A new polynomial-time algorithm for linear programming
Combinatorica
On the Newton Polytope of the Resultant
Journal of Algebraic Combinatorics: An International Journal
A polyhedral method for solving sparse polynomial systems
Mathematics of Computation
Sparse elimination and applications in kinematics
Sparse elimination and applications in kinematics
Finding all isolated zeros of polynomial systems in Cn via stable mixed volumes
Journal of Symbolic Computation - Special issue on polynomial elimination—algorithms and applications
A subdivision-based algorithm for the sparse resultant
Journal of the ACM (JACM)
An Efficient Algorithm for the Sparse Mixed Resultant
AAECC-10 Proceedings of the 10th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
An Algorithm for the Newton Resultant
An Algorithm for the Newton Resultant
Sparse Resultant under Vanishing Coefficients
Journal of Algebraic Combinatorics: An International Journal
Multihomogeneous resultant formulae by means of complexes
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
The resultant of an unmixed bivariate system
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
Sylvester-resultants for bivariate polynomials with planar newton polygons
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Expressing a fraction of two determinants as a determinant
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Multihomogeneous resultant formulae for systems with scaled support
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
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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 rational formulae. This is made possible by Macaulay's famous determinantal formula in the dense homogeneous case, extended by D'Andrea to the sparse case. However, the latter requires a lifting of the Newton polytopes, defined recursively on the dimension. Our main contribution is a single-lifting function of the Newton polytopes, which avoids recursion, and yields a simpler method for computing Macaulay-type formulae of sparse resultants. We focus on the case of generalized unmixed systems, where all Newton polytopes are scaled copies of each other, and sketch how our approach may extend to mixed systems of up to four polynomials, as well as those whose Newton polytopes have a sufficiently different face structure. In the mixed subdivision used to construct the matrices, our algorithm defines significantly fewer cells than D'Andrea's, though the matrix formulae are same. We discuss asymptotic complexity bounds and illustrate our results by fully studying a bivariate example.