Space Of Homogeneous Polynomials Research Articles

Hermite interpolation of type total degree associated with certain spaces of polynomials

Abstract We study Hermite interpolation for the space of polynomials of total degree in ℝ N and the space of homogeneous polynomials in ℝ N+1. We investigate the relations between the two types of Hermite interpolation. We show that they have the same regularity and continuity property.

Equations for GL Invariant Families of Polynomials

We provide an algorithm that takes as an input a given parametric family of homogeneous polynomials, which is invariant under the action of the general linear group, and an integer d. It outputs the ideal of that family intersected with the space of homogeneous polynomials of degree d. Our motivation comes from Question 7 in Ranestad and Sturmfels (Le Math. 75, 411–424, 2020) and Problem 13 in Sturmfels (2014), which ask to find equations for varieties of cubic and quartic symmetroids. The algorithm relies on a database of specific Young tableaux and highest weight polynomials. We provide the database and the implementation of the database construction algorithm. Moreover, we provide a Julia implementation to run the algorithm using the database, so that more varieties of homogeneous polynomials can easily be treated in the future.

Spherical (t,t)-designs with a small number of vectors

For t∈{1,2,…} fixed, a natural class of spherical designs is given by the vectors v1,…,vn in Fd=Rd,Cd (not all zero) which give equality in the bound∑j=1n∑k=1n|〈vj,vk〉|2t≥ct(Fd)(∑ℓ=1n‖vℓ‖2t)2, where ct(Fd) is a known constant. These spherical (t,t)-designs integrate a space of homogeneous polynomials of degree 2t, and are variously known as real spherical half-designs of order 2t, complex (projective) t-designs, complex spherical semi-designs, and as tight frames when t=1. Little is known about the minimal number of vectors n for such a design.Here we report on the results of a numerical search for (t,t)-designs with a minimal number of vectors. In some cases, we obtain the designs explicitly as an orbit of a unitary action of a finite group on the sphere. We also list all the currently known (t,t)-designs. It is shown that many of these belong to a family of designs which we construct from the complex reflection groups. This family includes several new spherical (t,t)-designs with a small number of vectors.

Majorana representation for mixed states

We generalize the Majorana stellar representation of spin-$s$ pure states to mixed states, and in general to any hermitian operator, defining a bijective correspondence between three spaces: the spin density-matrices, a projective space of homogeneous polynomials of four variables, and a set of equivalence classes of points (constellations) on spheres of different radii. The representation behaves well under rotations by construction, and also under partial traces where the reduced density matrices inherit their constellation classes from the original state $\rho$. We express several concepts and operations related to density matrices in terms of the corresponding polynomials, such as the anticoherence criterion and the tensor representation of spin-$s$ states described in [1].

Banach Lattice Structures and Concavifications in Banach Spaces

Let ( Ω , Σ , μ ) be a finite measure space and consider a Banach function space Y ( μ ) . We say that a Banach space E is representable by Y ( μ ) if there is a continuous bijection I : Y ( μ ) → E . In this case, it is possible to define an order and, consequently, a lattice structure for E in such a way that we can identify it as a Banach function space, at least regarding some local properties. General and concrete applications are shown, including the study of the notion of the pth power of a Banach space, the characterization of spaces of operators that are isomorphic to Banach lattices of multiplication operators, and the representation of certain spaces of homogeneous polynomials on Banach spaces as operators acting in function spaces.

Mean ergodic composition operators in spaces of homogeneous polynomials

We study some dynamical properties of composition operators defined on the space P(Xm) of m-homogeneous polynomials on a Banach space X when P(Xm) is endowed with two different topologies: the one of uniform convergence on compact sets and the one defined by the usual norm. The situation is quite different for both topologies: while in the case of uniform convergence on compact sets every power bounded composition operator is uniformly mean ergodic, for the topology of the norm there is no relation between the latter properties. Several examples are given.

Bases in the spaces of homogeneous polynomials and multilinear operators on Banach spaces

For Banach spaces $E_1, \dots ,E_m$, $E$ and $F$ with their bases, we show that a particular monomial sequence forms a basis of $\mathcal {P}(^mE; F)$, the space of continuous $m$-homogeneous polynomials from $E$ to $F$ (resp.\ a basis of $\mathcal {L}(E_1,\dots ,E_m;F)$, the space of continuous $m$-linear operators from $E_1\times \cdots \times E_m$ to $F$) if and only if the basis of $E$ (resp. the basis of $E_1,\dots ,E_m$) is a shrinking basis and every $P \in \mathcal {P}(^mE; F)$ (resp.\ every $T \in \mathcal {L}(E_1,\dots ,E_m;F)$) is weakly continuous on bounded sets.

