Homogeneous Polynomials Research Articles

Boundedness of metaplectic Toeplitz operators and Weyl symbols

We study Toeplitz operators on the Bargmann space, whose Toeplitz symbols are exponentials of complex inhomogeneous quadratic polynomials. Extending a result by Coburn–Hitrik–Sjöstrand [7], we show that the boundedness of such Toeplitz operators implies the boundedness of the corresponding Weyl symbols, thus completing the proof of the Berger–Coburn conjecture in this case. We also show that a Toeplitz operator is compact precisely when its Weyl symbol vanishes at infinity in this case.

Formula omitted]-Stirling numbers in type [formula omitted

Stirling numbers of the first and second kind, which respectively count permutations in the symmetric group and partitions of a set, have found extensive application in combinatorics, geometry, and algebra. We study analogues and q-analogues of these numbers corresponding to the Coxeter group of type B. In particular, we show how they are related to complete homogeneous and elementary symmetric polynomials; demonstrate how they q-count signed partitions and permutations; compute their ordinary, exponential, and q-exponential generating functions; and prove various identities about them. Ordered analogues of the q-Stirling numbers of the second kind have recently appeared in conjectures of Zabrocki and of Swanson–Wallach concerning the Hilbert series of certain super coinvariant algebras. We provide conjectural bases for these algebras and show that they have the correct Hilbert series.

-ZARISKI PAIRS OF SURFACE SINGULARITIES

Abstract Let $f_0$ and $f_1$ be two homogeneous polynomials of degree d in three complex variables $z_1,z_2,z_3$ . We show that the Lê–Yomdin surface singularities defined by $g_0:=f_0+z_i^{d+m}$ and $g_1:=f_1+z_i^{d+m}$ have the same abstract topology, the same monodromy zeta-function, the same $\mu ^*$ -invariant, but lie in distinct path-connected components of the $\mu ^*$ -constant stratum if their projective tangent cones (defined by $f_0$ and $f_1$ , respectively) make a Zariski pair of curves in $\mathbb {P}^2$ , the singularities of which are Newton non-degenerate. In this case, we say that $V(g_0):=g_0^{-1}(0)$ and $V(g_1):=g_1^{-1}(0)$ make a $\mu ^*$ -Zariski pair of surface singularities. Being such a pair is a necessary condition for the germs $V(g_0)$ and $V(g_1)$ to have distinct embedded topologies.

Hierarchy of Quadratic Lyapunov Functions for Linear Time-Varying and Related Systems

A method for constructing homogeneous polynomial Lyapunov functions is presented for linear time-varying or switched-linear systems and the class of nonlinear systems that can be represented as such. The method uses a simple recursion based on the Kronecker product to generate a hierarchy of related dynamical systems, whose first element is the system under study and the second element is the well-known Lyapunov differential equation. It is then proven that a quadratic Lyapunov function for the system at one level in the hierarchy, which can be found via semidefinite programming, is a homogeneous polynomial Lyapunov function for the system at the base level in the hierarchy. Searching for Lyapunov functions of the foregoing kind is equivalent to searching for homogeneous polynomial Lyapunov functions via the formulation of sum-of-squares programs. The quadratic perspective presented in this paper enables the easy development of procedures to compute bounds on pointwise-in-time system metrics, such as peak norms, system stability margins, and many other performance measures. The applications of the theory to analyzing an aircraft model, on the one hand, and an experimental aerospace vehicle, on the other hand, are presented. The theory can be comprehended with a first course on state-space control systems and an elementary knowledge of convex programming.

