In the recent paper by Pakovich and Muzychuk [Solution of the polynomial moment problem, Proc. Lond. Math. Soc. (3) 99 (2009), 633–657] it was shown that any solution of ‘the polynomial moment problem’, which asks to describe polynomials $Q$ orthogonal to all powers of a given polynomial $P$ on a segment, may be obtained as a sum of so-called ‘reducible’ solutions related to different decompositions of $P$ into a composition of two polynomials of lower degrees. However, the methods of that paper do not permit us to estimate the number of necessary reducible solutions or to describe them explicitly. In this paper we provide a description of polynomial solutions of the functional equation $P_1\circ W_1=P_2\circ W_2=\cdots =P_r\circ W_r,$and on this base describe solutions of the polynomial moment problem in an explicit form suitable for applications.

Let $\mathfrak{a}$ be a homogeneous ideal of a polynomial ring $R$ in $n$ variables over a field $\mathbb{k}$. Assume that $\mathrm{depth} (R/ \mathfrak{a})\geq t$, where $t$ is some number in $\{ 0, \ldots , n\} $. A result of Peskine and Szpiro says that if $\mathrm{char} (\mathbb{k})\gt 0$, then the local cohomology modules ${ H}_{\mathfrak{a}}^{i} (M)$ vanish for all $i\gt n- t$ and all $R$-modules $M$. In characteristic $0$, there are counterexamples to this for all $t\geq 4$. On the other hand, when $t\leq 2$, by exploiting classical results of Grothendieck, Lichtenbaum, Hartshorne and Ogus it is not difficult to extend the result to any characteristic. In this paper we settle the remaining case; specifically, we show that if $\mathrm{depth} (R/ \mathfrak{a})\geq 3$, then the local cohomology modules ${ H}_{\mathfrak{a}}^{i} (M)$ vanish for all $i\gt n- 3$ and all $R$-modules $M$, whatever the characteristic of $\mathbb{k}$ is.

For a $ \mathbb{Z} $-cover $\widetilde {M} \rightarrow M$ of a translation surface, which is a lattice surface, and which admits infinite strips, we prove that almost every direction for the straightline flow is ergodic.

We introduce an analogue in hyperkähler geometry of the symplectic implosion, in the case of $\mathrm{SU} (n)$ actions. Our space is a stratified hyperkähler space which can be defined in terms of quiver diagrams. It also has a description as a non-reductive geometric invariant theory quotient.

We show that the automorphism group of Drinfeld’s half-space over a finite field is the projective linear group of the underlying vector space. The proof of this result uses analytic geometry in the sense of Berkovich over the finite field equipped with the trivial valuation. We also take into account extensions of the base field.

Let ${F}_{BC} (\lambda , k; t)$ be the Heckman–Opdam hypergeometric function of type BC with multiplicities $k= ({k}_{1} , {k}_{2} , {k}_{3} )$ and weighted half-sum $\rho (k)$ of positive roots. We prove that ${F}_{BC} (\lambda + \rho (k), k; t)$ converges as ${k}_{1} + {k}_{2} \rightarrow \infty $ and ${k}_{1} / {k}_{2} \rightarrow \infty $ to a function of type A for $t\in { \mathbb{R} }^{n} $ and $\lambda \in { \mathbb{C} }^{n} $. This limit is obtained from a corresponding result for Jacobi polynomials of type BC, which is proven for a slightly more general limit behavior of the multiplicities, using an explicit representation of Jacobi polynomials in terms of Jack polynomials. Our limits include limit transitions for the spherical functions of non-compact Grassmann manifolds over one of the fields $ \mathbb{F} = \mathbb{R} , \mathbb{C} , \mathbb{H} $ when the rank is fixed and the dimension tends to infinity. The limit functions turn out to be exactly the spherical functions of the corresponding infinite-dimensional Grassmann manifold in the sense of Olshanski.

We show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about $0\in \mathbb{C} $ is free in the sense of pseudogroups even if the class of conjugacy of the generators is fixed. This result has a number of consequences on the topology of leaves for a (singular) holomorphic foliation defined on a neighborhood of an invariant curve. In particular, in the classical and simplest case arising from local nilpotent foliations possessing a unique separatrix which is given by a cusp of the form $\{ {y}^{2} - {x}^{2n+ 1} = 0\} $, our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves.