# Mixed-volume computation by dynamic lifting applied to polynomial system solving

@article{Verschelde1996MixedvolumeCB, title={Mixed-volume computation by dynamic lifting applied to polynomial system solving}, author={Jan Verschelde and Karin Gatermann and Ronald Cools}, journal={Discrete \& Computational Geometry}, year={1996}, volume={16}, pages={69-112} }

The aim of this paper is to present a flexible approach for the efficient computation of the mixed volume of a tuple of polytopes. In order to compute the mixed volume, a mixed subdivision of the tuple of polytopes is needed, which can be obtained by embedding the polytopes in a higher-dimensional space, i.e., by lifting them. Dynamic lifting is opposed to the static approach. This means that one considers one point at a time and only fixes the value of the lifting function when the pointâ€¦Â

## 73 Citations

### Balancing the lifting values to improve the numerical stability of polyhedral homotopy continuation methods

- MathematicsAppl. Math. Comput.
- 2000

### Distributed Computation of Mixed Volume

- Computer Science
- 1997

A parallel algorithm for computing the mixed volume of n convex polytopes in n-dimensional space, which modiies the sequential algorithm Lift-Prune, which computes mixed volume in arbitrary dimension and is the fastest algorithm today for general inputs.

### Unmixing the Mixed Volume Computation

- MathematicsDiscret. Comput. Geom.
- 2019

It is demonstrated through problems from real world applications that substantial reduction in computational costs can be achieved via this transformation in situations where the convex hull of the union of the polytopes has less complex geometry than the original poly topes.

### How to Count Efficiently all Affine Roots of a Polynomial System

- Computer Science, MathematicsDiscret. Appl. Math.
- 1999

### Computing Mixed Volume and All Mixed Cells in Quermassintegral Time

- Computer Science, MathematicsFound. Comput. Math.
- 2017

A geometric algorithm is introduced in this paper: its complexity is bounded in the average and probability-one settings in terms of some geometric invariants: quermassintegrals associated with the tuple of convex hulls of the support of each polynomial.

### Parallel implementation of the polyhedral homotopy method

- Computer Science, Mathematics2006 International Conference on Parallel Processing Workshops (ICPPW'06)
- 2006

A static workload distribution algorithm is used and a good speedup is achieved on the cyclic n-roots benchmark systems and dynamic workload balancing leads to reduced wall times on large polynomial systems which arise in mechanism design.

### Finding All Isolated Zeros of Polynomial Systems inCnvia Stable Mixed Volumes

- MathematicsJ. Symb. Comput.
- 1999

A new strategy using a single lifting is presented which quickly (and simultaneously) builds the stable mixed subdivision, the fine mixed subdivisions of thestable mixed cells, and the necessary subsystems.

### Parallel implementation of polyhedral homotopy methods

- Computer Science
- 2007

This thesis is the development of "parallel PHCpack", a project which started a couple of years ago developed by my advisor Jan Verschelde and his student Yusong Wang and which continues with Anton Leykin (parallel irreducible decomposition).

### Counterexamples to the Connectivity Conjecture of the Mixed Cells

- MathematicsDiscret. Comput. Geom.
- 1998

It turns out that a positive confirmation of this conjecture can substantially speed up the algorithm for the ``dynamical lifting'' developed in [4], when the polyhedral method is used for solving polynomial systems by homotopy continuation methods.

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