On the leading coefficients of real many-variable polynomials (Q1332285)

Let \(P\) denote a homogeneous polynomial of degree \(k\) in \(N\) variables, \(x_ 1, x_ 2, \dots, x_ N\). The leading terms of \(P\) are those which contain only one variable raised to the power \(k\). In continuation of his interests on the topic, the author studies here again the problem as to how large the leading coefficients can be in the case when \(0 \leq P \leq 1\) and all variables satisfy \(0 \leq x_ i \leq 1\). He solves the problem completely for \(k = 2\) and \(k = 3\) and for other cases obtains better estimates than those provided recently in \textit{R. Aron}, \textit{B. Beauzamy} and \textit{P. Enflo} [J. Approximation Theory 74, No. 2, 181-198 (1993)].