Related papers: The height distribution in root systems
We construct parametric families of (monic) reducible polynomials having two roots very close to each other.
We propose a modification of the predictions of the Cohen--Lenstra heuristic for class groups of number fields in the case where roots of unity are present in the base field. As evidence for this modified formula we provide a large set of…
We give an optimal necessary and sufficient condition for the quotient polynomial and remainder in the division algorithm to have positive coefficients.
In this short note we discuss recent results on hook length formulas of trees unifying some earlier results, and explain hook length formulas naturally associated to families of increasingly labelled trees.
We present two results, the first on the distribution of the roots of a polynomial over the ring of integers modulo $n$ and the second on the distribution of the roots of the Sylvester resultant of two multivariate polynomials. The second…
We show the existence of canonical heights of subvarieties for bounded sequences of morphisms and give some applications.
We prove several fundamental results about divisorial integral domains in the setup of multiplicative lattices.
We present a generalization of a formula of higher order derivatives and give a short proof.
We introduce the concept of completely positive roots of completely positive maps on operator algebras. We do this in different forms: as asymptotic roots, proper discrete roots and as continuous one-parameter semigroups of roots. We…
Motivated by the properties of the descent polynomials, which enumerate permutations of $S_n$ with a fixed descent set, we define descent polynomials for labeled rooted trees. We give recursive and explicit formulas for these polynomials…
A set of real $n$th roots that is pairwise linearly independent over the rationals must also be linearly independent. We show how this result may be extended to more general fields.
A short proof is given for the well-known Choi-Effros theorem on the structure of ranges of completely positive projections.
Let m be a fixed positive integer. It is well-known that a permutation $\sigma$ may have one, many, or no mth roots. In this note we provide an explicit expression and a generating function for the number of mth roots of \sigma. Let p_m(n)…
We generalize the definition and properties of root systems to complex reflection groups - roots become rank one projective modules over the ring of integers of a number field k. In the irreducible case, we provide a classification of root…
In this paper, we redesign and simplify an algorithm due to Remy et al. for the generation of rooted planar trees that satisfies a given partition of degrees. This new version is now optimal in terms of random bit complexity, up to a…
Linearized polynomials appear in many different contexts, such as rank metric codes, cryptography and linear sets, and the main issue regards the characterization of the number of roots from their coefficients. Results of this type have…
Selected examples of possible origins of power-law distributions are presented.
We give two determinantal representations for a bivariate polynomial. They may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two-parameter eigenvalue problem. The first determinantal…
We offer some summation formulas that appear to have great utility in probability theory. The proofs require some recent results from analysis that have thus far been applied to basic hypergeometric functions.
We describe positive generalized functionals in Gaussian Analysis. We focus on distribution spaces larger than the space of Hida Distributions. It is shown that a positive distribution is represented by a measure with specific growth of its…