Report
Uniform determinantal representations
| العنوان: | Uniform determinantal representations |
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| المؤلفون: | Boralevi, Ada, van Doornmalen, Jasper, Draisma, Jan, Hochstenbach, Michiel E., Plestenjak, Bor |
| المصدر: | SIAM Journal on Applied Algebra and Geometry 1 (2017) 415-441 |
| سنة النشر: | 2016 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Commutative Algebra, Mathematics - Algebraic Geometry, Mathematics - Numerical Analysis, 13P15, 65H04, 65F15, 65F50 |
| الوصف: | The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimisation, complexity theory, and scientific computing. Motivated by recent developments in this last area, we introduce the notion of a uniform determinantal representation, not of a single polynomial but rather of all polynomials in a given number of variables and of a given maximal degree. We derive a lower bound on the size of the matrix, and present a construction achieving that lower bound up to a constant factor as the number of variables is fixed and the degree grows. This construction marks an improvement upon a recent construction due to Plestenjak-Hochstenbach, and we investigate the performance of new representations in their root-finding technique for bivariate systems. Furthermore, we relate uniform determinantal representations to vector spaces of singular matrices, and we conclude with a number of future research directions. Comment: 23 pages, 3 figures, 4 tables |
| نوع الوثيقة: | Working Paper |
| DOI: | 10.1137/16M108565 |
| URL الوصول: | http://arxiv.org/abs/1607.04873 |
| رقم الانضمام: | edsarx.1607.04873 |
| قاعدة البيانات: | arXiv |
| DOI: | 10.1137/16M108565 |
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