Algorithms in algebraic geometry by Alicia Dickenstein, Frank-Olaf Schreyer, Andrew J. Sommese PDF

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

ISBN-10: 0387751548

ISBN-13: 9780387751542

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 software 12 months in functions of Algebraic Geometry through the Institute for arithmetic and Its functions on September 18-22, 2006 on the collage of Minnesota is one tangible indication of the curiosity. 110 individuals from 11 international locations and twenty states got here to hear the numerous talks; talk 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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In Fin. ) = {G. I dim(Fi n G j ) 2: rkw[i,j]} . 1) If the flags F. and G. 5, Ex. 10, 11]. Of course this allows one in theory to solve all Schubert problems, but the number and complexity of the equations conditions grows quickly to make this prohibitive for large n or d. 2 for more details. 1) are typically written in terms of an increasing rank function in the literature as we have done . However , when one wants to write down polynomial equations which vanish on this set , one must use a decreasing rank function .

P is rankable of rank r ifrkjP = r for all I ~ j ~ d. P is totally rankable if every principal subarray of P is rankable. For example, with n = 4, d = 3 the following example is a totally rankable dot array: {(3,4,1 ),(4,2 ,2 ), (1,4 ,3 ), (3,3 ,3) ,(2,3,4),(3,2 ,4), (4,1 ,4 )}. INTERSECTIONS OF SCHUBERT VARIETIES 27 We picture this as four 2-dimensional slices, where the first one is "slice 1" and the last is "slice 4": Thus (3,4,1) corresponds to the dot in the first slice on the left . The array {(3, 4, 1), (4,2,2), (1, 4, 3)} is not rankable since it has only two distinct values appearing in the second index and three in the first and third.

In Section 5, we describe how to use permutation arrays to solve Schubert problems and give equations for certain intersections of Schubert varieties. In Section 6, we give two examples of an algorithm for computing triple intersections of Schubert varieties and thereby computing the cup product in the cohomology ring of the flag manifold . The equations we give also allow us to compute Galois and monodromy groups for intersections of Schubert varieties; we describe this application in Section 7.

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Algorithms in algebraic geometry by Alicia Dickenstein, Frank-Olaf Schreyer, Andrew J. Sommese


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