Symmetric Newton Polytopes for Solving Sparse Polynomial Systems
نویسندگان
چکیده
The aim of this paper is to compute all isolated solutions to symmetric polynomial systems. Recently, it has been proved that modelling the sparse structure of the system by its Newton polytopes leads to a computational breakthrough in solving the system. In this paper, it will be shown how the Lifting Algorithm, proposed by Huber and Sturmfels, can be applied to symmetric Newton polytopes. This symmetric version of the Lifting Algorithm enables the eecient construction of the symmetric subdivision, giving rise to a symmetric homotopy, so that only the generating solutions have to be computed. EEciency is obtained by combination with the product homotopy. Applications illustrate the practical signiicance of the presented approach.
منابع مشابه
Study of multihomogeneous polynomial systems via resultant matrices
Resultants provide conditions for the solvability of polynomial equations and allow reducing polynomial system solving to linear algebra computations. Sparse resultants depend on the Newton polytopes of the input equations. This polytope is the convex hull of the exponent vectors corresponding to the nonzero monomials of the equations (viewed as lattice points in the Cartesian space of dimensio...
متن کاملA Polyhedral Method for Solving Sparse Polynomial Systems
A continuation method is presented for computing all isolated roots of a semimixed sparse system of polynomial equations. We introduce mixed subdivisions of Newton polytopes, and we apply them to give a new proof and algorithm for Bernstein's theorem on the expected number of roots. This results in a numerical homotopy with the optimal number of paths to be followed. In this homotopy there is o...
متن کاملGlobal Residues for Sparse Polynomial Systems
We consider families of sparse Laurent polynomials f1, . . . , fn with a finite set of common zeroes Zf in the torus T n = (C − {0}) . The global residue assigns to every Laurent polynomial g the sum of its Grothendieck residues over Zf . We present a new symbolic algorithm for computing the global residue as a rational function of the coefficients of the fi when the Newton polytopes of the fi ...
متن کاملSingle-lifting Macaulay-type formulae of generalized unmixed sparse resultants
Resultants are defined in the sparse (or toric) context in order to exploit the structure of the polynomials as expressed by their Newton polytopes. Since determinantal formulae are not always possible, the most efficient general method for computing resultants is as the ratio of two determinants. This is made possible by Macaulay’s seminal result [15] in the dense homogeneous case, extended by...
متن کاملPolynomial system solving: the case of a six-atom molecule
A relatively new branch of computational biology and chemistry has been emerging as an e ort to apply successful paradigms and algorithms from geometry and robot kinematics to predicting the structure of molecules, embedding them in Euclidean space, and nding the energetically favorable con gurations. We illustrate several e cient algebraic algorithms for enumerating all possible conformations ...
متن کامل