Solving algebraic problems with high accuracy
Proc. of the symposium on A new approach to scientific computation
Solving systems of polynomial inequalities in subexponential time
Journal of Symbolic Computation
Counting connected components of a semialgebraic set in subexponential time
Computational Complexity
On the combinatorial and algebraic complexity of quantifier elimination
Journal of the ACM (JACM)
Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation
ACM Transactions on Mathematical Software (TOMS)
Numerical homotopies to compute generic points on positive dimensional algebraic sets
Journal of Complexity
Computing Roadmaps of General Semi-Algebraic Sets
AAECC-9 Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Hauptvortrag: Quantifier elimination for real closed fields by cylindrical algebraic decomposition
Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages
Real solving for positive dimensional systems
Journal of Symbolic Computation
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Numerical Polynomial Algebra
Newton's method with deflation for isolated singularities of polynomial systems
Theoretical Computer Science
Existence of Solutions to Systems of Underdetermined Equations and Spherical Designs
SIAM Journal on Numerical Analysis
The voronoi diagram of three lines
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals
Foundations of Computational Mathematics
On the geometry of polar varieties
Applicable Algebra in Engineering, Communication and Computing
Computational existence proofs for spherical t-designs
Numerische Mathematik
Deciding reachability of the infimum of a multivariate polynomial
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Deflation and certified isolation of singular zeros of polynomial systems
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Algorithms of Intrinsic Complexity for Point Searching in Compact Real Singular Hypersurfaces
Foundations of Computational Mathematics
Algorithm 921: alphaCertified: Certifying Solutions to Polynomial Systems
ACM Transactions on Mathematical Software (TOMS)
Computing real solutions of polynomial systems via low-rank moment matrix completion
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
Verified error bounds for isolated singular solutions of polynomial systems: Case of breadth one
Theoretical Computer Science
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In this paper, we propose two algorithms for verifying the existence of real solutions of positive-dimensional polynomial systems. The first one is based on the critical point method and the homotopy continuation method. It targets for verifying the existence of real roots on each connected component of an algebraic variety V ∩ Rn defined by polynomial equations. The second one is based on the low-rank moment matrix completion method and aims for verifying the existence of at least one real roots on V ∩ Rn. Combined both algorithms with the verification algorithms for zero-dimensional polynomial systems, we are able to find verified real solutions of positive-dimensional polynomial systems very efficiently for a large set of examples.