On polynomials of least deviation from zero in several variables

Details On polynomials of least deviation from zero in several variables On polynomials of least deviation from zero in several variables

2004-01-29

arxiv.org/abs/math/0401416v1

Mathematics

Classical Analysis and ODEs

14 pages, 1 figure

Scientific paper

A polynomial of the form $x^\alpha - p(x)$, where the degree of $p$ is less than the total degree of $x^\alpha$, 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 $d=3$, extremal polynomial for $(x_1x_2x_3)^k$ on the ball and the sphere is found for $k=2$ and 4. For $d \ge 3$, a family of polynomials of the form $(x_1... x_d)^2 - p(x)$ 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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