Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
On the Newton Polytope of the Resultant
Journal of Algebraic Combinatorics: An International Journal
Polynomial and matrix computations (vol. 1): fundamental algorithms
Polynomial and matrix computations (vol. 1): fundamental algorithms
Programming with algebraic structures: design of the MAGMA language
ISSAC '94 Proceedings of the international symposium on Symbolic and algebraic computation
Polynomial-time computation of the dimension of algebraic varieties in zero-characteristic
Journal of Symbolic Computation
A computational method for diophantine approximation
Algorithms in algebraic geometry and applications
The MAGMA algebra system I: the user language
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
Modern computer algebra
A new algorithm for the geometric decomposition of a variety
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
Complexity of computing the local dimension of a semialgebraic set
Journal of Symbolic Computation
Computing an equidimensional decomposition of an algebraic variety by means of geometric resolutions
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
The projective noether maple package: computing the dimension of a projective variety
Journal of Symbolic Computation
Numerical homotopies to compute generic points on positive dimensional algebraic sets
Journal of Complexity
A Gröbner free alternative for polynomial system solving
Journal of Complexity
Kronecker's and Newton's approaches to solving: a first comparison
Journal of Complexity
Computing Chow forms and some applications
Journal of Algorithms
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components
SIAM Journal on Numerical Analysis
Symmetric Functions Applied to Decomposing Solution Sets of Polynomial Systems
SIAM Journal on Numerical Analysis
How Lower and Upper Complexity Bounds Meet in Elimination Theory
AAECC-11 Proceedings of the 11th 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
Degree bounds and lifting techniques for triangular sets
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Randomized and deterministic algorithms for the dimension of algebraic varieties
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Algebraic Complexity Theory
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Numerical algebraic geometry and symbolic computation
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Complexity issues in bivariate polynomial factorization
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Symbolic-numeric completion of differential systems by homotopy continuation
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Multivariate power series multiplication
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Application of numerical algebraic geometry and numerical linear algebra to PDE
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Improved dense multivariate polynomial factorization algorithms
Journal of Symbolic Computation
Numerical algebraic geometry and kinematics
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Component-level parallelization of triangular decompositions
Proceedings of the 2007 international workshop on Parallel symbolic computation
Change of order for regular chains in positive dimension
Theoretical Computer Science
Computing the global optimum of a multivariate polynomial over the reals
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Efficient Computations of Irredundant Triangular Decompositions with the RegularChains Library
ICCS '07 Proceedings of the 7th international conference on Computational Science, Part II
Efficient Algorithms for Computing Nœther Normalization
Computer Mathematics
Evaluation techniques for zero-dimensional primary decomposition
Journal of Symbolic Computation
Evaluation properties of invariant polynomials
Journal of Symbolic Computation
A parametric representation of totally mixed Nash equilibria
Computers & Mathematics with Applications
Computing isolated roots of sparse polynomial systems in affine space
Theoretical Computer Science
On a generalization of Stickelberger's Theorem
Journal of Symbolic Computation
Algorithms for computing triangular decompositions of polynomial systems
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Deciding reachability of the infimum of a multivariate polynomial
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Algorithms for computing triangular decomposition of polynomial systems
Journal of Symbolic Computation
Variant quantifier elimination
Journal of Symbolic Computation
Affine solution sets of sparse polynomial systems
Journal of Symbolic Computation
Hi-index | 0.00 |
We present a new probabilistic method for solving systems of polynomial equations and inequations. Our algorithm computes the equidimensional decomposition of the Zariski closure of the solution set of such systems. Each equidimensional component is encoded by a generic fiber, that is a finite set of points obtained from the intersection of the component with a generic transverse affine subspace. Our algorithm is incremental in the number of equations to be solved. Its complexity is mainly cubic in the maximum of the degrees of the solution sets of the intermediate systems counting multiplicities.Our method is designed for coefficient fields having characteristic zero or big enough with respect to the number of solutions. If the base field is the field of the rational numbers then the resolution is first performed modulo a random prime number after we have applied a random change of coordinates. Then we search for coordinates with small integers and lift the solutions up to the rational numbers. Our implementation is available within our package Kronecker from version 0.166, which is written in the Magma computer algebra system.