## Polynomial identity testing for depth 3 circuits (2006)

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Venue: | in Proceedings of the twenty-first Annual IEEE Conference on Computational Complexity (CCC |

Citations: | 26 - 5 self |

### BibTeX

@INPROCEEDINGS{Kayal06polynomialidentity,

author = {Neeraj Kayal and Shubhangi Saraf},

title = {Polynomial identity testing for depth 3 circuits},

booktitle = {in Proceedings of the twenty-first Annual IEEE Conference on Computational Complexity (CCC},

year = {2006}

}

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### Abstract

Abstract — We study ΣΠΣ(k) circuits, i.e., depth three arithmetic circuits with top fanin k. We give the first deterministic polynomial time blackbox identity test for ΣΠΣ(k) circuits over the field Q of rational numbers, thus resolving a question posed by Klivans and Spielman (STOC 2001). Our main technical result is a structural theorem for ΣΠΣ(k) circuits that compute the zero polynomial. In particular we show that if a ΣΠΣ(k) circuit C = ∑ i∈[k] Ai

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Citation Context ...w that the rank of a minimal and simple ΣΠΣ circuit with bounded top fanin, computing zero, can be unbounded. These results answer the open questions posed by Klivans-Spielman [KS01] and Dvir-Shpilka =-=[DS05]-=-. 1 Introduction Electronic Colloquium on Computational Complexity, Report No. 150 (2005) Polynomial Identity Testing (PIT) is the following problem: given an arithmetic circuit C computing a polynomi... |

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Citation Context ...wing problem: given an arithmetic circuit1 computing a multivariate polynomial f(X1, . . . , Xn) over a field F, determine if the polynomial is identically zero. Algorithms for primality testing [2], =-=[3]-=-, perfect matching [23] and some fundamental structural results in complexity such as the PCP Theorem and IP=PSPACE involve testing if a particular polynomial is zero. Schwartz [25] and Zippel [28] ob... |

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Citation Context ...of k over these fields, see Appendix A. 2.3. From the Rank Bound to Identity Testing We give below the construction of Karnin and Shpilka [16] which used ideas from an earlier work of Gabizon and Raz =-=[12]-=- to show how the rank bound of Theorem 2.2 translates into the blackbox identity testing algorithm of Theorem 2.1. Lemma 2.3: [Translating rank bounds into a blackbox identity test.] [16] Let F be a f... |

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Citation Context ... No Unbounded top-fanin ΣΠΣ circuits, if they are large enough, can compute all polynomials. The best known lowerbound for the size of a general ΣΠΣ circuit is quadratic, due to Shpilka and Wigderson =-=[27]-=-. After introducing the requisite terminology, we state the two main ingredients leading to this theorem - our proof of a conjecture by Dvir and Shpilka and a construction of rank preserving subspaces... |

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Citation Context ...n). The challenge posed by Klivans and Spielman was taken up by Dvir and Shpilka [9] and then by Kayal and Saxena [19] and a non-blackbox deterministic polynomial-time algorithm was devised (see also =-=[5]-=-). Recently Karnin and Shpilka [16] obtained a quasi-polynomial time blackbox identity test for ΣΠΣ(k, d, n) circuits. Despite the progress made on this question, a deterministic polynomial-time black... |

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Citation Context ...n such that dim(V ) ≥ 2. Then there exists a line L ⊆ V such that |L ∩ S| = 2. We state below the high dimensional Sylvester–Gallai Theorem. It was first proved in a slightly different form by Hansen =-=[14]-=-. The version below is a slightly refined version of Hansen’s result, and was obtained by Bonnice and Edelstein [6, Theorem 2.1]. Theorem 6.2: [Generalized Sylvester–Gallai for high dimensions] ([14],... |

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Citation Context ... and Spielman was taken up by Dvir and Shpilka [9] and then by Kayal and Saxena [19] and a non-blackbox deterministic polynomial-time algorithm was devised (see also [5]). Recently Karnin and Shpilka =-=[16]-=- obtained a quasi-polynomial time blackbox identity test for ΣΠΣ(k, d, n) circuits. Despite the progress made on this question, a deterministic polynomial-time blackbox test had remained elusive. In t... |

