A convex geometric approach to counting the roots of a polynomial system

DOI10.1016/0304-3975(93)00062-AzbMath0812.65040OpenAlexW2143881994MaRDI QIDQ1338221

J. Maurice Rojas

Publication date: 14 May 1995

Published in: Theoretical Computer Science (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/0304-3975(93)00062-a

zbMATH Keywords

computational complexityupper boundpolynomial systemcomplex coefficientsroot countingBézout's theoremnumber of isolated roots

Mathematics Subject Classification ID

Related Items (22)

Mixed-volume computation by dynamic lifting applied to polynomial system solving ⋮ Computing isolated roots of sparse polynomial systems in affine space ⋮ Sparse trace tests ⋮ Efficiently counting affine roots of mixed trigonometric polynomial systems ⋮ Generic properties of dispersion relations for discrete periodic operators ⋮ Polynomial maps with maximal multiplicity and the special closure ⋮ Bernstein's theorem in affine space ⋮ Elimination for generic sparse polynomial systems ⋮ Mixed volume of small reaction networks ⋮ Numerical algebraic geometry: a new perspective on gauge and string theories ⋮ Matrices in elimination theory ⋮ Solving degenerate sparse polynomial systems faster ⋮ Finding all isolated zeros of polynomial systems in \(\mathbb{C}^n\) via stable mixed volumes ⋮ The BKK root count in $\mathbf {C}^n$ ⋮ How to count efficiently all affine roots of a polynomial system ⋮ Curve valuations and mixed volumes in the implicitization of rational varieties ⋮ MULTIPLICITY AND ŁOJASIEWICZ EXPONENT OF GENERIC LINEAR SECTIONS OF MONOMIAL IDEALS ⋮ Toric intersection theory for affine root counting ⋮ On the multiplicity of isolated roots of sparse polynomial systems ⋮ Criteria for strict monotonicity of the mixed volume of convex polytopes ⋮ Numerical homotopies to compute generic points on positive dimensional algebraic sets ⋮ Enumerating a subset of the integer points inside a Minkowski sum

Uses Software

• LAPACK

Cites Work

• Bézout number calculations for multi-homogeneous polynomial systems
• A homotopy algorithm for a symmetric generalized eigenproblem
• Multigraded resultants of Sylvester type
• On the Newton polytope of the resultant
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• Homotopies for solving polynomial systems within a bounded domain
• Discriminants of polynomials in several variables
• Numerical Solution of a Class of Deficient Polynomial Systems
• Angular momentum, convex Polyhedra and Algebraic Geometry
• THE GEOMETRY OF TORIC VARIETIES
• `` Strong NP-Completeness Results
• Homotopies Exploiting Newton Polytopes for Solving Sparse Polynomial Systems
• The $GBQ$-Algorithm for Constructing Start Systems of Homotopies for Polynomial Systems
• Minkowski Addition of Polytopes: Computational Complexity and Applications to Gröbner Bases
• On a theory of computation and complexity over the real numbers: 𝑁𝑃- completeness, recursive functions and universal machines
• A Polyhedral Method for Solving Sparse Polynomial Systems
• Convex Polytopes
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