ON A COMPARISON BETWEEN DWORK AND RIGID COHOMOLOGIES OF PROJECTIVE COMPLEMENTS

Abstract For homogeneous polynomials $G_1,\ldots ,G_k$ over a finite field, their Dwork complex is defined by Adolphson and Sperber, based on Dwork’s theory. In this article, we will construct an explicit cochain map from the Dwork complex of $G_1,\ldots ,G_k$ to the Monsky–Washnitzer complex associated with some affine bundle over the complement $\mathbb {P}^n\setminus X_G$ of the common zero $X_G$ of $G_1,\ldots ,G_k$ , which computes the rigid cohomology of $\mathbb {P}^n\setminus X_G$ . We verify that this cochain map realizes the rigid cohomology of $\mathbb {P}^n\setminus X_G$ as a direct summand of the Dwork cohomology of $G_1,\ldots ,G_k$ . We also verify that the comparison map is compatible with the Frobenius and the Dwork operator defined on both complexes, respectively. Consequently, we extend Katz’s comparison results in [19] for projective hypersurface complements to arbitrary projective complements.

Graphical methods and rings of invariants on the symmetric algebra

Abstract Let G be a complex classical group, and let V be its defining representation (possibly plus a copy of the dual). A foundational problem in classical invariant theory is to write down generators and relations for the ring of G-invariant polynomial functions on the space $\mathcal P^m(V)$ of degree-m homogeneous polynomial functions on V. In this paper, we replace $\mathcal P^m(V)$ with the full polynomial algebra $\mathcal P(V)$ . As a result, the invariant ring is no longer finitely generated. Hence, instead of seeking generators, we aim to write down linear bases for bigraded components. Indeed, when G is of sufficiently high rank, we realize these bases as sets of graphs with prescribed number of vertices and edges. When the rank of G is small, there arise complicated linear dependencies among the graphs, but we remedy this setback via representation theory: in particular, we determine the dimension of an arbitrary component in terms of branching multiplicities from the general linear group to the symmetric group. We thereby obtain an expression for the bigraded Hilbert series of the ring of invariants on $\mathcal P(V)$ . We conclude with examples using our graphical notation, several of which recover classical results.

Terracini Loci: Dimension and Description of Its Components

We study the Terracini loci of an irreducible variety X embedded in a projective space: non-emptiness, dimensions and the geometry of their maximal dimension’s irreducible components. These loci were studied because they describe where the differential of an important geometric map drops rank. Our best results are if X is either a Veronese embedding of a projective space of arbitrary dimension (the set-up for the additive decomposition of homogeneous polynomials) or a Segre–Veronese embedding of a multiprojective space (the set-up for partially symmetric tensors). For an arbitrary X, we give several examples in which all Terracini loci are empty, several criteria for non-emptiness and examples with the maximal defect possible a priori of an element of a minimal Terracini locus. We raise a few open questions.

Projective joint spectra and characters of representations of [formula omitted

For a tuple of square complex-valued N×N matrices A1,…,An the determinant of their linear combination x1A1+⋯+xnAn, which is called a pencil, is a homogeneous polynomial of degree N in C[x1,...xn]. Zero-set of this polynomial is an algebraic set in the projective space CPn−1. This set is called the determinantal hypersurface or determinantal manifold of the tuple (A1,...,An). It was shown in [7] that determinantal hypersurfaces contain substantial information about representations of finite Coxeter groups. Namely, if G is a non-special Coxeter group of type A,B, or D, ρ1 and ρ2 are two linear representations of G, and the determinantal hypersurfaces of images of the Coxeter generators of G under ρ1 and ρ2 coincide as divisors in the projective space, the characters of ρ1 and ρ2 are equal, and, therefore, ρ1 and ρ2 are equivalent. In [30] this result was extended in the characters part to affine Coxeter groups of types B,C, and D. It was shown there that each such group contains a finite subset such that, if the determinantal hypersurfaces of the images of this set under two finite-dimensional representations coincide as divisors in the projective space, the characters of these representations are equal. Notably, the affine Coxeter groups of A type are not covered by this result, as their combinatorics is quite different. In this paper we explicitly construct a finite set in A˜n having the same property. We also show that every group which is a semidirect product of a finite group and a finitely generated abelian group contains a finite subset with the similar property: for every finite-dimensional representation of the group, the determinantal hypersurface of images of the set determines the representation character.

