### Abstract

Polynomial ranges are commonly used for numerically solving polynomial systems with interval Newton solvers. Often ranges are computed using the convex hull property of the tensorial Bernstein basis, which is exponential size in the number n of variables. In this paper, we consider methods to compute tight bounds for polynomials in n variables by solving two linear programming problems over a polytope. We formulate several polytopes based on the tensorial Bernstein basis, and we formulate a polytope for the quadratic patch Q _{n}:= (x_{1}, ..., x_{n}, x^{2}_{1}, ..., x^{2}_{n}, x_{1}x_{2}, ..., x _{n-1}^{x}_{n}) by projections. This Bernstein polytope has Θ(n^{2}) hyperplanes. We give the number of vertices, the number of hyperplanes, and the volume of each polytope for n = 1, 2, 3, 4, and we compare the computed range widths for random n-variate polynomials for n ≤ 10. The Bernstein polytope of polynomial size gives only marginally worse range bounds compared to the range bounds obtained with the tensorial Bernstein basis of exponential size.

Original language | English (US) |
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Title of host publication | APPLIED COMPUTING 2010 - The 25th Annual ACM Symposium on Applied Computing |

Pages | 1247-1252 |

Number of pages | 6 |

DOIs | |

State | Published - Jul 23 2010 |

Event | 25th Annual ACM Symposium on Applied Computing, SAC 2010 - Sierre, Switzerland Duration: Mar 22 2010 → Mar 26 2010 |

### Other

Other | 25th Annual ACM Symposium on Applied Computing, SAC 2010 |
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Country | Switzerland |

City | Sierre |

Period | 3/22/10 → 3/26/10 |

### Fingerprint

### Keywords

- Bernstein polynomials
- multivariate polynomials
- polynomial ranges
- polytopes

### ASJC Scopus subject areas

- Software

### Cite this

*APPLIED COMPUTING 2010 - The 25th Annual ACM Symposium on Applied Computing*(pp. 1247-1252) https://doi.org/10.1145/1774088.1774353

**Polytope-based computation of polynomial ranges.** / Fünfzig, Christoph; Michelucci, Dominique; Foufou, Sebti.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*APPLIED COMPUTING 2010 - The 25th Annual ACM Symposium on Applied Computing.*pp. 1247-1252, 25th Annual ACM Symposium on Applied Computing, SAC 2010, Sierre, Switzerland, 3/22/10. https://doi.org/10.1145/1774088.1774353

}

TY - GEN

T1 - Polytope-based computation of polynomial ranges

AU - Fünfzig, Christoph

AU - Michelucci, Dominique

AU - Foufou, Sebti

PY - 2010/7/23

Y1 - 2010/7/23

N2 - Polynomial ranges are commonly used for numerically solving polynomial systems with interval Newton solvers. Often ranges are computed using the convex hull property of the tensorial Bernstein basis, which is exponential size in the number n of variables. In this paper, we consider methods to compute tight bounds for polynomials in n variables by solving two linear programming problems over a polytope. We formulate several polytopes based on the tensorial Bernstein basis, and we formulate a polytope for the quadratic patch Q n:= (x1, ..., xn, x21, ..., x2n, x1x2, ..., x n-1xn) by projections. This Bernstein polytope has Θ(n2) hyperplanes. We give the number of vertices, the number of hyperplanes, and the volume of each polytope for n = 1, 2, 3, 4, and we compare the computed range widths for random n-variate polynomials for n ≤ 10. The Bernstein polytope of polynomial size gives only marginally worse range bounds compared to the range bounds obtained with the tensorial Bernstein basis of exponential size.

AB - Polynomial ranges are commonly used for numerically solving polynomial systems with interval Newton solvers. Often ranges are computed using the convex hull property of the tensorial Bernstein basis, which is exponential size in the number n of variables. In this paper, we consider methods to compute tight bounds for polynomials in n variables by solving two linear programming problems over a polytope. We formulate several polytopes based on the tensorial Bernstein basis, and we formulate a polytope for the quadratic patch Q n:= (x1, ..., xn, x21, ..., x2n, x1x2, ..., x n-1xn) by projections. This Bernstein polytope has Θ(n2) hyperplanes. We give the number of vertices, the number of hyperplanes, and the volume of each polytope for n = 1, 2, 3, 4, and we compare the computed range widths for random n-variate polynomials for n ≤ 10. The Bernstein polytope of polynomial size gives only marginally worse range bounds compared to the range bounds obtained with the tensorial Bernstein basis of exponential size.

KW - Bernstein polynomials

KW - multivariate polynomials

KW - polynomial ranges

KW - polytopes

UR - http://www.scopus.com/inward/record.url?scp=77954701892&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954701892&partnerID=8YFLogxK

U2 - 10.1145/1774088.1774353

DO - 10.1145/1774088.1774353

M3 - Conference contribution

AN - SCOPUS:77954701892

SN - 9781605586380

SP - 1247

EP - 1252

BT - APPLIED COMPUTING 2010 - The 25th Annual ACM Symposium on Applied Computing

ER -