Projective Reed–Muller type codes on higher dimensional scrolls

In 1988 Lachaud introduced the class of projective Reed–Muller codes, defined by evaluating the space of homogeneous polynomials of a fixed degree d on the points of $$\mathbb {P}^n(\mathbb {F}_q)$$ . In this paper we evaluate the same space of polynomials on the points of a higher dimensional scroll, defined from a set of rational normal curves contained in complementary linear subspaces of a projective space. We determine a formula for the dimension of the codes, and the exact value of the dimension and the minimum distance in some special cases.

Duality results in Banach and quasi-Banach spaces of homogeneous polynomials and applications

Spaces of homogeneous polynomials on a Banach space are frequently equipped with quasinorms instead of norms. In this paper we develop a technique to replace the original quasi-norm by a norm in a dual preserving way, in the sense that the dual of the space with the new norm coincides with the dual of the space with the original quasi-norm. Applications to problems on the existence and approximation of solutions of convolution equations and on hypercyclic convolution operators on spaces of entire functions are provided.

Kahane–Salem–Zygmund polynomial inequalities via Rademacher processes

We prove new abstract inequalities for the expectation of the supremum norm of homogeneous Bernoulli polynomials on the unit ball of a Banach space. The development of this type of estimates was stimulated by the classical Kahane–Salem–Zygmund inequality and its recent extensions. We combine ideas from stochastic processes and interpolation theory to control increments of a Rademacher process in an Orlicz space via entropy integrals. To do this we prove general estimates for entropy integrals. We discuss applications to the study of multidimensional Bohr radii and unconditionality in spaces of homogeneous polynomials on Banach spaces. We specialize our results beyond ℓp-spaces and are able to prove variants of Bayart's results in the setting of Orlicz sequence spaces.

Symmetric Powers of Nat

We identify the spaces of homogeneous polynomials in two variables 𝕂[Yk, XYk−1, ⋅, Xk] among representations of the Lie ring . This amounts to constructing a compatible 𝕂-linear structure on some abstract -modules, where is viewed as a Lie ring.

Induced operators on the space of homogeneous polynomials

Let [Formula: see text] be the complex vector space of homogeneous polynomials of degree [Formula: see text] with the independent variables [Formula: see text]. Let [Formula: see text] be the complex vector space of homogeneous linear polynomials in the variables [Formula: see text]. For any linear operator [Formula: see text] acting on [Formula: see text], there is a (unique) induced operator [Formula: see text] acting on [Formula: see text] satisfying [Formula: see text] In this paper, we study some algebraic and geometric properties of induced operator [Formula: see text]. Also, we obtain the norm of the derivative of the map [Formula: see text] in terms of the norm of [Formula: see text].

Banach spaces of homogeneous polynomials without the approximation property

We present simple proofs that spaces of homogeneous polynomials on Lp[0, 1] and lp provide plenty of natural examples of Banach spaces without the approximation property. By giving necessary and sufficient conditions, our results bring to completion, at least for an important collection of Banach spaces, a circle of results begun in 1976 by R. Aron and M. Schottenloher (1976).

Weak sequential completeness of spaces of homogeneous polynomials

Let Pw(En;F) be the space of all continuous n-homogeneous polynomials from a Banach space E into another F, that are weakly continuous on bounded sets. We give sufficient conditions for the weak sequential completeness of Pw(En;F). These sufficient conditions are also necessary if both E⁎ and F have the bounded compact approximation property. We also show that the weak sequential completeness and the reflexivity of Pw(En;F) are equivalent whenever both E and F are reflexive.

Maximal minors of a matrix with linear form entries

Let P be a matrix whose entries are homogeneous polynomials in n variables of degree one over an algebraically closed field. We show that the maximal minors, say m-minors, of P generate the linear space of homogeneous polynomials of degree m if P has the maximal rank m at every point of the affine n-space except the origin.

LIMITS OF THE QUANTUM SO(3) REPRESENTATIONS FOR THE ONE-HOLED TORUS

For N ≥ 2, we study a certain sequence [Formula: see text] of N-dimensional representations of the mapping class group of the one-holed torus arising from SO(3)-TQFT, and show that the conjecture of Andersen, Masbaum, and Ueno [Topological quantum field theory and the Nielsen–Thurston classification of M(0,4), Math. Proc. Cambridge Philos Soc.141 (2006) 477–488] holds for these representations. This is done by proving that, in a certain basis and up to a rescaling, the matrices of these representations converge as p tends to infinity. Moreover, the limits describe the action of SL2(ℤ) on the space of homogeneous polynomials of two variables of total degree N - 1.

