Distributions of Roots of Reduced Cubic Equations with Random Coefficients

Kerry M. Soileau

Home/Archive/arXiv

arXiv · 0704.2586

Distributions of Roots of Reduced Cubic Equations with Random Coefficients

Kerry M. Soileau

View original source ↗Read source document ↗Export metadata ↓

Abstract

If the coefficients of polynomials are selected by some random process, the zeros of the resulting polynomials are in some sense random. In this paper the author rephrases the above in more precise language, and calculates the joint conditional densities of a random vector whose values determine almost surely the zeros of a "random" reduced cubic.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Kerry M. Soileau. 2012-11-23. Distributions of Roots of Reduced Cubic Equations with Random Coefficients. https://arxiv.org/abs/0704.2586

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Random models on regularity-integrability structures

We prove a convergence result for a large class of random models that encompasses the case of the BPHZ models used in the study of singular stochastic PDEs. We introduce for that purpose a useful variation on the notion of regularity structure called a regularity-integrability structure. It allows to deal in a single elementary setting with models on a usual regularity structure and their first order Malliavin derivative.

math.PR↗

arXiv2026

Central Limit Theorems for Drift and Entropy of Random Walks on Free Products

In this article we consider a natural class of random walks on free products of graphs, which arise as convex combinations of random walks on the single factors. From the works of Gilch [1,2] it is well-known that for these random walks the asymptotic entropy as well as the drift w.r.t. the natural transition graph distance and also w.r.t. the word length exist. The aim of this article is to formulate three central limit theorems with respect to both drift definitions and also w.r.t. the entropy when seen as the limit of last visit generating functions. In the case that the random walk depends on finitely many parameters we show that the corresponding variances in the central limit theorems w.r.t. both drifts vary real-analytically in terms of these parameters, while the variance in the central limit theorem w.r.t. the entropy varies real-analytically in the case of free products of finite graphs.

math.PR↗

arXiv2026

Harnack inequality for $p$-harmonic functions: probabilistic and analytic approaches

We survey analytic and probabilistic approaches to the Harnack inequality for $p$-harmonic functions, $p>1$, with particular attention to the dependence of the constants on $p$ and the dimension. We explain the ideas of Moser iteration, the tug-of-war approach of Luiro, Parviainen and Saksman, and the log-gradient estimate of Kotschwar and Ni. Along the way, we give a quantitative refinement of the probabilistic argument and a direct planar proof.

math.PR↗