Combinatorial formula for Macdonald polynomials and generic Macdonald polynomials

We give a direct proof of the combinatorial formula for interpolation Macdonald polynomials by introducing certain polynomials, which we call generic Macdonald polynomials, and which depend on d additional parameters and specialize to all Macdonald polynomials of degree d. The form of these generic polynomials is that of a Bethe eigenfunction and they imitate, on a more elementary level, the R-matrix construction of quantum immanants.     

On cubic polynomials

It is shown that a system ofr homogeneous cubic equations with rational coefficients has a nontrivial solution in rational integers if the number of variables is at least (10r)⁵. For most such systems, an asymptotic formula holds for the numberz _P of solutions whose components have modulus <P.     

McMullen's root-finding algorithm for cubic polynomials

We show that a generally convergent root-finding algorithm for cubic polynomials defined by C. McMullen is of order 3, and we give generally convergent algorithms of order 5 and higher for cubic polynomials. We study the Julia sets for these algorithms and give a universal rational map and Julia set to explain the dynamics.

     

An analytical theory for Riemannian cubic polynomials

We establish some global existence and multiplicity resultsfor Riemannian cubic polynomials in a complete Riemannian manifold.     

Cotype 2 estimates for spaces of polynomials on sequence spaces

We give asymptotically correct estimations for the cotype 2 constant C₂(P(^m X _n )) ofthe spaceP(^m X _n ) of allm-homogenous polynomials onX _n , the span of the firstn sequencese _k =(\gd _kj ) _j in a Banach sequence spaceX. Applications to Minkowski, Orlicz and Lorentz sequence spaces are given. The second author was supported by DGESIC pr. no. 96-0758. The third author was supported by a grant of the Ministerio de Educación y Cultura (Spain), FP-97 29183763.     

Self-self-dual spaces of polynomials

A space of polynomials V of dimension 7 is called self-dual if the divided Wronskian of any 6-subspace is in V. A self-dual space V has a natural inner product. The divided Wronskian of any isotropic 3-subspace of V is a square of a polynomial. We call V self-self-dual if the square root of the divided Wronskian of any isotropic 3-subspace is again in V. We show that the self-self-dual spaces have a natural non-degenerate skew-symmetric 3-form defined in terms of Wronskians.We show that the self-self-dual spaces correspond to G₂-populations related to the Bethe Ansatz of the Gaudin model of type G₂ and prove that a G₂-population is isomorphic to the G₂ flag variety.     

Irreducible polynomials from a cubic transformation

Let $R(x)=g(x)/h(x)$ be a rational expression of degree three over the finite field ${\mathbb{F}}_q$. We count the irreducible polynomials in ${\mathbb{F}}_q[x]$, of a given degree, that have the form $h{(x)}^{\mathrm{deg}f}\cdot f(R(x))$ for some $f(x)\in {\mathbb{F}}_q[x]$. As an example of application of our results, we recover the number of irreducible transformation shift registers of order three, which were computed by Jiang and Yang in 2017.     

On primes represented by cubic polynomials

In this paper, we prove a theorem on the distribution of primes in cubic progressions on average.     

The combinatorial rigidity conjecture is false for cubic polynomials

We show that there exist two cubic polynomials with connected Julia sets which are combinatorially equivalent but not topologically conjugate on their Julia sets. This disproves a conjecture by McMullen from 1995.

     

Combinatorial models for the finite-dimensional Grassmannians

Let ? ⁿ be a linear hyperplane arrangement in ? ⁿ . We define two corresponding posetsG _k (? ⁿ andV _k (? ⁿ ) of oriented matroids, which approximate the GrassmannianG _k (? ⁿ ) and the Stiefel manifoldV _k (? ⁿ ). The basic conjectures are that the “OM-Grassmannian”G _k (? ⁿ ) has the homotopy type ofG _k (? ⁿ ), and that the “OM-Stiefel bundle” Δπ: ΔV _k (? ⁿ ) → ΔG _k (? ⁿ ) is a surjective map. These conjectures can be proved in some cases: we survey the known results and add some new ones. The conjectures fail if they are generalized to nonrealizable oriented matroids ? ⁿ .     

On ordered spaces of polynomials

We study quasimonotone increasing linear mappings in finite-dimensional spaces of real polynomials, ordered by the cone of nonnegative polynomials on . In particular, we prove a representation of the space of quasimonotone increasing and decreasing operators, which turns out to have dimension 3 or 4.     

Positive polynomials on Riesz spaces

We prove some properties of positive polynomial mappings between Riesz spaces, using finite difference calculus. We establish the polynomial analogue of the classical result that positive, additive mappings are linear. And we prove a polynomial version of the Kantorovich extension theorem.     

Bases in spaces of homogeneous polynomials on Banach spaces

In this work we present some conditions of equivalence for the existence of a monomial basis in spaces of homogeneous polynomials on Banach spaces.     

Chaotic polynomials on Banach spaces

We show the existence of chaotic (in the sense of Devaney) polynomials on Banach spaces of q-summable sequences. Such polynomials P consist of composition of the backward shift with a certain fixed polynomial p of one complex variable on each coordinate. In general we also prove that P is chaotic in the sense of Auslander and Yorke if and only if 0 belongs to the Julia set of p.     

Isometries between spaces of homogeneous polynomials

We derive Banach-Stone theorems for spaces of homogeneous polynomials. We show that every isometric isomorphism between the spaces of homogeneous approximable polynomials on real Banach spaces E and F is induced by an isometric isomorphism of E^′ onto F^′. With an additional geometric condition we obtain the analogous result in the complex case. Isometries between spaces of homogeneous integral polynomials and between the spaces of all n-homogeneous polynomials are also investigated.     

Positive bases in spaces of polynomials

For a nonempty compact set we determine the maximal possible dimension of a subspace of polynomial functions over Ω with degree at most m which possesses a positive basis. The exact value of this maximum depends on topological features of Ω, and we will see that in many of the cases m can be achieved. Whereas only for low m or finite sets Ω it is possible that we have a subspace X with positive basis and with dim X = m + 1. Hence there is no Ω for which a positive basis exists in for all . This work was accomplished during the 2^nd author’s stay in Paris under his Marie Curie fellowship, contract # MEIF-CT-2005-022927.