# 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 point…

## 76 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.

### Unmixing the Mixed Volume Computation

- MathematicsDiscrete & Computational Geometry
- 2019

Computing mixed volume of convex polytopes is an important problem in computational algebraic geometry. This paper establishes sufficient conditions under which the mixed volume of several convex…

### 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).

## References

SHOWING 1-10 OF 55 REFERENCES

### Homotopies exploiting Newton polytopes for solving sparse polynomial systems

- Mathematics, Computer Science
- 1994

The algorithm for computing the BKK bound for the number of isolated solutions of a polynomial system with a sparse monomial structure is described and the algorithmic construction of the cheater’s homotopy or the coefficienthomotopy is obtained.

### Symmetrical Newton Polytopes for Solving Sparse Polynomial Systems

- Mathematics, Computer Science
- 1995

This paper shows how the Lifting Algorithm, proposed by Huber and Sturmfels, can be applied to symmetric Newton polytopes and enables the efficient construction of the symmetric subdivision, giving rise to a symmetric homotopy.

### Homotopies for Solving Polynomial Systems Within a Bounded Domain

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 1994

### A polyhedral method for solving sparse polynomial systems

- Mathematics, Computer Science
- 1995

Mixed subdivisions of Newton polytopes are introduced, and they are applied to give a new proof and algorithm for Bernstein's theorem on the expected number of roots, which results in a numerical homotopy with the optimal number of paths to be followed.

### Minkowski Addition of Polytopes: Computational Complexity and Applications to Gröbner Basis

- MathematicsSIAM J. Discret. Math.
- 1992

Using the Minkowski addition of Newton polytopes, the authors show that the following problem can be solved in polynomial time for any finite set of polynomials $\mathcal{T} \subset K [ x_1, \ldots,x_d ]$, where d is fixed.

### Sparse elimination and applications in kinematics

- Computer Science, Mathematics
- 1994

Two methods based on the sparse resultant are implemented for calculating the common roots and then applied to concrete problems in robotics, vision and molecular kinematics, illustrating the claim that problems from these areas often exhibit structure that can be exploited in the context of sparse elimination.

### An Efficient Algorithm for the Sparse Mixed Resultant

- Mathematics, Computer ScienceAAECC
- 1993

The algorithm is the first to present a determinantal formula for arbitrary systems; moreover, its complexity for unmixed systems is polynomial in the resultant degree.

### On the Complexity of some Basic Problems in Computational Convexity: II. Volume and mixed volumes

- MathematicsUniversität Trier, Mathematik/Informatik, Forschungsbericht
- 1994

This paper is the second part of a broader survey of computational convexity, an area of mathematics that has crystallized around a variety of results, problems and applications involving…

### An optimal condition for determining the exact number of roots of a polynomial system

- Mathematics, Computer ScienceISSAC '91
- 1991

It is shown that the BKK bound is exact under much weaker assumptions: only coefficients corresponding to certain vertices of the Newton polytopes need be generic, which allows application of the BKR bound to many practical problems.