Products of many-variable polynomials: Pairs that are maximal in Bombieri's norm
For \(P(x_1, \dots, x_n)= \sum c_{i_1 \dots i_m} x_{i_1} x_{i_2} \dots x_{i_m}\) homogeneous of degree \(m\), define \[ [P]_2= (\sum |c_{i\dots i} |^2 )^{1/2}. \] The author characterizes \([PQ ]_2= [P ]_2 [Q ]_2\) and finds that \(P\) and \(Q\) have the form \(a( \sum t_i x_i)^m\) (with possibly different degrees). If the pair \((P, Q)\) is close to maximal (i.e., \[ [P Q]_2\geq \sqrt {1- \varepsilon} [P ]_2 [Q ]_2), \] then \(P\) and \(Q\) are close to being scalar multiples of powers of the same linear form. This continues work by the author and various collaborators [\textit{B. Beauzamy}, \textit{E. Bombieri}, \textit{P. Enflo} and \textit{H. L. Montgomery}, J. Number Theory 36, 219-245 (1990; Zbl 0729.30004); \textit{B. Beauzamy}, \textit{J. L. Frot} and \textit{C. Millour}, Massively parallel computations on many-variables polynomials (to appear)].
scientific article; zbMATH DE number 821868