## Solving Polynomial Equations: Foundations, Algorithms, and ApplicationsThe subject of this book is the solution of polynomial equations, that is, s- tems of (generally) non-linear algebraic equations. This study is at the heart of several areas of mathematics and its applications. It has provided the - tivation for advances in di?erent branches of mathematics such as algebra, geometry, topology, and numerical analysis. In recent years, an explosive - velopment of algorithms and software has made it possible to solve many problems which had been intractable up to then and greatly expanded the areas of applications to include robotics, machine vision, signal processing, structural molecular biology, computer-aided design and geometric modelling, as well as certain areas of statistics, optimization and game theory, and b- logical networks. At the same time, symbolic computation has proved to be an invaluable tool for experimentation and conjecture in pure mathematics. As a consequence, the interest in e?ective algebraic geometry and computer algebrahasextendedwellbeyonditsoriginalconstituencyofpureandapplied mathematicians and computer scientists, to encompass many other scientists and engineers. While the core of the subject remains algebraic geometry, it also calls upon many other aspects of mathematics and theoretical computer science, ranging from numerical methods, di?erential equations and number theory to discrete geometry, combinatorics and complexity theory. Thegoalofthisbookistoprovideageneralintroduction tomodernma- ematical aspects in computing with multivariate polynomials and in solving algebraic systems. |

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### Contents

Introduction to residues and resultants | 1 |

111 Local analytic residue | 3 |

12 Some applications of residues | 8 |

131 Definition | 16 |

141 Systems of equations in two variables | 21 |

15 Multidimensional residues | 27 |

16 Multivariate resultants | 44 |

17 Residues and resultants | 55 |

Tools for computing primary decompositions | 219 |

Putting it all together | 228 |

Algorithms and their complexities | 241 |

61 Statement of the problems | 242 |

62 Algorithms and complexity | 247 |

63 Dense encoding and algorithms | 248 |

641 Basic definitions and examples | 255 |

65 The NewtonHensel method | 263 |

Solving equations via algebras | 63 |

21 Solving equations | 64 |

22 Ideals defined by linear conditions | 78 |

23 Resultants | 91 |

24 Factoring | 100 |

25 Galois theory | 114 |

Symbolicnumeric methods for solving polynomial equations and applications | 125 |

31 Solving polynomial systems | 126 |

32 Structure of the quotient algebra | 131 |

33 Duality | 141 |

34 Resultant constructions | 145 |

35 Geometric solvers | 151 |

36 Applications | 158 |

An algebraists view on border bases | 169 |

41 Commuting endomorphisms | 172 |

42 Border prebases | 179 |

43 Border bases | 186 |

44 Application to statistics | 195 |

Tools for computing primary decompositions and applications to ideals associated to Bayesian networks | 203 |

Algebraic varieties and components | 205 |

Bayesian networks and Markov ideals | 212 |

66 Other trends | 266 |

Toric resultants and applications to geometric modelling | 269 |

71 Toric elimination theory | 270 |

72 Matrix formulae | 279 |

73 Implicitization with base points | 288 |

74 Implicit support | 292 |

75 Algebraic solving by linear algebra | 298 |

Introduction to numerical algebraic geometry | 301 |

80 Introduction | 302 |

81 Homotopy continuation methods an overview | 303 |

82 Homotopies to approximate all isolated solutions | 305 |

83 Homotopies for positive dimensional solution sets | 326 |

84 Software and applications | 335 |

Four lectures on polynomial absolute factorization | 339 |

Theorems of Hilbert and Bertini reduction to the bivariate case irreducibility tests 911 Hilberts irreducibility | 344 |

Factorization algorithms via computations in algebraic number fields | 351 |

Factorization algorithms via computations in the complex plane | 358 |

Reconstruction of the exact factors | 378 |

393 | |

419 | |

### Common terms and phrases

absolute factorization absolutely irreducible algebraic geometry algorithm apply Bayesian networks Bezoutian border bases bound Chapter coefficients common root complexity components consider coordinates corresponding defined definition denote dimension dimensional eigenvalue eigenvectors encoding example Exercise exists field finite formula Galois geometric given global residue Grobner basis Hence homogeneous polynomials homotopy ideal quotients implicit implies input integer intersection irreducible polynomial isomorphism K[xi Lemma linear form matrix maximal method minimal polynomial Minkowski sum mixed subdivision mixed volume monic monomials multivariate Newton polytopes non-derogatory nonzero normal form O-border basis obtain order ideal P-module parameters points polynomial f polynomial of degree polynomial system primary decomposition primary ideal prime problem projective Proof Proposition prove quotient residue classes Section solutions solve splitting splitting field subset Suppose Sylvester matrix takes distinct values term ordering Theorem toric resultant vanishing variables variety vector space zero