A new polynomial-time algorithm for linear programming
Combinatorica
The complexity of robot motion planning
The complexity of robot motion planning
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Asymptotically fast triangularization of matrices over rings
SIAM Journal on Computing
Multipolynomial resultant algorithms
Journal of Symbolic Computation
On the Newton Polytope of the Resultant
Journal of Algebraic Combinatorics: An International Journal
A note on testing the resultant
Journal of Complexity
A polyhedral method for solving sparse polynomial systems
Mathematics of Computation
Numeric-symbolic algorithms for evaluating one-dimensional algebraic sets
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Comparison of various multivariate resultant formulations
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Efficient incremental algorithms for the sparse resultant and the mixed volume
Journal of Symbolic Computation
On the complexity of sparse elimination
Journal of Complexity
Algorithms in algebraic geometry and applications
The structure of sparse resultant matrices
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Computing the isolated roots by matrix methods
Journal of Symbolic Computation - Special issue on symbolic numeric algebra for polynomials
Solving degenerate sparse polynomial systems faster
Journal of Symbolic Computation - Special issue on polynomial elimination—algorithms and applications
Fast Probabilistic Algorithms for Verification of Polynomial Identities
Journal of the ACM (JACM)
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Solving Systems of Polynomial Equations
IEEE Computer Graphics and Applications
Complexity of Quantifier Elimination in the Theory of Algebraically Closed Fields
Proceedings of the Mathematical Foundations of Computer Science 1984
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
When Polynomial Equation Systems Can Be "Solved" Fast?
AAECC-11 Proceedings of the 11th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
An Algorithm for the Newton Resultant
An Algorithm for the Newton Resultant
Hybrid sparse resultant matrices for bivariate systems
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Symbolic and numeric methods for exploiting structure in constructing resultant matrices
Journal of Symbolic Computation
Sparse resultant of composed polynomials II unmixed--mixed case
Journal of Symbolic Computation
Hybrid sparse resultant matrices for bivariate polynomials
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
On the efficiency and optimality of Dixon-based resultant methods
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Factoring sparse resultants of linearly combined polynomials
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Sparse Resultant under Vanishing Coefficients
Journal of Algebraic Combinatorics: An International Journal
Exact resultants for corner-cut unmixed multivariate polynomial systems using the Dixon formulation
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
Dense resultant of composed polynomials mixed-mixed case
Journal of Symbolic Computation
Inversion of parameterized hypersurfaces by means of subresultants
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Improved algorithms for computing determinants and resultants
Journal of Complexity - Special issue: Foundations of computational mathematics 2002 workshops
Conditions for determinantal formula for resultant of a polynomial system
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Resultants of skewly composed polynomials
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Computing multihomogeneous resultants using straight-line programs
Journal of Symbolic Computation
Rational Univariate Reduction via toric resultants
Journal of Symbolic Computation
Resultants of partially composed polynomials
Journal of Symbolic Computation
Implicit polynomial support optimized for sparseness
ICCSA'03 Proceedings of the 2003 international conference on Computational science and its applications: PartIII
Single-lifting Macaulay-type formulae of generalized unmixed sparse resultants
Journal of Symbolic Computation
The offset to an algebraic curve and an application to conics
ICCSA'05 Proceedings of the 2005 international conference on Computational Science and its Applications - Volume Part I
Enumerating a subset of the integer points inside a Minkowski sum
Computational Geometry: Theory and Applications
Implicitization of curves and (hyper)surfaces using predicted support
Theoretical Computer Science
Hi-index | 0.01 |
Multivariate resultants generalize the Sylvester resultant of two polynomials and characterize the solvability of a polynomial system. They also reduce the computation of all common roots to a problem in linear algebra. We propose a determinantal formula for the sparse resultant of an arbitrary system of n + 1 polynomials in n variables. This resultant generalizes the classical one and has significantly lower degree for polynomials that are sparse in the sense that their mixed volume is lower than their Bézout number. Our algorithm uses a mixed polyhedral subdivision of the Minkowski sum of the Newton polytopes in order to construct a Newton matrix. Its determinant is a nonzero multiple of the sparse resultant and the latter equals the GCD of at most n + 1 such determinants. This construction implies a restricted version of an effective sparse Nullstellensatz. For an arbitrary specialization of the coefficients, there are two methods that use one extra variable and yield the sparse resultant. This is the first algorithm to handle the general case with complexity polynomial in the resultant degree and simply exponential in n. We conjecture its extension to producing an exact rational expression for the sparse resultant.