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An Improved Construction of Variety-Evasive Subspace Families

Authors Robert Andrews , Abhibhav Garg

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LIPIcs.APPROX-RANDOM.2026.56.pdf
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ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • algebraic complexity
  • pseudorandomness
  • varieties
  • variety-evasive subspaces
  • Chow forms

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Abstract

We study the question of explicitly constructing variety-evasive subspace families, a pseudorandom primitive introduced by Guo (Computational Complexity 2024) that generalizes both hitting sets and lossless rank condensers. Roughly speaking, a variety-evasive subspace family ℋ is a collection of subspaces such that for every algebraic variety V in a fixed family ℱ, there is some subspace W ∈ ℋ that is in general position with respect to V.
We give an explicit construction of a subspace families that evade all degree-d varieties in an n-dimensional affine or projective space. Our construction improves on the size of the variety-evasive subspace families constructed by Guo and, for varieties of degree n^{1 + Ω(1)}, is polynomially close to Guo’s lower bound on the size of any such variety-evasive subspace family. Our variety-evasive subspace families rely on an improved construction of hitting sets for Chow forms of algebraic varieties.

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Robert Andrews and Abhibhav Garg. An Improved Construction of Variety-Evasive Subspace Families. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 56:1-56:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.56

Author Details

Robert Andrews
  • Cheriton School of Computer Science, University of Waterloo, Canada
Abhibhav Garg
  • Cheriton School of Computer Science, University of Waterloo, Canada

References

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