Abstract. We analyze the singularities of rational inner functions on the unit bidisk and study both when these functions belong to Dirichlet-type spaces and when their partial derivatives belong to Hardy spaces. We characterize derivative H p membership purely in terms of contact order, a measure of the rate at which the zero set of a rational inner function approaches the distinguished boundary of the bidisk. We also show that derivatives of rational inner functions with singularities fail to be in H p for p ≥ 3 2 and that higher nontangential regularity of a rational inner function paradoxically reduces the H p integrability of its derivative. We derive inclusion results for Dirichlet-type spaces from derivative inclusion for H p . Using Agler decompositions and local Dirichlet integrals, we further prove that a restricted class of rational inner functions fails to belong to the unweighted Dirichlet space.

We analyze the behavior of rational inner functions on the unit bidisk near singularities on the distinguished boundary T 2 using level sets. We show that the unimodular level sets of a rational inner function can be parametrized with analytic curves and connect the behavior of these analytic curves to that of the zero set of. We apply these results to obtain a detailed description of the fine numerical stability of : for instance, we show that @ @z 1 and @ @z 2 always possess the same L p-integrability on T 2 , and we obtain combinatorial relations between intersection multiplicities at singularities and vanishing orders for branches of level sets. We also present several new methods of constructing rational inner functions that allow us to prescribe properties of their zero sets, unimodular level sets, and singularities.

We establish an invertibility criterion for free polynomials and free functions evaluated on some tuples of matrices. We show that if the derivative is nonsingular on some domain closed with respect to direct sums and similarity, the function must be invertible. Thus, as a corollary, we establish the Jacobian conjecture in this context. Furthermore, our result holds for commutative polynomials evaluated on tuples of commuting matrices.Question 1.2. Let P : C N → C N be a polynomial map. If the Jacobian DP (x) is invertible for every x ∈ C N , is the map P itself invertible?

We extend the study of the Pick class, the set of complex analytic functions taking the upper half plane into itself, to the noncommutative setting. R. Nevanlinna showed that elements of the Pick class have certain integral representations which reflect their asymptotic behavior at infinity. Löwner connected the Pick class to matrix monotone functions. We generalize the Nevanlinna representation theorems and Löwner's theorem on matrix monotone functions to the free Pick class, the collection of functions that map tuples of matrices with positive imaginary part into the matrices with positive imaginary part which obey the free functional calculus. ContentsJ. E. PASCOE RYAN TULLY-DOYLE 3.1.1. The coefficient Hardy space of a general free vectorvalued function 23 3.2. Models 24 3.3. The lurking isometry argument for linear forms 24 4. Löwner's theorem 26 4.1. The Hamburger model 27 4.2. The Hamburger model construction 29 4.3. The localizing matrix construction of the Hamburger model 31 5.

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