Xu's conjecture on the bound for the recursively defined polynomial TdT_d

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For d≥3d\ge 3, let TdT_d be the recursively defined polynomial in the elementary symmetric functions on the standard simplex TdT^d, and write ∥Td∥C(Td)\|T_d\|_{C(T^d)} for its uniform norm. The theorem in the paper proves the relevant least-deviation equalities for d=3,4,5d=3,4,5. Xu's TdT_d norm conjecture. For d≥6d\ge 6, one has

∥Td(x)∥C(Td)≤1.\|T_d(x)\|_{C(T^d)}\le 1.

Consequently, the equality in the two preceding least-deviation theorems also holds for d≥6d\ge 6. The claim is presented as the missing step in extending the proved cases to higher dimensions, and the source gives no proof for d≥6d\ge 6.

References

Primary source

Yuan Xu, “On polynomials of least deviation from zero in several variables”, arXiv:math/0401416 (2004).

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