Abstract
We introduce the problem of constructing explicit variety evasive subspace families. Given a family \(\mathcal{F}\) of subvarieties of a projective or affine space, a collection \(\mathcal{H}\) of projective or affine \(k\)-subspaces is \((\mathcal{F},\epsilon)\)-evasive if for every \(\mathcal{V}\in\mathcal{F}\), all but at most \(\epsilon\)-fraction of \(W\in\mathcal{H}\) intersect every irreducible component of \(\mathcal{V}\) 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 low 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 tight lower bound for the size of \(k\)-subspace families that are evasive for degree-\(d\) varieties in a projective \(n\)-space. When \(n-k=n^{\Omega(1)}\), 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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Acknowledgements
A previous version of this paper has appeared as Guo (2021). We thank Nitin Saxena, Noga Ron-Zewi, Amit Sinhababu, and Suryajith Chillara for helpful discussions.
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Guo, Z. Variety Evasive Subspace Families. comput. complex. 33, 10 (2024). https://doi.org/10.1007/s00037-024-00256-1
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DOI: https://doi.org/10.1007/s00037-024-00256-1
Keywords
- algebraic complexity
- dimension reduction
- Noether normalization
- polynomial identity testing
- pseudorandomness
- varieties