2004/07/14 by Jean-Claude Evard, Evard, Jean-Claude
Computer Science · Mathematics · #11C08 #11D25 #11D41 #11D72 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic and Geometric Analysis #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.AG #math.NT #msc:11C08 #msc:11D25 #msc:11D41 #msc:11D72
paper · pdf · doi:10.48550/arxiv.math/0407256
24 pages,submitted to the American Mathematical Monthly on July 23, 2003, MS #03-491, rejected on July 8, 2004, as "Too long", will be resubmitted in a few days after having made a few improvements
arxiv created 2004/07/14 · openalex publication_date 2004/07/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Polynomials whose coefficients, roots, and critical points lie in the ring of rational integers are called nice polynomials. In this paper, we present a general method for investigating such polynomials. We extend our results from the ring of rational integers to rings of algebraic integers that are unique factorization domains, with special interest in the ring of Gaussian integers. We apply our method to establish strong properties of nice polynomials whose degree is a prime power. We present a considerable reduction of the system of equations for nice polynomials of arbitrary degree with three roots. We establish properties of nice antisymmetric polynomials, and properties of the averages of the roots of the derivatives of nice polynomials. Finally, we present new examples of nice polynomials obtained with the help of a computer, after a considerable simplification of the computation by our method.