内容

目次

Normal Forms for Vectors and Univariate Polynomials Standard Forms Normal Forms Noncommutative Associative Algebras Introduction Free Associative Algebras Normal Forms Computing Gröbner Bases Examples of Gröbner Bases and Their Applications Rewriting Systems and Gröbner Bases Exercises Nonsymmetric Operads Introduction Nonsymmetric Operads Free Nonsymmetric Operads Normal Forms Computing Gröbner Bases Examples of Gröbner Bases for Nonsymmetric Operads Normal Forms for Algebras over Nonsymmetric Operads Exercises Twisted Associative Algebras and Shuffle Algebras Introduction Twisted Associative Algebras and Shuffle Algebras Free Shuffle Algebras Normal Forms Computing Gröbner Bases Examples of Shuffle Algebras and their Applications Exercises Symmetric Operads and Shuffle Operads Introduction Symmetric Operads and Shuffle Operads Free Shuffle Operads Normal Forms Computing Gröbner Bases Examples of Gröbner Bases for Shuffle Operads Exercises Operadic Homological Algebra and Gröbner Bases Introduction First Instances of Koszul Signs for Graded Operads Koszul Duality for Operads Models for Operads from Gröbner Bases Exercises Commutative Gröbner Bases Introduction Commutative Associative Polynomials Equivalent Definitions of Commutative Gröbner Bases Classification of Commutative Monomial Orders Zero-Dimensional Ideals Complexity of Gröbner Bases: A Historical Survey Exercises Linear Algebra over Polynomial Rings Introduction Rank of a Polynomial Matrix; Determinantal Ideals Some Elementary Examples Algorithms for Linear Algebra over Polynomial Rings Bibliographical Comments Exercises Case Study of Nonsymmetric Binary Cubic Operads Introduction Toy Model: The Quadratic Case The Cubic Case Exercises Case Study of Nonsymmetric Ternary Quadratic Operads Introduction Generalities on Nonsymmetric Operads with One Generator Nonsymmetric Ternary Operads Further Directions Exercises Appendices: Maple Code for Buchberger’s Algorithm First Block: Initialization Second Block: Monomial Orders Third Block: Sorting Polynomials Fourth Block: Standard Forms of Polynomials Fifth Block: Reduce and Self-Reduce Sixth Block: Main Loop - Buchberger’s Algorithm