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Polynomial partitioning for a set of varieties
Algebraic Geometry · Submitted 30 Oct 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Given a set $\Gamma$ of low-degree k-dimensional varieties in $\mathbb{R}^n$, we prove that for any $D \ge 1$, there is a non-zero polynomial $P$ of degree at most $D$ so that each component of $\mathbb{R}^n \setminus Z(P)$ intersects $O(D^{k-n} |\Gamma|)$ varieties of $\Gamma$.
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The abstract, sentence by sentence
Given a set $\Gamma$ of low-degree k-dimensional varieties in $\mathbb{R}^n$, we prove that for any $D \ge 1$, there is a non-zero polynomial $P$ of degree at most $D$ so that each component of $\mathbb{R}^n \setminus Z(P)$ intersects $O(D^{k-n} |\Gamma|)$ varieties of $\Gamma$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Given a set $\Gamma$ of low-degree k-dimensional varieties in $\mathbb{R}^n$, we prove that for any $D \ge 1$, there is a non-zero polynomial $P$ of degree at most $D$ so that each component of $\mathbb{R}^n \setminus Z(P)$ intersects $O(D^{k-n} |\Gamma|)$ varieties of $\Gamma$. | result 0.91 | 0.93 | 0.93 | asserted 1.00 |
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The paper
Original abstract
Given a set $\Gamma$ of low-degree k-dimensional varieties in $\mathbb{R}^n$, we prove that for any $D \ge 1$, there is a non-zero polynomial $P$ of degree at most $D$ so that each component of $\mathbb{R}^n \setminus Z(P)$ intersects $O(D^{k-n} |\Gamma|)$ varieties of $\Gamma$.
Human review
How the literature uses it
8 arXiv papers cite this in 22 sentences
Papers citing it · 10 sentences without a confident label
Uses its method or tool 7 papers
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Using the polynomial partitioning technique of [ref] , we derive more efficient algorithms for the problems listed above.
On Ray Shooting for Triangles in 3-Space and Related Problems · Introduction · 1.00
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Our primary data structure consists of a partition tree $Ψ$ on $$, which is constructed using the polynomial-partitioning technique of Guth [ref] .
Intersection Queries for Flat Semi-Algebraic Objects in Three Dimensions and Related Problems · The overall data structure · 1.00
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The second ingredient, relevant to the proofs of Theorems [ref] and [ref], is the polynomial partitioning technique of Guth and Katz [ref] , and its more recent extension by Guth [ref] , which yields a divide-and-conquer mechanism via space decomposition by the zero set of a suitable polynomial.
Incidences with curves and surfaces in three dimensions, with applications to distinct and repeated distances · The main techniques · 0.99
What it cites · 4 sentences without a confident label
Uses the method or tool of 1 paper
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Katz and the author used Theorem [ref] in [ref] to study the incidence geometry of lines in $^3$, leading to new bounds for the distinct distance problem in the plane.
On the Erdos distinct distance problem in the plane · Introduction · 0.82
Cites as background 1 paper
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This question was studied by Barone and Basu [ref] .
Refined bounds on the number of connected components of sign conditions on a variety · Introduction · 0.88
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