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Citation Context ...ferent multiplication gates). As a step towards the conjecture, a poly(2k2 · log d) upper bound on the rank was obtained by Dvir and Shpilka [9]. This was subsequently improved by Saxena and Seshadri =-=[24]-=- to (poly(k) · log d). Over finite fields, this conjecture was disproved by Kayal and Saxena [19] but the situation over fields of characteristic zeroremained unclear. The conjecture soon revealed it... |

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Citation Context ...al results in complexity theory such as IP=PSPACE and the PCP theorem involve the use of identity testing. The first randomized algorithm for identity testing was discovered independently by Schwartz =-=[Sch80]-=- and Zippel [Zip79] and it involves evaluating the polynomial at a random point and accepting if and only if the polynomial evaluates to zero at that point. This was followed by randomized algorithms ... |

8 | Interpolation of depth-3 arithmetic circuits with two multiplication gates
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Citation Context ...polylogarithmic upper bound was used by Karnin and Shpilka[16] to give a quasipolynomial time deterministic blackbox identity test for ΣΠΣ circuits with bounded top fanin. It was also used by Shpilka =-=[26]-=- and by Karnin and Shpilka [17] to give a quasipolynomial time algorithm for reconstruction of ΣΠΣ circuits. In this paper, we prove the conjecture of Dvir and Shpilka over the field R of real numbers... |

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Citation Context ...nomial evaluates to zero at that point. This was followed by randomized algorithms that used fewer random bits [CK97, LV98, AB03] and a derandomization of the polynomial involved in primality testing =-=[AKS04]-=- but a complete derandomization remains distant. Recently, a surprising development was by Impaggliazzo and Kabanets [IK03] who showed that efficient deterministic algorithms for identity testing woul... |

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Citation Context ...of the problem research has focused on restricted models like monotone circuits and bounded depth circuits. For monotone arithmetic circuits, exponential lower bounds on the size (Jerrum & Snir 1982; =-=Shamir & Snir 1977-=-) and linear lower bounds on the depth (Shamir & Snir 1980; Tiwari & Tompa 1994) have been shown. However, only weak lower bounds are known for bounded depth arithmetic circuits (Pudlák 1994; Raz & Sh... |

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Citation Context ... The version below is a slightly refined version of Hansen’s result, and was obtained by Bonnice and Edelstein [6, Theorem 2.1]. Theorem 6.2: [Generalized Sylvester–Gallai for high dimensions] ([14], =-=[6]-=-) Let S be a finite set of points spanning an affine space V ⊆ R n such that dim(V ) ≥ 2t. Then, there exists a t dimensional hyperplane H such that |H ∩ S| = t + 1, and such that H is spanned by the ... |

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Citation Context ...1, x2, · · · , xn]. This gives a deterministic polynomial time algorithm for k = 3. Unfortunately, the ABC theorem for polynomials does not extend in the desired way to sums of more than 3 terms (see =-=[Pal93]-=-). In order to get an algorithm for larger values of k we generalize the above approach and go modulo products of linear forms. 3 The Algorithm In this section we give a deterministic polynomial time ... |

4 |
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Citation Context ...inear forms). Exponential lower bounds on the size of ΣΠΣ arithmetic circuits has been shown over finite fields [GK98]. For general ΣΠΣ circuits over infinite fields only the quadratic lower bound of =-=[SW99]-=- is known. No efficient algorithm for identity testing of ΣΠΣ circuits is known. Here we are interested in studying the identity testing problem for a restricted case of ΣΠΣ circuits – when the top fa... |

4 |
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Citation Context ...rved that a certain colorful analog of the Sylvester–Gallai Theorem would imply the rank bound for the special case of k = 3. Such a result had in fact been proved much earlier by Edelstein and Kelly =-=[10]-=-. Unfortunately, such a direct approach does not generalize for higher values of k. For more discussion about these results, see Appendix A. ⇐⇒ C is minimal, π(C) ≡ 0 ⇐⇒ C ≡ 0, and rank(π(C)) = rank(C... |

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1 | Ming Yang Kao. Reducing Randomness via irrational numbers - Chen - 1997 |

1 | Razborov (2000). Exponential Lower Bounds for Depth 3 Arithmetic Circuits in Algebras of Functions over Finite Fields - Grigoriev, A |