Numerical homotopies to compute generic points on positive dimensional algebraic sets

DOI10.1006/jcom.2000.0554zbMath0982.65070arXivmath/9906198OpenAlexW2033244327MaRDI QIDQ1594830

Jan Verschelde, Andrew John Sommese

Publication date: 26 March 2002

Published in: Journal of Complexity (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/9906198

zbMATH Keywords

algorithmhomotopy continuationnumerical examplesnumerical algebraic geometrysystem of polynomial equationsgeneric pointspolynomial continuation

Mathematics Subject Classification ID

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Uses Software

• FGb
• PHCpack
• Projective Noether
• mctoolbox

Cites Work

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• Bernstein's theorem in affine space
• An application of the Runge-Kutta space
• Some examples for solving systems of algebraic equations by calculating Gröbner bases
• A homotopy for solving general polynomial systems that respects m- homogeneous structures
• Computing all solutions to polynomial systems using homotopy continuation
• Coefficient-parameter polynomial continuation
• On the computation of multi-dimensional solution manifolds of parametrized equations
• A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic \(n\)-roots
• Complex analytic geometry
• How to count efficiently all affine roots of a polynomial system
• Toric intersection theory for affine root counting
• A new efficient algorithm for computing Gröbner bases \((F_4)\)
• Some complexity results for polynomial ideals
• A convex geometric approach to counting the roots of a polynomial system
• The adjunction theory of complex projective varieties
• Techniques of scientific computing (Part 2)
• Polar varieties, real equation solving, and data structures: the hypersurface case
• The Projective Noether Maple Package: Computing the dimension of a projective variety
• Critical and near-critical values in polynomial control problems. I: One- dimensional case
• Matrices in elimination theory
• On the theories of triangular sets
• Triangular sets for solving polynomial systems: a comparative implementation of four methods
• Solving degenerate sparse polynomial systems faster
• Finding all isolated zeros of polynomial systems in \(\mathbb{C}^n\) via stable mixed volumes
• Efficient incremental algorithms for the sparse resultant and the mixed volume
• Counting affine roots of polynomial systems via pointed Newton polytopes
• The Cheater’s Homotopy: An Efficient Procedure for Solving Systems of Polynomial Equations
• Algorithm 795
• The BKK root count in $\mathbf {C}^n$
• What can be computed in algebraic geometry?
• On multiplicities in polynomial system solving

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