Polynomial time deterministic identity testingalgorithm for $Σ^{[3]}ΠΣΠ^{[2]}$ circuits via Edelstein-Kelly type theorem for quadratic polynomials

Peleg, Shir; Shpilka, Amir

doi:10.48550/arxiv.2006.08263

Cited by 2 publications

(13 citation statements)

References 16 publications

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“…This motivated [BMS13,Gup14] to raise Problem 1.4 and Conjecture 1.6. In [PS20b] we gave a positive answer to Conjecture 1.6 for the case of degree-2 polynomials (r = 2). Interestingly, Theorem 1.2 played a crucial role in the proof, as well as in the proofs of [Shp20,PS20a].…”

Section: Pit and Sylvester-gallai Type Theoremsmentioning

confidence: 94%

“…Interestingly, Theorem 1.2 played a crucial role in the proof, as well as in the proofs of [Shp20,PS20a]. Studying the proofs of [Shp20,PS20a,PS20b] leads to the conclusion that in order to solve Problem 1.4 and Conjecture 1.6 for degrees larger than 2, we must first obtain a result analogous to Theorem 1.2.…”

Section: Pit and Sylvester-gallai Type Theoremsmentioning

confidence: 99%

“…The high level idea in the proof of Theorem 1.5 (which was generalized in [PS20a,PS20b]), includes two steps; The first step constructs a linear space of linear forms V, and a subset J ⊂ Q, both of constant dimension such that a vast majority of the polynomials in Q are in span{J , Q ∩ V }. 1 Implementing this idea requires a lot of case analysis, according to Theorem 1.13.…”

Section: Proof Ideamentioning

confidence: 99%

“…The motivation for proving this result, besides its own appeal, is two fold: a similar theorem played an important role in the polynomial identity testing (PIT for short) problem for small depth algebraic circuits, one of the fundamental open problems in theoretical computer science, see [PS20b]; and it is also related to a long line of work extending and generalizing the original Sylvester-Gallai theorem [Mel41,Gal44] such as the results in [Chv04, PS09, DH16, BDSW19]. In particular, our result builds and generalizes a result of [BDWY13,DSW14], that can be viewed as proving an analogous claim for the case of degree-1 polynomials.…”

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confidence: 99%

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Robust Sylvester-Gallai type theorem for quadratic polynomials

Peleg¹,

Shpilka²

2022

Preprint

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In this work we extend the robust version of the Sylvester-Gallai theorem, obtained by Barak, Dvir, Wigderson and Yehudayoff, and by Dvir, Saraf and Wigderson, to the case of quadratic polynomials. Specifically, we prove that if Q ⊂ C[x 1 . . . . , x n ] is a finite set, |Q| = m, of irreducible quadratic polynomials that satisfy the following condition:• There is δ > 0 such that for every Q ∈ Q there are at least δm polynomials P ∈ Q such that whenever Q and P vanish then so does a third polynomial in Q \ {Q, P}, then dim(span{Q}) = poly(1/δ). The work of Barak et al. and Dvir et al. studied the case of linear polynomials and proved an upper bound of O(1/δ) on the dimension (in the first work an upper bound of O(1/δ 2 ) was given, which was improved to O(1/δ) in the second work).

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Section: Pit and Sylvester-gallai Type Theoremsmentioning

confidence: 94%

Section: Pit and Sylvester-gallai Type Theoremsmentioning

confidence: 99%

Section: Proof Ideamentioning

confidence: 99%

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confidence: 99%

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Robust Sylvester-Gallai type theorem for quadratic polynomials

Peleg¹,

Shpilka²

2022

Preprint

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“…Extending this approach, Gupta [Gup14] formulated a conjecture of Sylvester-Gallai type and proved that his conjecture implies a complete derandomization of black-box PIT for depth-4 circuits with bounded top fan-in and bottom fan-in (also called ΣΠΣΠ(k, r) circuits, where k, r = O(1)). In a recent breakthrough (built on [Shp19,PS20a]), Peleg and Shpilka [PS20b] proved that this conjecture holds for k = 3 and r = 2, and used it to give a polynomial-time black-box PIT algorithm for ΣΠΣΠ(3, 2) circuits.…”

Section: Depth-4 Polynomial Identity Testingmentioning

confidence: 99%

Variety Evasive Subspace Families

Guo

2021

Preprint

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We introduce the problem of constructing explicit variety evasive subspace families. Given a family F of subvarieties of a projective or affine space, a collection H of projective or affine k-subspaces is (F , ǫ)evasive if for every V ∈ F, all but at most ǫ-fraction of W ∈ H intersect every irreducible component of V with (at most) the expected dimension. The problem of constructing such an explicit subspace family generalizes both deterministic black-box polynomial identity testing (PIT) and the problem of constructing explicit (weak) lossless rank condensers.Using Chow forms, we construct explicit k-subspace families of polynomial size that are evasive for all varieties of bounded degree in a projective or affine n-space. As one application, we obtain a complete derandomization of Noether's normalization lemma for varieties of bounded degree in a projective or affine n-space. In another application, we obtain a simple polynomial-time black-box PIT algorithm for depth-4 arithmetic circuits with bounded top fan-in and bottom fan-in that are not in the Sylvester-Gallai configuration, improving and simplifying a result of Gupta (ECCC TR 14-130).As a complement of our explicit construction, we prove a lower bound for the size of k-subspace families that are evasive for degree-d varieties in a projective n-space. When n − k = Ω(n), the lower bound is superpolynomial unless d is bounded. The proof uses a dimension-counting argument on Chow varieties that parametrize projective subvarieties.

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Polynomial time deterministic identity testingalgorithm for $Σ^{[3]}ΠΣΠ^{[2]}$ circuits via Edelstein-Kelly type theorem for quadratic polynomials

Cited by 2 publications

References 16 publications

Robust Sylvester-Gallai type theorem for quadratic polynomials

Robust Sylvester-Gallai type theorem for quadratic polynomials

Variety Evasive Subspace Families

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