Let Smn be the set of all polynomial vectors f(x)=(f1(x),…,fn(x)) of length n with components of degree at most m that are not identically zero. Further, set M(f)=∑h=1nM(fh),N(f)=∑h=1n∑k=1nM(fh−fk) and Q(f)=N(f)/M(f). The quantity of concern is Cmn:=supf∈SmnQ(f). In this paper, Mahler shows that Cmn≤2(n2−n)λm, where λ<1.91. This is a significant improvement over the trivial bound of Cmn≤2m+1(n−1).
Reprint of the author's paper [Ill. J. Math. 8, 1--4 (1964; Zbl 0128.07101)].