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Algorithms in Algebraic Geometry by Daniel J. Bates, Chris Peterson, Andrew J. Sommese (auth.),

By Daniel J. Bates, Chris Peterson, Andrew J. Sommese (auth.), Alicia Dickenstein, Frank-Olaf Schreyer, Andrew J. Sommese (eds.)

In the decade, there was a burgeoning of task within the layout and implementation of algorithms for algebraic geometric compuation. a few of these algorithms have been initially designed for summary algebraic geometry, yet now are of curiosity to be used in purposes and a few of those algorithms have been initially designed for purposes, yet now are of curiosity to be used in summary algebraic geometry.

The workshop on Algorithms in Algebraic Geometry that was once held within the framework of the IMA Annual software 12 months in purposes of Algebraic Geometry by means of the Institute for arithmetic and Its functions on September 18-22, 2006 on the collage of Minnesota is one tangible indication of the curiosity. a hundred and ten contributors from 11 international locations and twenty states got here to hear the numerous talks; speak about arithmetic; and pursue collaborative paintings at the many faceted difficulties and the algorithms, either symbolic and numberic, that light up them.

This quantity of articles captures many of the spirit of the IMA workshop.

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Define to be a collection of non-zero vectors chosen such that V x E E~l n E;2 n· . n E~d' These lines will provide a "skeleton" for the given Schubert problem. lThe name Richard P. Stanley has an unusually high number of interesting anagrams. Stanley has a long list of such anagrams on his office door. They are also available on his homepage by clicking on his name. 2. Let X = X W 1(E; ) n X w 2(E; ) n··· n Xwd(E~) be a O-dimensional intersection, with E; , . . ,E~ general. Let P c [n]d+l be the unique permutation array associated to this intersection.

These equations may be used to compute Galois and monodromy groups of intersect ions of Schubert varieties. We are able to limit the number of equations by using the permutation arrays of Eriksson and Linusson, and their permutation array varieties, introduced as generalizations of Schubert varieties. We show that there exists a unique permutation array corresponding to each realizable Schubert problem and give a simple recurrence to compute the corresponding rank table, giving in particular a simple criterion for a Littlewood-Richardson coefficient to be O.

The number of flags in a triple intersection is also a structure constant for the cohomology ring of the flag manifold . Our algorithm is based on solving a limited number of determinantal equations for each int ersect ion (far fewer than the naive approach in the case of triple intersections). These equations may be used to compute Galois and monodromy groups of intersect ions of Schubert varieties. We are able to limit the number of equations by using the permutation arrays of Eriksson and Linusson, and their permutation array varieties, introduced as generalizations of Schubert varieties.

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