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The first author recently introduced an integer sequence now numbered A355519 in OEIS. This sequence arose from counting bracket tournaments; its study evokes the analysis of the Catalan triangle (sequence A009766 in OEIS) and the related Catalan numbers, sequence A000108 in OEIS. We therefore introduce a general construction that places these sequences on the same footing and suggests many new integer sequences. We provide code for performing this construction and a lengthy list of examples. This construction determines a function from input sequences to output sequences. Some of the resulting output sequences are in OEIS and others are not., Comment: 14 pages

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

Let $f:{\mathbb B}^n \to {\mathbb B}^N$ be a holomorphic map. We study subgroups $\Gamma_f \subseteq {\rm Aut}({\mathbb B}^n)$ and $T_f \subseteq {\rm Aut}({\mathbb B}^N)$. When $f$ is proper, we show both these groups are Lie subgroups. When $\Gamma_f$ contains the center of ${\bf U}(n)$, we show that $f$ is spherically equivalent to a polynomial. When $f$ is minimal we show that there is a homomorphism $\Phi:\Gamma_f \to T_f$ such that $f$ is equivariant with respect to $\Phi$. To do so, we characterize minimality via the triviality of a third group $H_f$. We relate properties of ${\rm Ker}(\Phi)$ to older results on invariant proper maps between balls. When $f$ is proper but completely non-rational, we show that either both $\Gamma_f$ and $T_f$ are finite or both are noncompact.

We prove that a certain positivity condition, considerably more general than pseudoconvexity, enables one to conclude that the regular order of contact and singular order of contact agree when these numbers are $4$.

We prove a new classification result for (CR) rational maps from the unit sphere in some ${\mathbb C}^n$ to the unit sphere in ${\mathbb C}^N$. To so so, we work at the level of Hermitian forms, and we introduce ancestors and descendants., Comment: to appear in Illinois Journal of Math

We introduce the Hermitian-invariant group $\Gamma_f$ of a proper rational map $f$ between the unit ball in complex Euclidean space and a generalized ball in a space of typically higher dimension. We use properties of the groups to define the crucial new concepts of essential map and the source rank of a map. We prove that every finite subgroup of the source automorphism group is the Hermitian-invariant group of some rational proper map between balls. We prove that $\Gamma_f$ is non-compact if and only if $f$ is a totally geodesic embedding. We show that $\Gamma_f$ contains an $n$-torus if and only if $f$ is equivalent to a monomial map. We show that $\Gamma_f$ contains a maximal compact subgroup if and only if $f$ is equivalent to the juxtaposition of tensor powers. We also establish a monotonicity result; the group, after intersecting with the unitary group, does not decrease when a tensor product operation is applied to a polynomial proper map. We give a necessary condition for $\Gamma_f$ (when the target is a generalized ball) to contain automorphisms that move the origin., Comment: 30 pages. To appear in Advances in Mathematics

We study proper rational maps from the unit disk to balls in higher dimensions. After gathering some known results, we study the moduli space of unitary equivalence classes of polynomial proper maps from the disk to a ball, and we establish a normal form for these equivalence classes. We also prove that all rational proper maps from the disk to a ball are homotopic in target dimension at least $2$., Comment: 12 pages; to appear in Proceedings AMS

We introduce several homotopy equivalence relations for proper holomorphic mappings between balls. We provide examples showing that the degree of a rational proper mapping between balls (in positive codimension) is not a homotopy invariant. In domain dimension at least 2, we prove that the set of homotopy classes of rational proper mappings from a ball to a higher dimensional ball is finite. By contrast, when the target dimension is at least twice the domain dimension, it is well known that there are uncountably many spherical equivalence classes. We generalize this result by proving that an arbitrary homotopy of rational maps whose endpoints are spherically inequivalent must contain uncountably many spherically inequivalent maps. We introduce Whitney sequences, a precise analogue (in higher dimensions) of the notion of finite Blaschke product (in one dimension). We show that terms in a Whitney sequence are homotopic to monomial mappings, and we establish an additional result about the target dimensions of such homotopies., Comment: 18 pages, references, spelling, and doubled words words corrected, to appear in Advances in Mathematics

We pose and discuss several Hermitian analogues of Hilbert's $17$-th problem. We survey what is known, offer many explicit examples and some proofs, and give applications to CR geometry. We prove one new algebraic theorem: a non-negative Hermitian symmetric polynomial divides a nonzero squared norm if and only if it is a quotient of squared norms. We also discuss a new example of Putinar-Scheiderer., Comment: to appear in Advances in Math

We show that the Hermitian analogue of a famous result of Pfister fails. To do so we provide a Hermitian symmetric polynomial $r$ of total degree 2d such that any non-zero multiple of it cannot be written as a Hermitian sum of squares with fewer than $d+1$ squares., Comment: 7 pages, to appear in Proc. Amer. Math. Soc., improved exposition in the proof

Properties of Hermitian forms are used to investigate several natural questions from CR Geometry. To each Hermitian symmetric polynomial we assign a Hermitian form. We study how the signature pairs of two Hermitian forms behave under the polynomial product. We show, except for three trivial cases, that every signature pair can be obtained from the product of two indefinite forms. We provide several new applications to the complexity theory of rational mappings between hyperquadrics, including a stability result about the existence of non-trivial rational mappings from a sphere to a hyperquadric with a given signature pair., Comment: 19 pages, latex, fixed typos, to appear in Journal of Geometric Analysis