Ebook: Computations in Algebraic Geometry with Macaulay 2

Systems of polynomial equations arise throughout mathematics, science, and engineering. Algebraic geometry provides powerful theoretical techniques for studying the qualitative and quantitative features of their solution sets. Re­ cently developed algorithms have made theoretical aspects of the subject accessible to a broad range of mathematicians and scientists. The algorith­ mic approach to the subject has two principal aims: developing new tools for research within mathematics, and providing new tools for modeling and solv­ ing problems that arise in the sciences and engineering. A healthy synergy emerges, as new theorems yield new algorithms and emerging applications lead to new theoretical questions. This book presents algorithmic tools for algebraic geometry and experi­ mental applications of them. It also introduces a software system in which the tools have been implemented and with which the experiments can be carried out. Macaulay 2 is a computer algebra system devoted to supporting research in algebraic geometry, commutative algebra, and their applications. The reader of this book will encounter Macaulay 2 in the context of concrete applications and practical computations in algebraic geometry. The expositions of the algorithmic tools presented here are designed to serve as a useful guide for those wishing to bring such tools to bear on their own problems. A wide range of mathematical scientists should find these expositions valuable. This includes both the users of other programs similar to Macaulay 2 (for example, Singular and CoCoA) and those who are not interested in explicit machine computations at all.

---

This book presents algorithmic tools for algebraic geometry and experimental applications of them. It also introduces a software system in which the tools have been implemented and with which the experiments can be carried out. Macaulay 2 is a computer algebra system devoted to supporting research in algebraic geometry, commutative algebra, and their applications. The reader of this book will encounter Macaulay 2 in the context of concrete applications and practical computations in algebraic geometry. The expositions of the algorithmic tools presented here are designed to serve as a useful guide for those wishing to bring such tools to bear on their own problems. These expositions will be valuable to both the users of other programs similar to Macaulay 2 (for example, Singular and CoCoA) and those who are not interested in explicit machine computations at all. The first part of the book is primarily concerned with introducing Macaulay2, whereas the second part emphasizes the mathematics.

---

This book presents algorithmic tools for algebraic geometry and experimental applications of them. It also introduces a software system in which the tools have been implemented and with which the experiments can be carried out. Macaulay 2 is a computer algebra system devoted to supporting research in algebraic geometry, commutative algebra, and their applications. The reader of this book will encounter Macaulay 2 in the context of concrete applications and practical computations in algebraic geometry. The expositions of the algorithmic tools presented here are designed to serve as a useful guide for those wishing to bring such tools to bear on their own problems. These expositions will be valuable to both the users of other programs similar to Macaulay 2 (for example, Singular and CoCoA) and those who are not interested in explicit machine computations at all. The first part of the book is primarily concerned with introducing Macaulay2, whereas the second part emphasizes the mathematics.
Content:
Front Matter....Pages i-xv
Front Matter....Pages 1-1
Ideals, Varieties and Macaulay 2 ....Pages 3-15
Projective Geometry and Homological Algebra....Pages 17-40
Data Types, Functions, and Programming....Pages 41-53
Teaching the Geometry of Schemes....Pages 55-70
Front Matter....Pages 71-71
Monomial Ideals....Pages 73-100
From Enumerative Geometry to Solving Systems of Polynomial Equations....Pages 101-129
Resolutions and Cohomology over Complete Intersections....Pages 131-178
Algorithms for the Toric Hilbert Scheme....Pages 179-214
Sheaf Algorithms Using the Exterior Algebra....Pages 215-249
Needles in a Haystack: Special Varieties via Small Fields....Pages 251-279
D-modules and Cohomology of Varieties....Pages 281-323
Back Matter....Pages 325-329

---

This book presents algorithmic tools for algebraic geometry and experimental applications of them. It also introduces a software system in which the tools have been implemented and with which the experiments can be carried out. Macaulay 2 is a computer algebra system devoted to supporting research in algebraic geometry, commutative algebra, and their applications. The reader of this book will encounter Macaulay 2 in the context of concrete applications and practical computations in algebraic geometry. The expositions of the algorithmic tools presented here are designed to serve as a useful guide for those wishing to bring such tools to bear on their own problems. These expositions will be valuable to both the users of other programs similar to Macaulay 2 (for example, Singular and CoCoA) and those who are not interested in explicit machine computations at all. The first part of the book is primarily concerned with introducing Macaulay2, whereas the second part emphasizes the mathematics.
Content:
Front Matter....Pages i-xv
Front Matter....Pages 1-1
Ideals, Varieties and Macaulay 2 ....Pages 3-15
Projective Geometry and Homological Algebra....Pages 17-40
Data Types, Functions, and Programming....Pages 41-53
Teaching the Geometry of Schemes....Pages 55-70
Front Matter....Pages 71-71
Monomial Ideals....Pages 73-100
From Enumerative Geometry to Solving Systems of Polynomial Equations....Pages 101-129
Resolutions and Cohomology over Complete Intersections....Pages 131-178
Algorithms for the Toric Hilbert Scheme....Pages 179-214
Sheaf Algorithms Using the Exterior Algebra....Pages 215-249
Needles in a Haystack: Special Varieties via Small Fields....Pages 251-279
D-modules and Cohomology of Varieties....Pages 281-323
Back Matter....Pages 325-329
....