Heading control based on extended homogeneous polynomial Lyapunov function

AbstractFor the nonlinear parameter‐varying (NPV) model of unmanned surface vehicle with the consideration of the velocities on yaw and surge as well as wave disturbances, a robust control method is proposed based on extended homogeneous polynomial Lyapunov function (EHPLF) to regulate heading for the superior performance on the rapidity, accuracy, and robustness. First, a NPV model of heading error is established to design a general form of a state feedback controller with a robust performance. Second, a Lyapunov matrix with full states and varying parameter is constructed to derive the robust global exponential stability conditions by Euler's homogeneity relation for the NPV system, known as the EHPLF stability conditions. Third, since the EHPLF stability conditions consist of a set of nonlinear coupled inequalities that cannot be directly solved by sum of squares (SOS) toolboxes, they are decoupled with matrix transformations to obtain the EHPLF‐SOS stability conditions, which is solved for the parameters of the state feedback controller. Finally, the simulation results indicate that EHPLF method exhibits a superior performance on dynamic, steady‐state, and robustness.

A Fischer type decomposition theorem from the apolar inner product

We continue the study initiated by H. S. Shapiro on Fischer decompositions of entire functions, showing that such decomposition exist in a weak sense (we do not prove uniqueness) under hypotheses regarding the order of the entire function f to be expressed as f=P·q+r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f= P\\cdot q+r$$\\end{document}, the polynomial P, and bounds on the apolar norm of homogeneous polynomials of degree m. These bounds, previously used by Khavinson and Shapiro, and by Ebenfelt and Shapiro, can be interpreted as a quantitative, asymptotic strengthening of Bombieri’s inequality. In the special case where both the dimension of the space and the degree of P are two, we characterize for which polynomials P such bounds hold.

Co-positivity of tensors and boundedness-from-below conditions of CP conserving two-Higgs-doublet potential

In this paper, the analytic sufficient and necessary conditions are obtained for the CP conserving two-Higgs-doublet potential to be bounded from below by using the co-positivity of tensors. This is achieved by treating the potential as a fourth-order homogeneous polynomial about the moduli of the two Higgs doublet fields, where the “angles” is described as the misalignment of the two doublets, then solving three minimum problems with respect to the misalignment. Finally, the analytic conditions are established with the help of the corresponding theory and methods of higher-order tensors.

Calibration and fast evaluation algorithms for homogeneous orthotropic polynomial yield functions

Calibration and fast evaluation algorithms for homogeneous orthotropic polynomial yield functions

Necessary and Sufficient Conditions for Stabilizing an Uncertain Stochastic Multidelay System

We focus on the problem of asymptotically mean-square stabilization in discrete-time stochastic systems that exhibit plant uncertainty, multiple input delays, and multiplicative noises. Our innovative contributions are described as follows. First, we employ a reduction method to transform the original model into a delay-free auxiliary system, and establish an equivalent proposition for stabilization based on this reformulation. On the basis of the reformulated model, we propose two stabilization criteria for the uncertainty-free case, including both Lyapunov-type and Riccati-type criteria. More generally, we extend the stabilization result to the uncertain model, and propose a necessary and sufficient stabilization criterion utilizing matrix homogeneous polynomials. Finally, we explore the existence and uniqueness of a delay margin under certain structural restrictions, and provide a closed-form representation of this margin.

Schmidt rank and singularities

UDC 512.5 We revisit Schmidt's theorem connecting the Schmidt rank of a tensor with the codimension of a certain variety and adapt the proof to the case of arbitrary characteristic. We also find a sharper result of this kind for homogeneous polynomials, assuming the characteristic does not divide the degree. Further, we use this to relate the Schmidt rank of a homogeneous polynomial (resp., a collection of homogeneous polynomials of the same degree) with the codimension of the singular locus of the corresponding hypersurface (resp., intersection of hypersurfaces). This gives an effective version of Ananyan--Hochster's theorem [J. Amer. Math. Soc., <strong>33</strong>, No. 1, 291–309 (2020), Theorem A].

