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 distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".