Related papers: Topology of polar weighted homogeneous hypersurfac…

The Jacobi polynomials on the simplex are orthogonal polynomials with respect to the weight function $W_\bg(x) = x_1^{\g_1} ... x_d^{\g_d} (1- |x|)^{\g_{d+1}}$ when all $\g_i > -1$ and they are eigenfunctions of a second order partial…

We show that polar actions of cohomogeneity two on simple compact Lie groups of higher rank, endowed with a biinvariant Riemannian metric, are hyperpolar. Combining this with a recent result of the second-named author, we are able to prove…

Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W, such that S (but not necessarily R) is reduced. For each such pair (R,S) we construct a family of W-invariant orthogonal…

We discuss Poisson structures on a weighted polynomial algebra $A:=\Bbbk[x, y, z]$ defined by a homogeneous element $\Omega\in A$, called a potential. We start with classifying potentials $\Omega$ of degree deg$(x)+$deg$(y)+$deg$(z)$ with…

Hyperbolic polynomials are real multivariate polynomials with only real roots along a fixed pencil of lines. Testing whether a given polynomial is hyperbolic is a difficult task in general. We examine different ways of translating…

The nonzero level sets of a homogeneous, logarithmically homogeneous, or translationally homogeneous function are affine spheres if and only if the Hessian determinant of the function is a multiple of a power or an exponential of the…

We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are…

We study the orthogonal projection of homogeneous polynomials onto the space of homogeneous polyharmonic polynomials. To do this we derive the decomposition of homogeneous polynomials in terms of the Kelvin transform of derivatives of the…

Hexagon relations are combinatorial or algebraic realizations of four-dimensional Pachner moves. We introduce some simple set-theoretic hexagon relations and then `quantize' them using what we call `polynomial hexagon cohomologies'. Based…

A finite-dimensional algebra $A$ over an algebraically closed field $K$ is called periodic if it is periodic under the action of the syzygy operator in the category of $A-A-$ bimodules. The periodic algebras are self-injective and occur…

We classify all the centers of a planar weight-homogeneous polynomial vector field of weight degree 1, 2, 3 and 4.

We define twisted Alexander polynomials of a complex hypersurface with arbitrary singularities. These generalize the classical Alexander polynomials of high dimensional hypersurfaces and the twisted Alexander polynomial of plane curves. We…

A hyperbolic polynomial (HP) is a real univariate polynomial with all roots real. By Descartes' rule of signs a HP with all coefficients nonvanishing has exactly $c$ positive and exactly $p$ negative roots counted with multiplicity, where…

The purpose of this note is to extend in a simple and unified way the known results on interlacing of zeros of paraorthogonal polynomials on the unit circle. These polynomials can be regarded as the characteristic polynomials of any matrix…

Orthogonal polynomials for the multivariate hypergeometric distribution are defined on lattices in polyhedral domains in $\RR^d$. Their structures are studied through a detailed analysis of classical Hahn polynomials with negative integer…

A general problem is to classify the real forms of a complex variety up to isomorphism. This paper introduces the polar group of a real form $X$ of a complex variety $Y$ as a tool to distinguish such real forms. This group is an invariant…

Any homogeneous polynomial $P(x, y, z)$ of degree $d$, being restricted to a unit sphere $S^2$, admits essentially a unique representation of the form $\lambda_0 + \sum_{k = 1}^d \lambda_k [\prod_{j = 1}^k L_{kj}]$, where $L_{kj}$'s are…

We consider random orthonormal polynomials $$ P_{n}(x)=\sum_{i=0}^{n}\xi_{i}p_{i}(x), $$ where $\xi_{0}$, . . . , $\xi_{n}$ are independent random variables with zero mean, unit variance and uniformly bounded $(2+\ep_0)$-moments, and…

The existing classification of homogeneous quaternionic spaces is not complete. We study these spaces in the context of certain $N=2$ supergravity theories, where dimensional reduction induces a mapping between {\em special} real, K\"ahler…

We investigate the degree of the polar transformations associated to a certain class of multi-valued homogeneous functions. In particular we prove that the degree of the pre-image of generic linear spaces by a polar transformation…