On Extension of Multilinear Operators and Homogeneous Polynomials in Vector Lattices

On Extension of Multilinear Operators and Homogeneous Polynomials in Vector Lattices

Segre-driven radicality testing

We present a probabilistic algorithm to test if a homogeneous polynomial ideal I defining a scheme X in Pn is radical using Segre classes and other geometric notions from intersection theory which is applicable for certain classes of ideals. If all isolated primary components of the scheme X are reduced and it has no embedded components outside of the singular locus of Xred=V(I), then the algorithm is not applicable and will return that it is unable to decide radically; in all the other cases it will terminate successfully and in either case its complexity is singly exponential in n. The realm of the ideals for which our radical testing procedure is applicable and for which it requires only single exponential time includes examples which are often considered pathological, such as the ones drawn from the famous Mayr-Meyer set of ideals which exhibit doubly exponential complexity for the ideal membership problem.

The quantum harmonic oscillator with icosahedral symmetry and some explicit wavefunctions

The Dunkl Laplacian is used to define the Hamiltonian of a modified quantum harmonic oscillator, associated with any finite reflection group. The potential is a sum of the inverse squares of the linear functions whose zero sets are the mirrors of the group’s reflections. The symmetric group version of this is known as the Calogero-Moser model of N identical particles on a line. This paper focuses on the group of symmetries of the regular icosahedron, associated to the root system of type H3. Special wavefunctions are defined by a generating function arising from the vertices of the icosahedron and have the key property of allowing easy calculation of the effect of the Dunkl Laplacian. The ground state is the product of a Gaussian function with powers of linear functions coming from the root system. Two types of wavefunctions are considered, inhomogeneous polynomials with specified top-degree part, and homogeneous harmonic polynomials. The squared norms for both types are explicitly calculated. Symmetrization is applied to produce the invariant polynomials of both types, as well as their squared norms. The action of the angular momentum square on the harmonic homogeneous polynomials is determined. There is also a sixth-order operator commuting with the Hamiltonian and the group action.

On forms in prime variables

Let F 1 F_1 , …, F R F_R be homogeneous polynomials of degree d ⩾ 2 d\geqslant 2 with integer coefficients in n n variables, and let F = ( F 1 , … , F R ) \mathbf {F}=(F_1,\ldots ,F_R) . Suppose that F 1 F_1 , …, F R F_R is a non-singular system and n ⩾ 4 d + 2 d 2 R 5 n\geqslant 4^{d+2}d^2R^5 . We prove that there are infinitely many solutions to F ( x ) = 0 \mathbf {F}(\mathbf {x})=\mathbf {0} in prime coordinates if (i) F ( x ) = 0 \mathbf {F}(\mathbf {x})=\mathbf {0} has a non-singular solution over the p p -adic units U p \mathbb {U}_p for all prime numbers p p , and (ii) F ( x ) = 0 \mathbf {F}(\mathbf {x})=\mathbf {0} has a non-singular solution in the open cube ( 0 , 1 ) n (0,1)^n .

Expansion of the fundamental solution of a second‐order elliptic operator with analytic coefficients

AbstractLet be a second‐order elliptic operator with analytic coefficients defined in . We construct explicitly and canonically a fundamental solution for the operator, that is, a function such that . As a consequence of our construction, we obtain an expansion of the fundamental solution in homogeneous terms (homogeneous polynomials divided by a power of , plus homogeneous polynomials multiplied by if the dimension is even) which improves the classical result of [6]. The control we have on the complexity of each homogeneous term is optimal and in particular, when is the Laplace–Beltrami operator of an analytic Riemannian manifold, we recover the construction of the fundamental solution due to Kodaira [8]. The main ingredients of the proof are a harmonic decomposition for singular functions and the reduction of the convergence of our construction to a nontrivial estimate on weighted paths on a graph with vertices indexed by .

Reduced‐quaternion inframonogenic functions on the ball

A function from a domain in to the quaternions is said to be inframonogenic if , where . All inframonogenic functions are biharmonic. In the context of functions taking values in the reduced quaternions, we show that the inframonogenic homogeneous polynomials of degree form a subspace of dimension . We use the homogeneous polynomials to construct an explicit, computable orthogonal basis for the Hilbert space of square‐integrable inframonogenic functions defined in the ball in .