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Electronic Colloquium on Computational Complexity

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Reports tagged with varieties:
TR08-042 | 6th April 2008
Zeev Dvir

Deterministic Extractors for Algebraic Sources

An algebraic source is a random variable distributed
uniformly over the set of common zeros of one or more multivariate
polynomials defined over a finite field $F$. Our main result is
the construction of an explicit deterministic extractor for
algebraic sources over exponentially large prime fields. More
precisely, we give ... more >>>

TR21-067 | 6th May 2021
Zeyu Guo

Variety Evasive Subspace Families

Revisions: 1

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 $\mathcal{V}$ ... more >>>

TR22-169 | 26th November 2022
Zeyu Guo, Ben Lee Volk, Akhil Jalan, David Zuckerman

Extractors for Images of Varieties

Revisions: 1

We construct explicit deterministic extractors for polynomial images of varieties, that is, distributions sampled by applying a low-degree polynomial map $f : \mathbb{F}_q^r \to \mathbb{F}_q^n$ to an element sampled uniformly at random from a $k$-dimensional variety $V \subseteq \mathbb{F}_q^r$. This class of sources generalizes both polynomial sources, studied by Dvir, ... more >>>

TR23-140 | 20th September 2023
Eshan Chattopadhyay, Jesse Goodman, Mohit Gurumukhani

Extractors for Polynomial Sources over $\mathbb{F}_2$

We explicitly construct the first nontrivial extractors for degree $d \ge 2$ polynomial sources over $\mathbb{F}_2^n$. Our extractor requires min-entropy $k\geq n - \frac{\sqrt{\log n}}{(d\log \log n)^{d/2}}$. Previously, no constructions were known, even for min-entropy $k\geq n-1$. A key ingredient in our construction is an input reduction lemma, which allows ... more >>>

ISSN 1433-8092 | Imprint