Document

# Factors of Low Individual Degree Polynomials

## File

LIPIcs.CCC.2015.198.pdf
• Filesize: 0.54 MB
• 19 pages

## Cite As

Rafael Oliveira. Factors of Low Individual Degree Polynomials. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 198-216, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.CCC.2015.198

## Abstract

In [Kaltofen, 1989], Kaltofen proved the remarkable fact that multivariate polynomial factorization can be done efficiently, in randomized polynomial time. Still, more than twenty years after Kaltofen's work, many questions remain unanswered regarding the complexity aspects of polynomial factorization, such as the question of whether factors of polynomials efficiently computed by arithmetic formulas also have small arithmetic formulas, asked in [Kopparty/Saraf/Shpilka,CCC'14], and the question of bounding the depth of the circuits computing the factors of a polynomial. We are able to answer these questions in the affirmative for the interesting class of polynomials of bounded individual degrees, which contains polynomials such as the determinant and the permanent. We show that if P(x_1, ..., x_n) is a polynomial with individual degrees bounded by r that can be computed by a formula of size s and depth d, then any factor f(x_1, ..., x_n) of P(x_1, ..., x_n) can be computed by a formula of size poly((rn)^r, s) and depth d+5. This partially answers the question above posed in [Kopparty/Saraf/Shpilka,CCC'14], that asked if this result holds without the exponential dependence on r. Our work generalizes the main factorization theorem from Dvir et al. [Dvir/Shpilka/Yehudayoff,SIAM J. Comp., 2009], who proved it for the special case when the factors are of the form f(x_1, ..., x_n) = x_n - g(x_1, ..., x_n-1). Along the way, we introduce several new technical ideas that could be of independent interest when studying arithmetic circuits (or formulas).
##### Keywords
• Arithmetic Circuits
• Factoring
• Algebraic Complexity

## Metrics

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

## References

1. Benny Chor and Ronald L. Rivest. A knapsack-type public key cryptosystem based on arithmetic in finite fields. IEEE Transactions on Information Theory, 34(5):901-909, 1988.
2. Z. Dvir, A. Shpilka, and A. Yehudayoff. Hardness-randomness tradeoffs for bounded depth arithmetic circuits. SIAM J. on Computing, 39(4):1279-1293, 2009.
3. J. von zur Gathen and J. Gerhard. Modern computer algebra. Cambridge University Press, 1999.
4. J. Von Zur Gathen and E. Kaltofen. Factoring sparse multivariate polynomials. Journal of Computer and System Sciences, 31(2):265-287, 1985.
5. V. Guruswami and M. Sudan. Improved decoding of reed-solomon and algebraic-geometry codes. IEEE Trans. Inf. Theor., 45(6):1757-1767, September 2006.
6. V. Kabanets and R. Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity, 13(1-2):1-46, 2004.
7. E. Kaltofen. Polynomial-time reductions from multivariate to bi- and univariate integral polynomial factorization. SIAM J. on computing, 14(2):469-489, 1985.
8. 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, 1989.
9. E. Kaltofen. Polynomial factorization: a success story. In ISSAC, pages 3-4, 2003.
10. Swastik Kopparty, Shubhangi Saraf, and Amir Shpilka. Equivalence of polynomial identity testing and deterministic multivariate polynomial factorization. In IEEE 29th Conference on Computational Complexity, CCC 2014, Vancouver, BC, Canada, June 11-13, 2014, pages 169-180, 2014.
11. A. K. Lenstra, H. W. Lenstra, and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4):515-534, 1982.
12. Hendrik W Lenstra Jr. Finding small degree factors of lacunary polynomials. Number theory in progress, 1:267-276, 1999.
13. R. Oliveira. Factors of low individual degree polynomials. http://www.cs.princeton.edu/~rmo/papers/small-depth-factors.pdf, 2015.
14. A. Shpilka and I. Volkovich. On the relation between polynomial identity testing and finding variable disjoint factors. In ICALP (1), pages 408-419, 2010.
15. 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.
16. Madhu Sudan. Decoding of reed solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180-193, 1997.