Mahler, Kurt

Reprint: A remark on a paper of mine on polynomials (1964)

Doc. Math. Extra Vol. Mahler Selecta, 625-629 (2019)
DOI: 10.25537/dm.2019.SB-625-629


Let \(S_{mn}\) be the set of all polynomial vectors \[\boldsymbol{f}(x)=(f_1(x),\ldots,f_n(x))\] of length \(n\) with components of degree at most \(m\) that are not identically zero. Further, set \[M(\boldsymbol{f})=\sum_{h=1}^n M(f_h),\qquad N(\boldsymbol{f})=\sum_{h=1}^n\sum_{k=1}^n M(f_h-f_k)\] and \(Q(\boldsymbol{f})=N(\boldsymbol{f})/M(\boldsymbol{f})\). The quantity of concern is \(C_{mn}:=\sup_{\boldsymbol{f}\in S_{mn}}Q(\boldsymbol{f}). \) In this paper, Mahler shows that \[C_{mn}\le 2(n^2-n)\lambda^m,\] where \(\lambda<1.91\). This is a significant improvement over the trivial bound of \(C_{mn}\le 2^{m+1}(n-1)\). \par Reprint of the author's paper [Ill. J. Math. 8, 1--4 (1964; Zbl 0128.07101)].

Mathematics Subject Classification