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.

Inhalt
Joint spectrum and discriminant varieties of commuting nonselfadjoint operators.- 1. Introduction.- 2. Joint spectra of commuting operators with compact imaginary parts.- 3. Colligations and vessels.- 4. The discriminant varieties.- References.- On the differential structure of matrix-valued rational inner functions.- 1. Introduction and preliminaries.- 2. The differential structure of Inp.- 3. Charts using Schur algorithm.- 4. Conclusion.- References.- Conservative dynamical systems and nonlinear Livsic-Brodskii nodes.- 1. Conservative systems.- 2. Nonlinear Livsic-Brodskii nodes: models for a given dynamics up to energy preserving diffeomorphic change of variable.- 3. Other partionings of the cast of characters into knowns and unknowns.- References.- Orthogonal polynomials over Hilbert modules.- 1. Introduction.- 2. Orthogonalization with invertible squares.- 3. Preliminaries on inertia theorems for unilateral shifts.- 4. The main result.- References.- Relations of linking and duality between symmetric gauge functions.- 1. Introduction.- 2. Linked symmetric gauge functions.- 3. Quotient of symmetric gauge functions.- 4. Q-norms.- References.- Julia operators and coefficient problems.- 1. Introduction.- 2. Julia operators for triangular matrices.- 3. Multiplication transformations on power series.- 4. Extension problem for substitution transformations.- Appendix. Formal algebra.- References.- Shifts, realizations and interpolation, Redux.- 1. Introduction.- 2. Formulas and facts.- 3. R? variance.- 4. Realizations.- 5. Reproducing kernel spaces.- 6. H(S) spaces.- 7. A basic interpolation problem.- 8. Factorization and recursive methods.- 9. Characteristic functions.- References.- Arveson's distance formulae and robust stabilization for linear time-varying systems.-1. Introduction.- 2. Preliminaries.- 3. Stabilization and proper representations.- 4. Robust stabilization: Proper representation uncertainty.- 5. Gap metric robustness.- Entire cyclic cohomology of Banach algebras.- 1. Background.- 2. Definitions.- 3. Results.- References.- The bounded real characteristic function and Nehari extensions.- 1. Introduction.- 2. Bounded real functions.- 3. Hankel operators.- 4. State space realizations.- 5. Suboptimal Nehari extensions.- References.- On isometric isomorphism between the second dual to the small Lipschitz space and the big Lipschitz space.- The Kantorovich-Rubinstein norm.- Completion of the space of measures in the KR norm.- Critical and noncritical metric spaces.- References.- Rules for computer simplification of the formulas in operator model theory and linear systems.- I. Introduction.- II. The reduction and basis algorithms.- III. Operator relations with finite basis for rules.- IV. Operator relations with infinite basis for rules.- V. A new algebra containing the functional calculus of operator theory.- VI. Gröbner basis property.- VII. Summary of practical rules you might use.- References.- Some global properties of fractional-linear transformations.- Preliminaries.- 1. The case of invertible plus-operators.- 2. The general case of a non-invertible operator U.- References.- Boundary values of Berezin symbols.- 1. Introduction.- 2. Compactness criterion.- 3. Continuous Berezin symbols.- 4. Two questions.- References.- Generalized Hermite polynomials and the bose-like oscillator calculus.- 1. Introduction.- 2. Generalized Hermite polynomials.- 3. The generalized Fourier transform.- 4. Generalized translation.- 5. The Bose-like oscillator.- References.- A general theory of sufficient collections of normswith a prescribed semigroup of contractions.- 1. Formulation of the problem.- 2. Notions.- 3. Formulations of results.- 4. Proofs of results.- References.