Overview
- Editors:
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A. Feintuch
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I. Gohberg
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Table of contents (16 papers)
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- M. S. Livšic, A. S. Markus
Pages 1-29
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- Daniel Alpay, Laurent Baratchart, Andrea Gombani
Pages 30-66
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- Asher Ben-Artzi, Israel Gohberg
Pages 96-126
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- Rajendra Bhatia, Chandler Davis
Pages 127-137
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- Gene Christner, Kin Y. Li, James Rovnyak
Pages 138-181
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- Peter Fillmore, Masoud Khalkhali
Pages 256-263
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- J. William Helton, John J. Wavrik
Pages 325-354
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- Eric Nordgren, Peter Rosenthal
Pages 362-368
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- Nahum Zobin, Veronica Zobina
Pages 397-416
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Back Matter
Pages 417-422
About this book
Our goal is to find Grabner bases for polynomials in four different sets of expressions: 1 x- , (1 - x)-1 (RESOL) X, 1 x- (1 - xy)-1 (EB) X, , y-1, (1-yx)-1 y, (1_y)-1 (1-x)-1 (preNF) (EB) plus and (1 - xy)1/2 (1 - yx )1/2 (NF) (preNF) plus and Most formulas in the theory of the Nagy-Foias operator model [NF] are polynomials in these expressions where x = T and y = T*. Complicated polynomials can often be simplified by applying "replacement rules". For example, the polynomial (1 - xy)-2 - 2xy(1-xy)-2 + xy2 (1 - xy)-2 -1 simplifies to O. This can be seen by three applications of the replacement rule (1-xy) -1 xy -t (1 - xy)-1 -1 which is true because of the definition of (1-xy)-1. A replacement rule consists of a left hand side (LHS) and a right hand side (RHS). The LHS will always be a monomial. The RHS will be a polynomial whose terms are "simpler" (in a sense to be made precise) than the LHS. An expression is reduced by repeatedly replacing any occurrence of a LHS by the corresponding RHS. The monomials will be well-ordered, so the reduction procedure will terminate after finitely many steps. Our aim is to provide a list of substitution rules for the classes of expressions above. These rules, when implemented on a computer, provide an efficient automatic simplification process. We discuss and define the ordering on monomials later.