Emerging Applications of Algebraic GeometryMihai Putinar, Seth Sullivant Springer Science & Business Media, 10. dec. 2008 - 376 sider Recent advances in both the theory and implementation of computational algebraic geometry have led to new, striking applications to a variety of fields of research. The articles in this volume highlight a range of these applications and provide introductory material for topics covered in the IMA workshops on "Optimization and Control" and "Applications in Biology, Dynamics, and Statistics" held during the IMA year on Applications of Algebraic Geometry. The articles related to optimization and control focus on burgeoning use of semidefinite programming and moment matrix techniques in computational real algebraic geometry. The new direction towards a systematic study of non-commutative real algebraic geometry is well represented in the volume. Other articles provide an overview of the way computational algebra is useful for analysis of contingency tables, reconstruction of phylogenetic trees, and in systems biology. The contributions collected in this volume are accessible to non-experts, self-contained and informative; they quickly move towards cutting edge research in these areas, and provide a wealth of open problems for future research. |
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... section . 2. Preliminaries . Let Cd denote complex Euclidean space with Eu- clidean norm given by | z | 2 = Σ = 1 | zj | 2 . The unit , odd dimensional sphere Σ = 1z ; 2 . S2d - 1 { z E Cd ; | z | = 1 } € is a particularly important ...
... section we naturally ask what is the minimum number of terms in the repre- sentation as a squared norm . This difficult question is particularly natural for at least three reasons : there is considerable literature on this problem in ...
... section we consider how many terms are needed to write R as an Hermitian sum of squares on the unit sphere . As we mentioned in the introduction , the real variables analogue of this problem is already quite appealing . It corresponds ...
... section spectral values . Given the polynomial R ( z , z ) , we solve the problem R = | ƒ | 2 on the sphere as in [ 6 ] | ƒ2 on the sphere as in [ 6 ] or [ 8 ] . We add a variable t to homogenize R ; call the result Rh . We may choose C ...
... Section 9 . This survey is not intended to be comprehensive . Rather its purpose is to provide some snippets of results in non - commutative semi - algebraic geometry and their related computer algebra and numerical algorithms , and of ...
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ALGEBRAIC STATISTICS AND CONTINGENCY TABLE PROBLEMS LOGLINEAR MODELS LIKELIHOOD ESTIMATION AND DISCLOSUR... | 63 |
USING INVARIANTS FOR PHYLOGENETIC TREE CONSTRUCTION | 89 |
ON THE ALGEBRAIC GEOMETRY OF POLYNOMIAL DYNAMICAL SYSTEMS | 109 |
SUMS OF SQUARES MOMENT MATRICES AND OPTIMIZATION OVER POLYNOMIALS | 157 |
POSITIVITY AND SUMS OF SQUARESA GUIDE TO RECENT RESULTS | 271 |
NONCOMMUTATIVE REAL ALGEBRAIC GEOMETRY SOMEBASIC CONCEPTS AND FIRST IDEAS | 325 |
OPEN PROBLEMS IN ALGEBRAIC STATISTICS | 351 |
LIST OF WORKSHOP PARTICIPANTS | 365 |