By Alicia Dickenstein, Frank-Olaf Schreyer, Andrew J. Sommese

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 functions 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 application yr in functions of Algebraic Geometry by means of the Institute for arithmetic and Its purposes on September 18-22, 2006 on the college of Minnesota is one tangible indication of the curiosity. a hundred and ten members from 11 international locations and twenty states got here to hear the various talks; speak about arithmetic; and pursue collaborative paintings at the many faceted difficulties and the algorithms, either symbolic and numberic, that remove darkness from them.

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

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2 for a detailed analysis. Second, we see the permutation arrays as a complete flag analog of the checkerboards in the geometric Littlewood-Richardson rule of [Vakil, 2006a]. More specifically, checker boards are two nested [n]2 permutation arrays. 3. 2). 2, but we don't know a general rule at this time . A two-step version of such a rule is given in [Coskun], see also [Coskun and VakilJ . 2. Algorithmic complexity. It is well known that solving Schubert problems are "hard". To our knowledge, no complete analysis of the algorithmic complexity is known.

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. We describe pathologies of Eriksson and Linusson's permutation array varieties (failure of existence, ir reducibility, equidimensionality, and reducedness of equations) , and define the more natural permutation array schemes.

1. 1 for a given collection of permutations w l, .. , w d such that I:i l( wi) = ( '2 ). 3. If P; i= Tn ,d' then X is the empty set. 2. This criterion catches 7 of the 8 zero coefficients in 3 dimensions, 373 of the 425 in 4 dimensions, and 28920 of the 33265 in dimension 5. The dimension 3 case missed by this criterion is presumably typical of what the criterion fails to see: there are no 2-planes in 3-space containing three general l-dimensional subspaces. However, given a 2-plane V, three general flags with l-subspaces contained in V are indeed transverse.