Constructive Polynomial Partitioning for Algebraic Curves in\n $\\mathbb{R}^3$ with Applications
Bibliographic record
Abstract
In 2015, Guth proved that for any set of $k$-dimensional bounded complexity\nvarieties in $\\mathbb{R}^d$ and for any positive integer $D$, there exists a\npolynomial of degree at most $D$ whose zero set divides $\\mathbb{R}^d$ into\nopen connected sets, so that only a small fraction of the given varieties\nintersect each of these sets. Guth's result generalized an earlier result of\nGuth and Katz for points.\n Guth's proof relies on a variant of the Borsuk-Ulam theorem, and for $k>0$,\nit is unknown how to obtain an explicit representation of such a partitioning\npolynomial and how to construct it efficiently. In particular, it is unknown\nhow to effectively construct such a polynomial for bounded-degree algebraic\ncurves (or even lines) in $\\mathbb{R}^3$.\n We present an efficient algorithmic construction for this setting. Given a\nset of $n$ input algebraic curves and a positive integer $D$, we efficiently\nconstruct a decomposition of space into $O(D^3\\log^3{D})$ open "cells," each of\nwhich meets $O(n/D^2)$ curves from the input. The construction time is\n$O(n^2)$. For the case of lines in $3$-space we present an improved\nimplementation, whose running time is $O(n^{4/3} \\log^{O(1)} n)$. The constant\nof proportionality in both time bounds depends on $D$ and the maximum degree of\nthe polynomials defining the input curves.\n As an application, we revisit the problem of eliminating depth cycles among\nnon-vertical lines in $3$-space, recently studied by Aronov and Sharir (2018),\nand show an algorithm that cuts $n$ such lines into $O(n^{3/2+\\epsilon})$\npieces that are depth-cycle free, for any $\\epsilon > 0$. The algorithm runs in\n$O(n^{3/2+\\epsilon})$ time, which is a considerable improvement over the\npreviously known algorithms.\n
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.009 | 0.003 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".