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TitleFrom oil fields to Hilbert schemes.
Author(s) Martin Kreuzer, Hennie Poulisse, Lorenzo Robbiano
TypeBook, Chapter in Book, Conference Proceeding
AbstractNew techniques for dealing with problems of numerical stability in computations involving multivariate polynomials allow a new approach to real world problems. Using a modelling problem for the optimization of oil production as a motivation, we present several recent developments involving border bases of polynomial ideals. After recalling the foundations of border basis theory in the exact case, we present a number of approximate techniques such as the eigenvalue method for polynomial system solving, the AVI algorithm for computing approximate border bases, and the SOI algorithm for computing stable order ideals. To get a deeper understanding for the algebra underlying this approximate world, we present recent advances concerning border basis and Gröbner basis schemes. They are open subschemes of Hilbert schemes and parametrize flat families of border bases and Gröbner bases. For the reader it will be a long, tortuous, sometimes dangerous, and hopefully fascinating journey from oil fields to Hilbert schemes.
Keywordsoil field polynomial system solving, eigenvalue method, Buchberger-Möller algorithm, border basis, approximate algorithm, border basis scheme
ISBN978-3-211-99313-2/pbk; 978-3-2
URL http://link.springer.com/chapter/10.1007%2F978-3-211-99314-9_1
PublisherBerlin: Springer
Translation No
Refereed No