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Given a proper, rational map of balls, D'Angelo and Xiao introduced five natural groups encoding properties of the map. We study these groups using a recently discovered normal form for rational maps of balls. Using this normal form, we also provide several new groups associated to the map., Comment: 11 pages. Fixed typos

We construct CR mappings between spheres that are invariant under actions of finite unitary groups. In particular, we combine a tensoring procedure with D'Angelo's construction of a canonical group-invariant CR mapping to obtain new invariant mappings. We also explore possible gap phenomena in this setting., Comment: 11 pages

We study real bihomogeneous polynomials $r(z,\bar{z})$ in $n$ complex variables for which $r(z,\bar{z}) \|z\|^2$ is the squared norm of a holomorphic polynomial mapping. Such polynomials are the focus of the Sum of Squares Conjecture, which describes the possible ranks for the squared norm $r(z,\bar{z}) \|z\|^2$ and has important implications for the study of proper holomorphic mappings between balls in complex Euclidean spaces of different dimension. Questions about the possible signatures for $r(z,\bar{z})$ and the rank of $r(z,\bar{z}) \|z\|^2$ can be reformulated as questions about polynomial ideals. We take this approach and apply purely algebraic tools to obtain constraints on the signature of $r$., Comment: 5 pages

A smooth, strongly $\mathbb{C}$-convex, real hypersurface $S$ in $\mathbb{CP}^n$ admits a projective dual CR structure in addition to the standard CR structure. Given a smooth function $u$ on $S$, we provide characterizations for when $u$ can be decomposed as a sum of a CR function and a dual CR function. Following work of Lee on pluriharmonic boundary values, we provide a characterization using differential forms. We further provide a characterization using tangential vector fields in the style of Audibert and Bedford., Comment: 19 pages. To appear in Pure and Applied Mathematics Quarterly

The goal of this article is to prove the Sum of Squares Conjecture for real polynomials $r(z,\bar{z})$ on $\mathbb{C}^3$ with diagonal coefficient matrix. This conjecture describes the possible values for the rank of $r(z,\bar{z}) \|z\|^2$ under the hypothesis that $r(z,\bar{z})\|z\|^2=\|h(z)\|^2$ for some holomorphic polynomial mapping $h$. Our approach is to connect this problem to the degree estimates problem for proper holomorphic monomial mappings from the unit ball in $\mathbb{C}^2$ to the unit ball in $\mathbb{C}^k$. D'Angelo, Kos, and Riehl proved the sharp degree estimates theorem in this setting, and we give a new proof using techniques from commutative algebra. We then complete the proof of the Sum of Squares Conjecture in this case using similar algebraic techniques., Comment: 17 pages. 6 figures

We develop a link between degree estimates for rational sphere maps and compressed sensing. We provide several new ideas and many examples, both old and new, that amplify connections with linear programming. We close with a list of ten open problems., Comment: 22 pages. To appear in a special issue of Complex Analysis and its Synergies entitled "Geometric Analysis of PDEs and Several Complex Variables". The changes from the first version are minimal, consisting primarily of two new sentences at the end of the introduction and a bit more discussion of Problem 4. We also corrected two typos, made two small changes for clarity, and added a reference

We show that the boundary of any bounded strongly pseudoconvex complete circular domain in $\mathbb C^2$ must contain points that are exceptionally tangent to a projective image of the unit sphere., Comment: Some additional material

We develop a method for proving sup-norm and H\"older estimates for $\overline{\partial}$ on wide class of finite type pseudoconvex domains in $\mathbb{C}^n$. A fundamental obstruction to proving sup-norm estimates is the possibility of singular complex curves with exceptionally high order of contact with the boundary. Our method handles this problem, and in $\mathbb{C}^3$, we prove sup-norm and H\"older estimates for all bounded, pseudoconvex domains with real-analytic boundary., Comment: 29 pages

We analyze a canonical construction of group-invariant CR Mappings between hyperquadrics due to D'Angelo. Given source hyperquadric of $Q(1,1)$, we determine the signature of the target hyperquadric for all finite subgroups of $SU(1,1)$. We also extend combinatorial results proven by Loehr, Warrington, and Wilf on determinants of sparse circulant determinants. We apply these results to study CR mappings invariant under finite subgroups of $U(1,1)$., Comment: 17 pages

In this paper we characterize sums of CR functions from competing CR structures in two scenarios. In one scenario the structures are conjugate and we are adding to the theory of pluriharmonic boundary values. In the second scenario the structures are related by projective duality considerations. In both cases we provide explicit vector field-based characterizations for two-dimensional circular domains satisfying natural convexity conditions., Comment: 17 pages

We study a new biholomorphic invariant of holomorphic maps between domains in different dimensions based on generic initial ideals. We start with the standard generic monomial ideals to find invariants for rational maps of spheres and hyperquadrics, giving a readily computable invariant in this important case. For example, the generic initial monomials distinguish all four inequivalent rational proper maps from the two to the three dimensional ball. Next, we associate to each subspace $X \subset {\mathcal O}(U)$ a generic initial monomial subspace, which is invariant under biholomorphic transformations and multiplication by nonzero functions. The generic initial monomial subspace is a biholomorphic invariant for holomorphic maps if the target automorphism is linear fractional as in the case of automorphisms of spheres or hyperquadrics., Comment: 16 pages, fixed typos, accepted to Math. Z

Several questions in CR geometry lead naturally to the study of bihomogeneous polynomials $r(z,\bar{z})$ on $\C^n \times \C^n$ for which $r(z,\bar{z})\norm{z}^{2d}=\norm{h(z)}^2$ for some natural number $d$ and a holomorphic polynomial mapping $h=(h_1,..., h_K)$ from $\C^n$ to $\C^K$. When $r$ has this property for some $d$, one seeks relationships between $d$, $K$, and the signature and rank of the coefficient matrix of $r$. In this paper, we reformulate this basic question as a question about the growth of the Hilbert function of a homogeneous ideal in $\C[z_1,...,z_n]$ and apply a well-known result of Macaulay to estimate some natural quantities., Comment: 8 pages

Using Green's hyperplane restriction theorem, we prove that the rank of a Hermitian form on the space of holomorphic polynomials is bounded by a constant depending only on the maximum rank of the form restricted to affine manifolds. As an application we prove a rigidity theorem for CR mappings between hyperquadrics in the spirit of the results of Baouendi-Huang and Baouendi-Ebenfelt-Huang. Given a real-analytic CR mapping of a hyperquadric (not equivalent to a sphere) to another hyperquadric $Q(A,B)$, either the image of the mapping is contained in a complex affine subspace, or $A$ is bounded by a constant depending only on $B$. Finally, we prove a stability result about existence of nontrivial CR mappings of hyperquadrics. That is, as long as both $A$ and $B$ are sufficiently large and comparable, then there exist CR mappings whose image is not contained in a hyperplane. The rigidity result also extends when mapping to hyperquadrics in infinite dimensional Hilbert-space., Comment: 25 pages, 1 figure, obsoletes arXiv:1006.0186. Provide a new complete proof in section 5. Accepted to Math. Ann

This paper has been withdrawn as the proof of Lemma 4.1 is incomplete., Comment: This paper has been withdrawn

We study the signature pair for certain group-invariant Hermitian polynomials arising in CR geometry. In particular, we determine the signature pair for the finite subgroups of $SU(2)$. We introduce the asymptotic positivity ratio and compute it for cyclic subgroups of $U(2)$. We calculate the signature pair for dihedral subgroups of $U(2)$., Comment: 28 pages. Accepted in the International Journal of Mathematics