The complexity of robot motion planning
The complexity of robot motion planning
Generalised characteristic polynomials
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Mathematics for computer algebra
Mathematics for computer algebra
Algorithms for polynomial real root isolation
Algorithms for polynomial real root isolation
Sparse elimination and applications in kinematics
Sparse elimination and applications in kinematics
Asymptotically fast computation of subresultants
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Fundamental problems of algorithmic algebra
Fundamental problems of algorithmic algebra
Counting real zeros
Almost tight recursion tree bounds for the Descartes method
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
On the complexity of real root isolation using continued fractions
Theoretical Computer Science
Complete subdivision algorithms, II: isotopic meshing of singular algebraic curves
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
On the asymptotic and practical complexity of solving bivariate systems over the reals
Journal of Symbolic Computation
Lower bounds for zero-dimensional projections
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Continued fraction expansion of real roots of polynomial systems
Proceedings of the 2009 conference on Symbolic numeric computation
New progress in real and complex polynomial root-finding
Computers & Mathematics with Applications
Theoretical Computer Science
Exact algorithms for solving stochastic games: extended abstract
Proceedings of the forty-third annual ACM symposium on Theory of computing
Univariate real root isolation in an extension field
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Root isolation of zero-dimensional polynomial systems with linear univariate representation
Journal of Symbolic Computation
Local generic position for root isolation of zero-dimensional triangular polynomial systems
CASC'12 Proceedings of the 14th international conference on Computer Algebra in Scientific Computing
On the complexity of solving a bivariate polynomial system
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Univariate real root isolation in multiple extension fields
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
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In this paper we derive aggregate separation bounds, named after Davenport-Mahler-Mignotte (DMM), on the isolated roots of polynomial systems, specifically on the minimum distance between any two such roots. The bounds exploit the structure of the system and the height of the sparse (or toric) resultant by means of mixed volume, as well as recent advances on aggregate root bounds for univariate polynomials, and are applicable to arbitrary positive dimensional systems. We improve upon Canny's gap theorem [7] by a factor of O(dn-1), where d bounds the degree of the polynomials, and n is the number of variables. One application is to the bitsize of the eigenvalues and eigenvectors of an integer matrix, which also yields a new proof that the problem is polynomial. We also compare against recent lower bounds on the absolute value of the root coordinates by Brownawell and Yap [5], obtained under the hypothesis there is a 0-dimensional projection. Our bounds are in general comparable, but exploit sparseness; they are also tighter when bounding the value of a positive polynomial over the simplex. For this problem, we also improve upon the bounds in [2, 16]. Our analysis provides a precise asymptotic upper bound on the number of steps that subdivision-based algorithms perform in order to isolate all real roots of a polynomial system. This leads to the first complexity bound of Milne's algorithm [22] in 2D.