On polynomials of least deviation from zero in several variables

2004/01/29 by Yuan Xu, Xu, Yuan
Mathematics · #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Mathematical functions and polynomials #math.CA

paper · pdf · doi:10.48550/arxiv.math/0401416

14 pages, 1 figure

arxiv created 2004/01/29 · openalex publication_date 2004/01/29 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A polynomial of the form x^α- p(x), where the degree of p is less than the total degree of x^α, is said to be least deviation from zero if it has the smallest uniform norm among all such polynomials. We study polynomials of least deviation from zero over the unit ball, the unit sphere and the standard simplex. For d=3, extremal polynomial for (x₁x₂x₃)^k on the ball and the sphere is found for k=2 and 4. For d ≥ 3, a family of polynomials of the form (x₁... x_d)² - p(x) is explicit given and proved to be the least deviation from zero for d =3,4,5, and it is conjectured to be the least deviation for all d.

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