Non-symmetric polarization

Some remarks on non-symmetric polarization

“…Defant and Schlüters achieved similar upper bounds by refining their original calculations from [2] as it was mentioned during a personal communication.…”

“…Finally, applying (1) yields the result. Changing only the way in which this symmetrization is carried out and using the same arguments as in [2], we obtain improved bounds for the constants C(m, n) and C p (m, n). Additionally, we provide lower bounds for these constants.…”

“…Following [2], we turn to study how the coefficients of the succesive shuffles of L P change. Let L : (C n ) m → C be an m-linear form given by L x (1)…”

“…|P (x)|, (1) for any norm · in C n . In [2], Defant and Schlüters defined a non-symmetric m-linear form L P arising from a given m-homogeneous polynomial P . More precisely, for an m-homogeneous polynomial P : C n → C defined by P (x) = Assuming unconditionality of the norm · in C n , Defant and Schlüters proved that a similar estimate as in (1) holds for L P .…”

“…for any norm • in C n . In [2], Defant and Schlüters defined a non-symmetric m-linear form L P arising from a given m-homogeneous polynomial P . More precisely, for an m-homogeneous polynomial P : C n → C defined by P (x) = 1≤j 1 ≤...≤jm≤n c j 1 ...jm x j 1 .…”