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

Abstract

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 $C_{mn}\le 2^{m+1}(n-1)$.

Reprint of the author's paper [Ill. J. Math. 8, 1--4 (1964; Zbl 0128.07101)].