducible factors of $f$ computable by constant-depth circuits. This list $L$ might also include circuits that are spurious: they either do not correspond to factors of $f$ or are not even well-defined, e.g. the input to a division gate is a sub-circuit that computes the identically zero polynomial. The key technical ingredient of our algorithm is a notion of the pseudo-resultant of $f$ and a factor $g$, which serves as a proxy for the resultant of $g$ and $f/g$, with the advantage that the circuit complexity of the pseudo-resultant is comparable to that of the circuit complexity of $f$ and $g$. This notion, which might be of independent interest, together with the recent results of Limaye, Srinivasan and Tavenas, helps us derandomize one key step of multivariate polynomial factorization algorithms - that of deterministically finding a good starting point for Newton Iteration for the case when the input polynomial as well as the irreducible factor of interest have small constant-depth circuits.

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References (19)
  1. Deterministic Factorization of Sparse Polynomials with Bounded Individual Degree. J. ACM, 67(2):8:1–8:28, 2020.
  2. Peter Bürgisser. Completeness and Reduction in Algebraic Complexity Theory, volume 7 of Algorithms and computation in mathematics. Springer, 2000.
  3. Hardness-Randomness Tradeoffs for Bounded Depth Arithmetic Circuits. SIAM J. Comput., 39(4):1279–1293, 2009.
  4. Michael A. Forbes. Deterministic Divisibility Testing via Shifted Partial Derivatives. In Proceedings of the 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), FOCS ’15, page 451–465, USA, 2015. IEEE Computer Society.
  5. Complexity Theory Column 88: Challenges in Polynomial Factorization1. SIGACT News, 46(4):32–49, dec 2015.
  6. Improved decoding of Reed-Solomon and algebraic-geometry codes. IEEE Transactions on Information Theory, 45(6):1757–1767, 1999.
  7. Erich Kaltofen. Factorization of Polynomials Given by Straight-Line Programs. In Randomness and Computation, pages 375–412. JAI Press, 1989.
  8. Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds. Computational Complexity, 13(1-2):1–46, 2004. Preliminary version in the \nth35 Annual ACM Symposium on Theory of Computing (STOC 2003).
  9. Deterministic Algorithms for Low Degree Factors of Constant Depth Circuits. CoRR, abs/2309.09701, 2023. Pre-print available at arXiv:2309.09701.
  10. A super-polynomial lower bound for regular arithmetic formulas. In Proceedings of the \nth46 Annual ACM Symposium on Theory of Computing (STOC 2014), pages 146–153, 2014. Pre-print available at \StrSubstituteTR13/091TR[\tmpstring]\IfSubStr\tmpstring/ \StrBefore\tmpstring/[\ecccyear]\StrBehind\tmpstring/[\ecccreport]\StrBefore\tmpstring-[\ecccyear]\StrBehind\tmpstring-[\ecccreport]eccc:TR\ecccyear-\ecccreport.
  11. Equivalence of Polynomial Identity Testing and Polynomial Factorization. Computational Complexity, 24(2):295–331, 2015. Preliminary version in the \nth29 Annual IEEE Conference on Computational Complexity (CCC 2014).
  12. Computing with Polynomials Given By Black Boxes for Their Evaluation: Greatest Common Divisors, Factorization, Separation of Numerators and Denominators. In 29th Annual Symposium on Foundations of Computer Science, White Plains, New York, USA, 24-26 October 1988, pages 296–305. IEEE Computer Society, 1988.
  13. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4):515–534, 1982.
  14. Superpolynomial Lower Bounds Against Low-Depth Algebraic Circuits. In Proceedings of the \nth62 Annual IEEE Symposium on Foundations of Computer Science (FOCS 2021), pages 804–814. IEEE, 2021. Preliminary version in the Electronic Colloquium on Computational Complexity (ECCC), Technical Report TR21-081.
  15. Ramprasad Saptharishi. A survey of lower bounds in arithmetic circuit complexity. Github survey, 2015.
  16. Factorization of Polynomials Given by Arithmetic Branching Programs. Comput. Complex., 30(2):15, 2021.
  17. Madhu Sudan. Decoding of Reed Solomon Codes beyond the Error-Correction Bound. J. Complexity, 13(1):180–193, 1997.
  18. On the Relation between Polynomial Identity Testing and Finding Variable Disjoint Factors. In Automata, Languages and Programming, 37th International Colloquium, ICALP 2010, Bordeaux, France, July 6-10, 2010, Proceedings, Part I, volume 6198 of Lecture Notes in Computer Science, pages 408–419. Springer, 2010.
  19. Joachim von zur Gathen and Jürgen Gerhard. Modern Computer Algebra. Cambridge University Press, 3 edition, 2013.
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