Abstract
New 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.
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Acknowledgements
The first and third author gratefully acknowledge the financial support of the Algebraic Oil Project provided by the Stichting Shell Research Foundation. All three authors are grateful to J. Abbott, C. Fassino and M.L. Torrente for allowing them to use their paper [2] as a basis for Section 5 above. Further thanks are due to D. Heldt and J. Limbeck for the implementation of preliminary versions of the AVI algorithm and to J. Abbott and M.L. Torrente for the implementation of the SOI algorithm which both aided the understanding of the peculiarities of the approximate world substantially. Finally, the authors thank D. Heldt and M. Popoviciu for useful discussions about the subjects of this paper.
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Kreuzer, M., Poulisse, H., Robbiano, L. (2009). From Oil Fields to Hilbert Schemes. In: Robbiano, L., Abbott, J. (eds) Approximate Commutative Algebra. Texts and Monographs in Symbolic Computation. Springer, Vienna. https://doi.org/10.1007/978-3-211-99314-9_1
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