Document

# On Hardness of Testing Equivalence to Sparse Polynomials Under Shifts

## File

LIPIcs.STACS.2023.22.pdf
• Filesize: 0.82 MB
• 20 pages

## Cite As

Suryajith Chillara, Coral Grichener, and Amir Shpilka. On Hardness of Testing Equivalence to Sparse Polynomials Under Shifts. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.STACS.2023.22

## Abstract

We say that two given polynomials f, g ∈ R[x_1, ..., x_n], over a ring R, are equivalent under shifts if there exists a vector (a_1, ..., a_n) ∈ Rⁿ such that f(x_1+a_1, ..., x_n+a_n) = g(x_1, ..., x_n). This is a special variant of the polynomial projection problem in Algebraic Complexity Theory. Grigoriev and Karpinski (FOCS 1990), Lakshman and Saunders (SIAM J. Computing, 1995), and Grigoriev and Lakshman (ISSAC 1995) studied the problem of testing polynomial equivalence of a given polynomial to any t-sparse polynomial, over the rational numbers, and gave exponential time algorithms. In this paper, we provide hardness results for this problem. Formally, for a ring R, let SparseShift_R be the following decision problem - Given a polynomial P(X), is there a vector 𝐚 such that P(X+𝐚) contains fewer monomials than P(X). We show that SparseShift_R is at least as hard as checking if a given system of polynomial equations over R[x_1,..., x_n] has a solution (Hilbert’s Nullstellensatz). As a consequence of this reduction, we get the following results. 1) SparseShift_ℤ is undecidable. 2) For any ring R (which is not a field) such that HN_R is NP_R-complete over the Blum-Shub-Smale model of computation, SparseShift_ is also NP_R-complete. In particular, SparseShift_ℤ is also NP_ℤ-complete. We also study the gap version of the SparseShift_R and show the following. 1) For every function β:ℕ → ℝ_+ such that β ∈ o(1), N^β-gap-SparseShift_ℤ is also undecidable (where N is the input length). 2) For R = 𝔽_p, ℚ, ℝ or ℤ_q and for every β > 1 the β-gap-SparseShift_R problem is NP-hard. Furthermore, there exists a constant α > 1 such that for every d = O(1) in the sparse representation model, and for every d ≤ n^O(1) in the arithmetic circuit model, the α^d-gap-SparseShift_R problem is NP-hard when given polynomials of degree at most d, in O(nd) many variables, as input.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Circuit complexity
• Theory of computation → Algebraic complexity theory
##### Keywords
• algebraic complexity
• shift equivalence
• polynomial equivalence
• Hilbert’s Nullstellensatz
• hardness of approximation

