American Mathematical Monthly Contents—January 2011

ARTICLES

Why Eisenstein Proved the Eisenstein Criterion and Why Schönemann Discovered It First

David A. Cox

This article explores the history of the Eisenstein irreducibility criterion and explains how Theodor Schönemann discovered this criterion before Eisenstein. Both were inspired by Gauss's Disquisitiones Arithmeticae, though they took very different routes to their discoveries. The article will discuss a variety of topics from 19th-century number theory, including Gauss's lemma, finite fields, the lemniscate, elliptic integrals, abelian groups, the Gaussian integers, and Hensel's lemma.

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How Small Can a Polynomial Be Near Infinity?

Jennifer M. Johnson; János Kollár 

We give bounds for real polynomials in two variables near infinity and near a zero. The bounds depend only on the degree of the polynomial and are simple to state.

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Arnol'd, the Jacobi Identity, and Orthocenters

Nikolai V. Ivanov

The three altitudes of a plane triangle pass through a single point, called the orthocenter of the triangle. This property holds literally in Euclidean geometry, and, properly interpreted, also in hyperbolic and spherical geometries. Recently, V. I. Arnol'd offered a fresh look at this circle of ideas and connected it with the well-known Jacobi identity. The main goal of this article is to present an elementary version of Arnol'd's approach. In addition, several related ideas, including ones of M. Chasles, W. Fenchel and T. Jørgensen, and A. A. Kirillov, are discussed.

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NOTES

Two Extensions of Results of Archimedes

Nicholas Pippenger

We review the two results in the Method of Archimedes, show how they follow from Cavalieri's principle, and show how two extensions of them can also be proved using Cavalieri's principle.

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On Polynomial Rings with a Goldbach Property

Paul Pollack

David Hayes observed in 1965 that when $$R=Z$$, every element of $$R[T]$$ of degree $$n\geq1$$ is a sum of two irreducibles in $$R[T]$$ of degree $$n$$. We show that this result continues to hold for any Noetherian domain $$R$$ with infinitely many maximal ideals.

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Measurable Functions with a Given Set of Integrability Exponents

Alfonso Villani

Given a measure space $$(\Omega, \mathcal{A}, \mu)$$, it is well known that for every $$\mathcal{A}$$-measurable function $$ƒ : \Omega \rightarrow \mathbb{R}$$ the set $$\mathcal{E}(f)=\{p\in(0,+\infty):f\in \mathcal{L}^{p}(\mu)\}$$ is always an interval, possibly degenerate, but, in general, it cannot be any given interval $$I\subseteq(0,+\infty)$$. Thus we consider the problem of characterizing those measure spaces for which $$\mathcal{E}(f)$$ can be an arbitrary subinterval of $$(0, +\infty)$$. We show that they are precisely the measure spaces such that there is no inclusion between different $$\mathcal{L}^{p}$$ spaces.

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Operator Reverse Monotonicity of the Inverse

Alexis Akira Toda

In statistics and econometrics, the equivalence between matrix inequalities $$A\geq B\Leftrightarrow B-1\geq A-1$$ is used to obtain a lower bound on the variance matrix, where $$A$$ and $$B$$ are symmetric and positive definite. The same property holds for linear operators on Hilbert spaces that are bijective, self-adjoint, and positive definite. I give a short and elementary proof of this fact.

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