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2004 On Polynomials of Least Deviation from Zero in Several Variables
Yuan Xu
Experiment. Math. 13(1): 103-113 (2004).


A polynomial of the form {\small $x^\alpha - p(x)$}, where the degree of p is less than the total degree of {\small $x^\alpha$}, 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 {\small $d=3$}, extremal polynomial for {\small $(x_1x_2x_3)^k$} on the ball and the sphere is found for {\small $k=2$} and 4. For {\small $d \ge 3$}, a family of polynomials of the form {\small $(x_1\cdots x_d)^2 - p(x)$} is explicitly given and proved to be the least deviation from zero for {\small $d =3,4,5$}, and it is conjectured to be the least deviation for all d.


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Yuan Xu. "On Polynomials of Least Deviation from Zero in Several Variables." Experiment. Math. 13 (1) 103 - 113, 2004.


Published: 2004
First available in Project Euclid: 10 June 2004

zbMATH: 1058.41005
MathSciNet: MR2065570

Primary: 41A10 , 41A50 , 41A63
Secondary: 65D20

Keywords: Best approximation , Chevshev polynomial , Least deviation from zero , several variables

Rights: Copyright © 2004 A K Peters, Ltd.


Vol.13 • No. 1 • 2004
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