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John P. D'Angelo

Publications and source records attributed to John P. D'Angelo.

At least 19 recordsLinked to original sources
arXiv2025

Invariant polynomials, gaps, and sparseness

We consider each of the three classes of representations of cyclic groups that arise in the study of rational sphere maps. We study the possible number of terms for invariant polynomials with non-negative coefficients that are constant on the appropriate line or hyperplane. Our result provides crucial information about gaps in the possible target dimensions for certain invariant polynomial sphere maps. We interpret our results in terms of sparseness for solutions of certain linear systems.

math.CV↗
arXiv2022

Integer Sequences and Output Arrays

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.

math.NT↗
arXiv2017

Symmetries and regularity for holomorphic maps between balls

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

math.CV↗
arXiv2017

A remark on finite type conditions

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 $4$.

math.CV↗
arXiv2017

On the classification of rational sphere maps

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.

math.CV↗
arXiv2017

Symmetries in CR complexity theory

We introduce the Hermitian-invariant group $Γ_f$ 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 $Γ_f$ is non-compact if and only if $f$ is a totally geodesic embedding. We show that $Γ_f$ contains an $n$-torus if and only if $f$ is equivalent to a monomial map. We show that $Γ_f$ 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 $Γ_f$ (when the target is a generalized ball) to contain automorphisms that move the origin.

math.CV↗
arXiv2016

Proper holomorphic maps from the unit disk to some unit ball

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 $2$.

math.CV↗
arXiv2015

Homotopy equivalence for proper holomorphic mappings

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.

math.CV↗
arXiv2010

Pfister's theorem fails in the Hermitian case

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.

math.CV↗
arXiv2010

Hermitian analogues of Hilbert's 17-th problem

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.

math.CV↗
arXiv2010

Hermitian symmetric polynomials and CR complexity

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.

math.CV↗
arXiv2009

Invariant CR Mappings

We summarize some work on CR mappings invariant under a subgroup of U(n) and prove a result on the failure of rigidity.

math.CV↗
arXiv2008

Helical CR Structures and Sub-Riemannian Geodesics

A helical CR structure is a decomposition of a real Euclidean space into an even-dimensional horizontal subspace and its orthogonal vertical complement, together with an almost complex structure on the horizontal space and a marked vector in the vertical space. We prove an equivalence between such structures and step two Carnot groups equipped with a distinguished normal geodesic, and also between such structures and smooth real curves whose derivatives have constant Euclidean norm. As a consequence, we relate step two Carnot groups equipped with sub-Riemannian geodesics with this family of curves. The restriction to the unit circle of certain planar homogeneous polynomial mappings gives an instructive class of examples. We describe these examples in detail.

math.CV↗
arXiv2008

Complexity results for CR mappings between spheres

Using elementary number theory, we prove several results about the complexity of CR mappings between spheres. It is known that CR mappings between spheres, invariant under finite groups, lead to sharp bounds for degree estimates on real polynomials constant on a hyperplane. We show here that there are infinitely many degrees for which the uniqueness of sharp examples fails. The proof uses a Pell equation and complicated explicit computations. We also show that the so-called gap phenomenon for proper mappings between balls does not occur beyond a certain target dimension. This proof uses the solution of the postage stamp problem.

math.CV↗
arXiv2007

Degree Estimates for Polynomials Constant on a Hyperplane

The study of proper rational mappings between balls in complex Euclidean spaces naturally leads to the relationship between the degree and imbedding dimension of such a mapping. The special case for monomial mappings is equivalent to the question discussed in this paper. Estimate the degree $d$ of a polynomial in $n$ real variables, assumed to have non-negative coefficients and to be constant on a hyperplane, in terms of the number $N$ of its terms. No such estimate is possible when $n=1$. The sharp bound $d\le 2N-3$ is known when $n=2$. This paper includes two main results. The first provides a bound, not sharp for $n\ge 3$, for all $n\ge 2$. This bound implies the more easily stated bound $d\le {4(2N-3)\over 3(2n-3)}$ for $n\ge 3$. The second result is a stabilization theorem; if $n$ is sufficiently large given $d$, then the sharp bound $d \le {N-1 \over n-1}$ holds. In this situation we determine all polynomials for which the bound is sharp.

math.CV↗
arXiv2003

Positivity conditions for Hermitian symmetric functions

We introduce a countable collection of positivity classes for Hermitian symmetric functions on a complex manifold, and establish their basic properties. We study a related notion of stability. The first main result shows that, if the underlying matrix of coefficients of an entire Hermitian symmetric function has at most k positive eigenvalues, then it can lie in the k-th positivity class only if it is a squared norm. We establish a similar result for Hermitian symmetric functions on the total space of a holomorphic line bundle. Finally we study the positivity classes for a natural one-parameter family of Hermitian metrics on a power of the universal bundle over complex projective space; we obtain sharp information about the parameter values in order to be in the k-th class. The paper closes with some additional information about the case when k is 2, where a nonlinear version of the Cauchy-Schwarz inequality arises.

math.CV↗