Symmetric ideals and numerical primary decomposition

Numerical algebraic geometry;Commutative algebra;Algorithms

Krone, Robert Carlton ; Leykin, Anton Mathematics Yu, Josephine Blekherman, Greg Garoufalidis, Stavros Vempala, Santosh ; Leykin, Anton

University:Georgia Institute of Technology

Department:Mathematics

关键词: Numerical algebraic geometry;    Commutative algebra;    Algorithms;   

Others  :  https://smartech.gatech.edu/bitstream/1853/53907/1/KRONE-DISSERTATION-2015.pdf

美国|英语

来源: SMARTech Repository

【 摘 要 】

The thesis considers two distinct strategies for algebraic computation with polynomials in high dimension.The first concerns ideals and varieties with symmetry, which often arise in applications from areas such as algebraic statistics and optimization. We explore the commutative algebra properties of such objects, and work towards classifying when symmetric ideals admit finite descriptions including equivariant Gröbner bases and generating sets. Several algorithms are given for computing such descriptions. Specific focus is given to the case of symmetric toric ideals.A second area of research is on problems in numerical algebraic geometry. Numerical algorithms such as homotopy continuation can efficiently compute the approximate solutions of systems of polynomials, but generally have trouble with multiplicity. We develop techniques to compute local information about the scheme structure of an ideal at approximate zeros. This is used to create a hybrid numeric-symbolic algorithm for computing a primary decomposition of the ideal.