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Michael Ben-Or and Prasoon Tiwari. A deterministic algorithm for sparse multivariate polynominal interpolation (extended abstract). In Janos Simon, editor, Proceedings of the 20th Annual ACM Symposium on Theory of Computing, May 2-4, 1988, Chicago, Illinois, USA, pages 301-309. ACM, 1988. URL: https://doi.org/10.1145/62212.62241.
2. Lenore Blum, Felipe Cucker, Michael Shub, and Steve Smale. Complexity and real computation. Springer, 1998. URL: https://link.springer.com/book/10.1007/978-1-4612-0701-6.
3. Lenore Blum, Mike Shub, and Steve Smale. On a theory of computation and complexity over the real numbers: NP- completeness, recursive functions and universal machines. Bulletin (New Series) of the American Mathematical Society, 21(1):1-46, 1989. URL: https://doi.org/bams/1183555121.
4. Allan Borodin and Prasoon Tiwari. On the decidability of sparse univariate polynomial interpolation. Comput. Complex., 1:67-90, 1991. URL: https://doi.org/10.1007/BF01200058.
5. Xi Chen, Neeraj Kayal, and Avi Wigderson. Partial derivatives in arithmetic complexity and beyond. Foundations and Trends® in Theoretical Computer Science, 6(1–2):1-138, 2011. URL: https://doi.org/10.1561/0400000043.
6. Michael Clausen, Andreas W. M. Dress, Johannes Grabmeier, and Marek Karpinski. On zero-testing and interpolation of k-sparse multivariate polynomials over finite fields. Theor. Comput. Sci., 84(2):151-164, 1991. URL: https://doi.org/10.1016/0304-3975(91)90157-W.
7. Martin Davis. Hilbert’s tenth problem is unsolvable. The American Mathematical Monthly, 80(3):233-269, 1973. URL: https://doi.org/10.1080/00029890.1973.11993265.
8. Zeev Dvir, Rafael Mendes de Oliveira, and Amir Shpilka. Testing equivalence of polynomials under shifts. In Javier Esparza, Pierre Fraigniaud, Thore Husfeldt, and Elias Koutsoupias, editors, Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, volume 8572 of Lecture Notes in Computer Science, pages 417-428. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-43948-7_35.
9. Vitaly Feldman, Parikshit Gopalan, Subhash Khot, and Ashok Kumar Ponnuswami. New results for learning noisy parities and halfspaces. In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), 21-24 October 2006, Berkeley, California, USA, Proceedings, pages 563-574. IEEE Computer Society, 2006. URL: https://doi.org/10.1109/FOCS.2006.51.
10. Dima Grigoriev. Testing shift-equivalence of polynomials by deterministic, probabilistic and quantum machines. Theor. Comput. Sci., 180(1-2):217-228, 1997. URL: https://doi.org/10.1016/S0304-3975(96)00188-0.
11. Dima Grigoriev and Marek Karpinski. Algorithms for sparse rational interpolation. In Stephen M. Watt, editor, Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation, ISSAC '91, Bonn, Germany, July 15-17, 1991, pages 7-13. ACM, 1991. URL: https://doi.org/10.1145/120694.120696.
12. Dima Grigoriev and Marek Karpinski. A zero-test and an interpolation algorithm for the shifted sparse polynominals. In Gérard D. Cohen, Teo Mora, and Oscar Moreno, editors, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 10th International Symposium, AAECC-10, San Juan de Puerto Rico, Puerto Rico, May 10-14, 1993, Proceedings, volume 673 of Lecture Notes in Computer Science, pages 162-169. Springer, 1993. URL: https://doi.org/10.1007/3-540-56686-4_41.
13. Dima Grigoriev, Marek Karpinski, and Michael F. Singer. Interpolation of sparse rational functions without knowing bounds on exponents. In 31st Annual Symposium on Foundations of Computer Science, St. Louis, Missouri, USA, October 22-24, 1990, Volume II, pages 840-846. IEEE Computer Society, 1990. URL: https://doi.org/10.1109/FSCS.1990.89616.
14. Dima Grigoriev and Yagati N. Lakshman. Algorithms for computing sparse shifts for multivariate polynomials. Appl. Algebra Eng. Commun. Comput., 11(1):43-67, 2000. URL: https://doi.org/10.1007/s002000050004.
15. Venkatesan Guruswami and Prasad Raghavendra. Hardness of learning halfspaces with noise. In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), 21-24 October 2006, Berkeley, California, USA, Proceedings, pages 543-552. IEEE Computer Society, 2006. URL: https://doi.org/10.1109/FOCS.2006.33.
16. Venkatesan Guruswami and Prasad Raghavendra. A 3-query PCP over integers. In David S. Johnson and Uriel Feige, editors, Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, 2007, pages 198-206. ACM, 2007. URL: https://doi.org/10.1145/1250790.1250819.
17. Johan Håstad. Some optimal inapproximability results. J. ACM, 48(4):798-859, 2001. URL: https://doi.org/10.1145/502090.502098.
18. Erich Kaltofen, Yagati N. Lakshman, and J.-M. Wiley. Modular rational sparse multivariate polynomial interpolation. In Shunro Watanabe and Morio Nagata, editors, Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC '90, Tokyo, Japan, August 20-24, 1990, pages 135-139. ACM, 1990. URL: https://doi.org/10.1145/96877.96912.
19. Neeraj Kayal. Affine projections of polynomials: extended abstract. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19-22, 2012, pages 643-662. ACM, 2012. URL: https://doi.org/10.1145/2213977.2214036.
20. Pascal Koiran. Hilbert’s nullstellensatz is in the polynomial hierarchy. J. Complex., 12(4):273-286, 1996. URL: https://doi.org/10.1006/jcom.1996.0019.
21. Yagati N. Lakshman and B. David Saunders. Sparse polynomial interpolation in nonstandard bases. SIAM J. Comput., 24(2):387-397, 1995. URL: https://doi.org/10.1137/S0097539792237784.
22. Yuri V. Matiyasevich. The diophantineness of enumerable sets. Dokl. Akad. Nauk SSSR, 191(2):279-283, 1970. URL: http://mi.mathnet.ru/dan35274.
23. Dori Medini and Amir Shpilka. Hitting sets and reconstruction for dense orbits in vp_e and ΣΠΣ circuits. In Valentine Kabanets, editor, 36th Computational Complexity Conference, CCC 2021, July 20-23, 2021, Toronto, Ontario, Canada (Virtual Conference), volume 200 of LIPIcs, pages 19:1-19:27. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.19.
24. Ramprasad Saptharishi. A survey of lower bounds in arithmetic circuit complexity, version 9.0.3. Github survey, 2021. URL: https://github.com/dasarpmar/lowerbounds-survey/releases/tag/v9.0.3.
25. Shubhangi Saraf and Sergey Yekhanin. Noisy interpolation of sparse polynomials, and applications. In Proceedings of the 26th Annual IEEE Conference on Computational Complexity, CCC 2011, San Jose, California, USA, June 8-10, 2011, pages 86-92. IEEE Computer Society, 2011. URL: https://doi.org/10.1109/CCC.2011.38.
26. Amir Shpilka and Amir Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Found. Trends Theor. Comput. Sci., 5(3-4):207-388, 2010. URL: https://doi.org/10.1561/0400000039.
27. Leslie G. Valiant. Completeness classes in algebra. In Michael J. Fischer, Richard A. DeMillo, Nancy A. Lynch, Walter A. Burkhard, and Alfred V. Aho, editors, Proceedings of the 11h Annual ACM Symposium on Theory of Computing, April 30 - May 2, 1979, Atlanta, Georgia, USA, pages 249-261. ACM, 1979. URL: https://doi.org/10.1145/800135.804419.