Building above Read-once Polynomials: Identity Testing and Hardness of Representation

Mahajan, Meena; Rao, B. V. Raghavendra; Sreenivasaiah, Karteek

doi:10.1007/978-3-319-08783-2_1

Cited by 2 publications

(1 citation statement)

References 18 publications

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“…Applying the hardness of representation approach of Shpilka and Volkovich [76], Mahajan-Rao-Sreenivasaiah [51] showed how to develop a deterministic black-box PIT algorithm for multilinear polynomials computed by ∑ ∑ ∏ O (1) formulas. Independently, Forbes [21] (following Forbes-Shpilka [26]) showed that this lower bound can again be made to apply to leading monomials 16 (which implies a deterministic black-box PIT algorithm for all ∑ ∑ ∏ O (1) formulas, with the same complexity as Mahajan-Rao-Sreenivasaiah [51]). This leading monomial lower bound, which we now state, is important for its applications to polynomials with hard multiples.…”

Section: Depth-3 Powering Formulasmentioning

confidence: 99%

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Forbes¹,

Shpilka²,

Tzameret³

et al. 2021

Theory of Comput.

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Section: Depth-3 Powering Formulasmentioning

confidence: 99%

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Forbes¹,

Shpilka²,

Tzameret³

et al. 2021

Theory of Comput.

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Complexity Theory Column 88

Forbes

Shpilka

2015

SIGACT News

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Algebraic complexity theory studies the complexity of computing (multivariate) polynomials efficiently using algebraic circuits. This succinct representation leads to fundamental algorithmic challenges such as the polynomial identity testing (PIT) problem (decide nonzeroness of the computed polynomial) and the polynomial factorization problem (compute succinct representations of the factors of the circuit). While the Schwartz-Zippel-DeMillo-Lipton Lemma [Sch80,Zip79,DL78] gives an easy randomized algorithm for PIT, randomized algorithms for factorization require more ideas as given by Kaltofen [Kal89]. However, even derandomizing PIT remains a fundamental problem in understanding the power of randomness. In this column, we survey the factorization problem, discussing the algorithmic ideas as well as the applications to other problems. We then discuss the challenges ahead, in particular focusing on the goal of obtaining deterministic factoring algorithms. While deterministic PIT algorithms have been developed for various restricted circuit classes, there are very few corresponding factoring algorithms. We discuss some recent progress on the divisibility testing problem (test if a given polynomial divides another given polynomial) which captures some of the difficulty of factoring. Along the way we attempt to highlight key challenges whose solutions we hope will drive progress in the area.

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Building above Read-once Polynomials: Identity Testing and Hardness of Representation

Cited by 2 publications

References 18 publications

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Complexity Theory Column 88

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