Courbes algébriques ordinaires et tissus associés

Let Γ be a complex algebraic curve of degree d, non-degenerate, reduced, and irreducible, in the complex projective space Pn(n⩾3,d⩾n). Denoting by c(n,h) the dimension of the vector space of homogeneous polynomials of degree h with respect to n variables, let k0 be the integer (⩾1) such that c(n,k0)⩽d<c(n,k0+1). The curve Γ is said to be ordinary if it has the following property: the set of algebraic hypersurfaces of a “generic” hyperplane H of Pn which have degree h and contain the hyperplane section Γ∩H, is empty if h⩽k0, and is a projective space of dimension c(n,h)−d−1 if h>k0. Equivalently when k0⩾2, the associated web in Pˇn of such a curve is “ordinary” in the sense of Cavalier and Lehmann (2012) [1]. The arithmetic genus of an ordinary curve is upper-bounded by the number π′(n,d)=∑h=1k0(d−c(n,h))(=k0d−c(n+1,k0)+1), and this bound is reached for these ordinary curves which are arithmetically Cohen–Macaulay. For n=3 and any d⩾3, the family of the ordinary curves of degree d which are arithmetically Cohen–Macaulay is non-empty, and is an irreducible component of the Hilbert scheme Hd,π′(3,d).By contrast, the complete intersection of n−1 algebraic hypersurfaces is never ordinary if k0⩾2.

Gel'fand-Tsetlin bases of orthogonal polynomials in Hermitean Clifford analysis

An explicit algorithmic construction is given for orthogonal bases for spaces of homogeneous polynomials, in the context of Hermitean Clifford analysis, which is a higher dimensional function theory centered around the simultaneous null solutions of two Hermitean conjugate complex Dirac operators. Copyright © 2011 John Wiley & Sons, Ltd.

Invariant norm quantifying nonlinear content of Hamiltonian systems

Given a Hamiltonian system, one can represent it using a symplectic map. This symplectic map is specified by a set of homogeneous polynomials which are uniquely determined by the Hamiltonian. In this paper, we construct an invariant norm in the space of homogeneous polynomials of a given degree. This norm is a function of parameters characterizing the original Hamiltonian system. Such a norm has several potential applications.

Rang et courbure de Blaschke des tissus holomorphes réguliers de codimension un

1 To any d-web of codimension one on a holomorphic n-dimensional manifold M ( d > n ), we associate an analytic subset S of M. We call regular the webs for which S has at most dimension n − 1 . This condition is generically satisfied. 1 We changed recently the terminology, and call now “ordinary” instead of “regular” the webs which are studied in this note. Denoting by c ( n , h ) the dimension of the vector space of homogeneous polynomials of degree h in n variables, we prove that the rank of a regular web has an upper-bound π ′ ( n , d ) equal to 0 for d < c ( n , 2 ) , and to ∑ h = 1 k 0 ( d − c ( n , h ) ) for d ⩾ c ( n , 2 ) , k 0 denoting the integer such that c ( n , k 0 ) ⩽ d < c ( n , k 0 + 1 ) . This bound π ′ ( n , d ) is optimal for regular webs. For n ⩾ 3 , it is strictly smaller than the bound π ( n , d ) of Chern–Castelnuovo. Moreover, if d is precisely equal to c ( n , k 0 ) , we define a holomorphic connection on some vector bundle E of rank π ′ ( n , d ) above M ∖ S , such that the vector space of germs of Abelian relation of the web at a point of M ∖ S is isomorphic to the vector space of germs at that point of holomorphic sections of E with vanishing covariant derivative: the curvature of this connection, which generalizes the curvature of Blaschke–Dubourdieu–Panzani–Hénaut, is then the obstruction for the rank of the web to reach the value π ′ ( n , d ) . [When n = 2 , S is always empty so that any web is regular, π ′ ( 2 , d ) = π ( 2 , d ) , any d may be written c ( 2 , k 0 ) : we recover the results given in Pantazi (1938) and Hénaut (2004).] To cite this article: V. Cavalier, D. Lehmann, C. R. Acad. Sci. Paris, Ser. I 346 (2008).