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dc.contributor.advisorLeykin, Anton
dc.contributor.authorKrone, Robert Carlton
dc.date.accessioned2015-09-21T14:25:39Z
dc.date.available2015-09-21T14:25:39Z
dc.date.created2015-08
dc.date.issued2015-07-27
dc.date.submittedAugust 2015
dc.identifier.urihttp://hdl.handle.net/1853/53907
dc.description.abstractThe 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.
dc.format.mimetypeapplication/pdf
dc.language.isoen_US
dc.publisherGeorgia Institute of Technology
dc.subjectNumerical algebraic geometry
dc.subjectCommutative algebra
dc.subjectAlgorithms
dc.titleSymmetric ideals and numerical primary decomposition
dc.typeDissertation
dc.description.degreePh.D.
dc.contributor.departmentMathematics
thesis.degree.levelDoctoral
dc.contributor.committeeMemberYu, Josephine
dc.contributor.committeeMemberBlekherman, Greg
dc.contributor.committeeMemberGaroufalidis, Stavros
dc.contributor.committeeMemberVempala, Santosh
dc.date.updated2015-09-21T14:25:39Z


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