Liftings of polynomial systems decreasing the mixed volume
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866909791996084224 |
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| author | Borger, Christopher Kahle, Thomas Kretschmer, Andreas Sager, Sebastian Schulze, Jonas |
| author_facet | Borger, Christopher Kahle, Thomas Kretschmer, Andreas Sager, Sebastian Schulze, Jonas |
| contents | The BKK theorem states that the mixed volume of the Newton polytopes of a system of polynomial equations upper bounds the number of isolated torus solutions of the system. Homotopy continuation solvers make use of this fact to pick efficient start systems. For systems where the mixed volume bound is not attained, such methods are still tracking more paths than necessary. We propose a strategy of improvement by lifting a system to an equivalent system with a strictly lower mixed volume at the expense of more variables. We illustrate this idea providing lifting constructions for arbitrary bivariate systems and certain dense-enough systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2105_10714 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Liftings of polynomial systems decreasing the mixed volume Borger, Christopher Kahle, Thomas Kretschmer, Andreas Sager, Sebastian Schulze, Jonas Algebraic Geometry Primary: 52A39, 65H20, Secondary: 65H04, 14M25, 13P15, 14Q65, 52B20 The BKK theorem states that the mixed volume of the Newton polytopes of a system of polynomial equations upper bounds the number of isolated torus solutions of the system. Homotopy continuation solvers make use of this fact to pick efficient start systems. For systems where the mixed volume bound is not attained, such methods are still tracking more paths than necessary. We propose a strategy of improvement by lifting a system to an equivalent system with a strictly lower mixed volume at the expense of more variables. We illustrate this idea providing lifting constructions for arbitrary bivariate systems and certain dense-enough systems. |
| title | Liftings of polynomial systems decreasing the mixed volume |
| topic | Algebraic Geometry Primary: 52A39, 65H20, Secondary: 65H04, 14M25, 13P15, 14Q65, 52B20 |
| url | https://arxiv.org/abs/2105.10714 |