Sylvester-Gallai type theorems for quadratic polynomials

Shpilka, Amir

doi:10.1145/3313276.3316341

Cited by 12 publications

(43 citation statements)

References 31 publications

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“…Theorem 1.7 (Theorem 1.8 of [Shp20]). There is a constant λ such that the following holds for every n ∈ N. Let T 1 , T 2 and T 3 be finite sets of homogeneous quadratic polynomials over C satisfying the following properties:…”

Section: Conjecture 16 (Conjecture 30 Of [Gup14]mentioning

confidence: 99%

“…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]. 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%

“…This problem was answered affirmatively, with a stronger conclusion, in the case of quadratic polynomials (r = 2) in [Shp20].…”

mentioning

confidence: 93%

“…Theorem 1.5 (Theorem 1.7 of [Shp20]). There is a constant λ such that the following holds for every n ∈ N. Let T ⊂ C[x 1 , .…”

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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: Conjecture 16 (Conjecture 30 Of [Gup14]mentioning

confidence: 99%

Section: Pit and Sylvester-gallai Type Theoremsmentioning

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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Variety Evasive Subspace Families

Guo

2024

comput. complex.

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We introduce the problem of constructing explicit variety evasive subspace families. Given a family $$\mathcal{F}$$ F of subvarieties of a projective or affine space, a collection $$\mathcal{H}$$ H of projective or affine $$k$$ k -subspaces is $$(\mathcal{F},\epsilon)$$ ( F , ϵ ) -evasive if for every $$\mathcal{V}\in\mathcal{F}$$ V ∈ F , all but at most $$\epsilon$$ ϵ -fraction of $$W\in\mathcal{H}$$ W ∈ H intersect every irreducible component of $$\mathcal{V}$$ 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 $$k$$ k -subspace families of polynomial size that are evasive for all varieties of bounded degree in a projective or affine $$n$$ n -space. As one application, we obtain a complete derandomization of Noether’s normalization lemma for varieties of low degree in a projective or affine $$n$$ 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 tight lower bound for the size of $$k$$ k -subspace families that are evasive for degree-$$d$$ d varieties in a projective $$n$$ n -space. When $$n-k=n^{\Omega(1)}$$ n - k = n Ω ( 1 ) , the lower bound is superpolynomial unless $$d$$ d is bounded. The proof uses a dimension counting argument on Chow varieties that parametrize projective subvarieties.

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Sylvester-Gallai type theorems for quadratic polynomials

Cited by 12 publications

References 31 publications

Robust Sylvester-Gallai type theorem for quadratic polynomials

Robust Sylvester-Gallai type theorem for quadratic polynomials

Variety Evasive Subspace Families

Variety Evasive Subspace Families

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