American Mathematical Monthly - April 2016

It's raining polynomials in this issue of the Monthly! What's an integer-valued polynomial? Find out in the lead article by Paul-Jean Cahen and Jean-Luc Chabert. 

Continuing on with this theme, find out what the ring of polynomials that map the Fibonacci number to the integers looks like in Keith Johnson and Ira Scheibelhut's paper "Rational Polynomials that take Integer Values at the Fibonacci numbers." 

Thomas Garrity reviews Steven Weintraub's book "Differential Forms: Theory and Practice." Our Problem Section will help steer you through your spring final exams. Stay tuned for the May issue when we will be counting Legos!

  - Scott T. Chapman, Editor

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Table of Contents

What You Should Know About Integer-Valued Polynomials

Paul-Jean Cahen and Jean-Luc Chabert

The authors wish to celebrate the centenary of Pólya′s paper Ueber ganzwertige ganze funktionen where first explicitly appeared the term “integer-valued polynomials.” This survey is focused on the emblematic example of the ring Int(ℤ) formed by the polynomials with rational coefficients taking integer values on the integers. This ring has surprising algebraic properties, often obtained by means of analytical properties. Yet, the article mentions also several extensions, either by considering integer-valued polynomials on a subset of ℤ, or by replacing ℤ by the ring of integers of a number field.DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.4.311

Rational Polynomials That Take Integer Values at the Fibonacci Numbers

Keith Johnson and Kira Scheibelhut

An integer-valued polynomial on a subset S of ℤ is a polynomial f(x) ∈ ℚ[x] with the property f(S) ⊆ ℤ. This article describes the ring of such polynomials in the special case that S is the Fibonacci numbers. An algorithm is described for finding a regular basis, i.e., an ordered sequence of polynomials, the nth one of degree n, with which any such polynomial can be expressed as a unique integer linear combination.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.4.338

Factoring Forms

Gary Brookfield

We provide necessary and sufficient conditions for the complete reducibility of ternary forms of degree three. Curiously, this result was well-known in the 19th century, but then forgotten.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.4.347

Conewise Linear Periodic Maps of the Plane with Integer Coefficients

Grant Cairns, Yuri Nikolayevsky and Gavin Rossiter

We study periodic, conewise linear maps of the plane with integer coefficients starting with Mort Brown′s map. We show that if the number of cones is two, there is only a short list of possible periods (this fact can be seen as the crystallographic restriction for this class of maps). Otherwise, without the restriction on the number of cones, a map can have any period. We show how to construct such maps using binary trees and so called admissible sequences.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.4.363

When Does a Given Polynomial with Integer Coefficients Divide Another?

Mohamed Ayad, Omar Kihel and Jesse Larone

Let f and g be polynomials with integer coefficients. In this paper, we improve upon a theorem of Nieto. We show that if the content of g divides the content of f and g(n) divides f(n) for an integer narbitrarily chosen larger than some explicit constant depending on the coefficients and the degrees of f and g, then g divides f in ℤ[x]. In addition, given a polynomial f with integer coefficients, we provide a method to determine if f is irreducible over ℤ, and if not, find one of its divisors in ℤ[x].

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.4.376

Notes

What Moser Could Have Asked: Counting Hamilton Cycles in Tournaments

Moser asked for a construction of explicit tournaments on n vertices having at leastHamilton cycles. We show that he could have asked for rather more.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.4.382

Some Generalizations of the Riemann–Lebesgue Lemma

Ovidiu Costin, Neil Falkner and Jeffery D. McNeal

We present several generalizations of the Riemann–Lebesgue lemma. Our approach highlights the role of cancellation in the Riemann–Lebesgue lemma.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.4.387

On Fiber Diameters of Continuous Maps

Peter S. Landweber, Emanuel A. Lazar and Neel Patel

We present a surprisingly short proof that for any continuous map f : ℝn → ℝm, if n > m, then there exists no bound on the diameter of fibers of f. Moreover, we show that when m = 1, the union of small fibers of f is bounded; when m > 1, the union of small fibers need not be bounded. Applications to data analysis are considered.

DOI: http://dx.doi.org/10.4169/amer.math.monthly.123.4.392

Problems and Solutions

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.4.399

Book Review

Differential Forms: Theory and Practice By Steven H. Weintraub

Reviewed by Thomas Garrity

To purchase the article from JSTOR: http://dx.doi.org/10.4169/amer.math.monthly.123.4.407

MathBits

A Proof of The Pythagorean Theorem After Descartes

Nuno Luzia

Irrationality via Maximality

​Daniel López-Aguayo

​Laurent Series and ℓp Sequences 

Dennis S. Bernstein, Khaled F. Aljanaideh and Arthur E. Frazho

100 Years Ago This Month in the American Mathematical Monthly

Edited by Vadim Ponomarenko