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Abstract
A polynomial of the form xa – p(x), where the degree of p is less than the total degree of xa , is said to be least deviation from zero if it has the smallest uniform norm among all such polynomials. We study polynomialsof least deviation from zero over the unit ball, the unit sphere, and the standard simplex. For d = 3, extremal polynomial for (x 1 x 2 x 3) k on the ball and the sphere is found for k = 2 and 4. For d ≥ 3, a family of polynomials of the form (x l… xd )2 – p(x) is explicitly 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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