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

Abstract

Let $S_{mn}$ be the set of all polynomial vectors $\mathbf{f}(x)=(f_1(x),\dots ,f_n(x))$ of length $n$ with components of degree at most $m$ that are not identically zero. Further, set $M(\mathbf{f})={\sum }_{h=1}^{n}M(f_h),N(\mathbf{f})={\sum }_{h=1}^n{\sum }_{k=1}^{n}M(f_h-f_k)$ and $Q(\mathbf{f})=N(\mathbf{f})/M(\mathbf{f})$. The quantity of concern is $C_{mn}:={\sup }_{\mathbf{f}\in S_{mn}}Q(\mathbf{f}).$ 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)].