Depth-4 Identity Testing and Noether’s Normalization Lemma

Mukhopadhyay, Partha

doi:10.1007/978-3-319-34171-2_22

Cited by 5 publications

(1 citation statement)

References 27 publications

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“…Remark. In [Muk16], Mukhopadhyay gave a deterministic polynomial-time black-box PIT algorithm for ΣΠΣΠ(k, r) circuits satisfying a variant of the non-SG assumption. (Its time complexity is similar to the time complexity in Theorem 1.10.)…”

Section: Theorem 110 ([Gup14]mentioning

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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Section: Theorem 110 ([Gup14]mentioning

confidence: 99%

Variety Evasive Subspace Families

Guo

2021

Preprint

View full text Add to dashboard Cite

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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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