目录 Chapter 1.Polynomials in One Variable1.1.The Fundamental Theorem of Algebra1.2.Numerical Root Finding1.3.Real Roots1.4.Puiseux Series1.5.Hypergeometric Series1.6.ExercisesChapter 2.GrSbner Bases of Zero-Dimensional Ideals2.1.Computing Standard Monomials and the Radical2.2.Localizing and Removing Known Zeros2.3.Companion Matrices2.4.The Trace Form2.5.Solving Polynomial Equations in Singular2.6.ExercisesChapter 3.Bernstein's Theorem and Fewnomials3.1.From Bzout's Theorem to Bernstein's Theorem3.2.Zero-dimensional Binomial Systems3.3.Introducing a Toric Deformation3.4.Mixed Subdivisions of Newton Polytopes3.5.Khovanskii's Theorem on Fewnomials3.6.ExercisesChapter 4.Resultants4.1.The Univariate Resultant4.2.The Classical Multivariate Resultant4.3.The Sparse Resultant4.4.The Unmixed Sparse Resultant4.5.The Resultant of Four Trilinear Equations4.6.ExercisesChapter 5.Primary Decomposition5.1.Prime Ideals, Radical Ideals and Primary Ideals5.2.How to Decompose a Polynomial System5.3.Adjacent Minors5.4.Permanental Ideals5.5.ExercisesChapter 6.Polynomial Systems in Economics6.1.Three-Person Games with Two Pure Strategies6.2.Two Numerical Examples Involving Square Roots6.3.Equations Defining Nash Equilibria6.4.The Mixed Volume of a Product of Simplices6.5.Computing Nash Equilibria with PHCpack6.6.ExercisesChapter 7.Sums of Squares7.1.Positive Semidefinite Matrices7.2.Zero-dimensional Ideals and SOStools7.3.Global Optimization7.4.The Real Nullstellensatz7.5.Symmetric Matrices with Double Eigenvalues7.6.ExercisesChapter 8.Polynomial Systems in Statistics8.1.Conditional Independence8.2.Graphical Models8.3.Random Walks on the Integer Lattice8.4.Maximum Likelihood Equations8.5.ExercisesChapter 9.Tropical Algebraic Geometry9.1.Tropical Geometry in the Plane9.2.Amoebas and their Tentacles9.3.The Bergman Complex of a Linear Space9.4.The Tropical Variety of an Ideal9.5.ExercisesChapter 10.Linear Partial Differential Equations with Constant Coefficients10.1.Why Differential Equations?10.2.Zero-dimensional Ideals10.3.Computing Polynomial Solutions10.4.How to Solve Monomial Equations10.5.The Ehrenpreis-Palamodov Theorem10.6.Noetherian Operators10.7.ExercisesBibliographyIndex 作者介绍
序言 ★在热带半环中求解多项式方程组的方法具有广泛的应用,并且之前没有在专著中讨论过。本书叙述生动,包括许多例子、评注、计算机代数会话和大量吸引人的练习。具有一定的基础代数知识的读者即可阅读本书,它也非常适合研究生水平的讲座课程。对希望获取一本介绍多项式方程组求解现今技术著作的研究人员,我们强烈推荐本书。 ——Zentralblatt Math
以下为对购买帮助不大的评价