### Abstract

In 2015, Guth proved that, for any set of k-dimensional varieties in R^{d}and for any positive integer D, there exists a polynomial of degree at most D whose zero-set divides R^{d}into open connected “cells,” so that only a small fraction of the given varieties intersect each cell. Guth’s result generalized an earlier result of Guth and Katz for points. Guth’s proof relies on a variant of the Borsuk-Ulam theorem, and for k > 0, it is unknown how to obtain an explicit representation of such a partitioning polynomial and how to construct it efficiently. In particular, it is unknown how to effectively construct such a polynomial for curves (or even lines) in R^{3}. We present an efficient algorithmic construction for this setting. Given a set of n input curves and a positive integer D, we efficiently construct a decomposition of space into O(D^{3}log^{3}D) open cells, each of which meets at most O(n/D^{2}) curves from the input. The construction time is O(n^{2}), where the constant of proportionality depends on D and the maximum degree of the polynomials defining the input curves. For the case of lines in 3-space we present an improved implementation, whose running time is O(n^{4}/^{3}polylog n). As an application, we revisit the problem of eliminating depth cycles among non-vertical pairwise disjoint triangles in 3-space, recently studied by Aronov et al. (2017) and De Berg (2017). Our main result is an algorithm that cuts n triangles into O(n^{3}/2+^{ε}) pieces that are depth cycle free, for any ε > 0. The algorithm runs in O(n^{3}/2+^{ε}) time, which is nearly worst-case optimal. We also sketch several other applications of our effective partitioning for curves in R^{3}

Original language | English (US) |
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Pages | 2636-2648 |

Number of pages | 13 |

State | Published - Jan 1 2019 |

Event | 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 - San Diego, United States Duration: Jan 6 2019 → Jan 9 2019 |

### Conference

Conference | 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 |
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Country | United States |

City | San Diego |

Period | 1/6/19 → 1/9/19 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*Constructive polynomial partitioning for algebraic curves in R*. 2636-2648. Paper presented at 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, United States.

^{3}with applications**Constructive polynomial partitioning for algebraic curves in R ^{3}with applications.** / Aronov, Boris; Ezra, Esther; Zahl, Joshua.

Research output: Contribution to conference › Paper

^{3}with applications' Paper presented at 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, United States, 1/6/19 - 1/9/19, pp. 2636-2648.

^{3}with applications. 2019. Paper presented at 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, United States.

}

TY - CONF

T1 - Constructive polynomial partitioning for algebraic curves in R3with applications

AU - Aronov, Boris

AU - Ezra, Esther

AU - Zahl, Joshua

PY - 2019/1/1

Y1 - 2019/1/1

N2 - In 2015, Guth proved that, for any set of k-dimensional varieties in Rdand for any positive integer D, there exists a polynomial of degree at most D whose zero-set divides Rdinto open connected “cells,” so that only a small fraction of the given varieties intersect each cell. Guth’s result generalized an earlier result of Guth and Katz for points. Guth’s proof relies on a variant of the Borsuk-Ulam theorem, and for k > 0, it is unknown how to obtain an explicit representation of such a partitioning polynomial and how to construct it efficiently. In particular, it is unknown how to effectively construct such a polynomial for curves (or even lines) in R3. We present an efficient algorithmic construction for this setting. Given a set of n input curves and a positive integer D, we efficiently construct a decomposition of space into O(D3log3D) open cells, each of which meets at most O(n/D2) curves from the input. The construction time is O(n2), where the constant of proportionality depends on D and the maximum degree of the polynomials defining the input curves. For the case of lines in 3-space we present an improved implementation, whose running time is O(n4/3polylog n). As an application, we revisit the problem of eliminating depth cycles among non-vertical pairwise disjoint triangles in 3-space, recently studied by Aronov et al. (2017) and De Berg (2017). Our main result is an algorithm that cuts n triangles into O(n3/2+ε) pieces that are depth cycle free, for any ε > 0. The algorithm runs in O(n3/2+ε) time, which is nearly worst-case optimal. We also sketch several other applications of our effective partitioning for curves in R3

AB - In 2015, Guth proved that, for any set of k-dimensional varieties in Rdand for any positive integer D, there exists a polynomial of degree at most D whose zero-set divides Rdinto open connected “cells,” so that only a small fraction of the given varieties intersect each cell. Guth’s result generalized an earlier result of Guth and Katz for points. Guth’s proof relies on a variant of the Borsuk-Ulam theorem, and for k > 0, it is unknown how to obtain an explicit representation of such a partitioning polynomial and how to construct it efficiently. In particular, it is unknown how to effectively construct such a polynomial for curves (or even lines) in R3. We present an efficient algorithmic construction for this setting. Given a set of n input curves and a positive integer D, we efficiently construct a decomposition of space into O(D3log3D) open cells, each of which meets at most O(n/D2) curves from the input. The construction time is O(n2), where the constant of proportionality depends on D and the maximum degree of the polynomials defining the input curves. For the case of lines in 3-space we present an improved implementation, whose running time is O(n4/3polylog n). As an application, we revisit the problem of eliminating depth cycles among non-vertical pairwise disjoint triangles in 3-space, recently studied by Aronov et al. (2017) and De Berg (2017). Our main result is an algorithm that cuts n triangles into O(n3/2+ε) pieces that are depth cycle free, for any ε > 0. The algorithm runs in O(n3/2+ε) time, which is nearly worst-case optimal. We also sketch several other applications of our effective partitioning for curves in R3

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M3 - Paper

AN - SCOPUS:85066856680

SP - 2636

EP - 2648

ER -