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

  • Kurt Mahler


Let SmnS_{mn} be the set of all polynomial vectors f(x)=(f1(x),,fn(x))\boldsymbol{f}(x)=(f_1(x),\ldots,f_n(x)) of length nn with components of degree at most mm that are not identically zero. Further, set M(f)=h=1nM(fh),N(f)=h=1nk=1nM(fhfk)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(f)=N(f)/M(f)Q(\boldsymbol{f})=N(\boldsymbol{f})/M(\boldsymbol{f}). The quantity of concern is Cmn:=supfSmnQ(f).C_{mn}:=\sup_{\boldsymbol{f}\in S_{mn}}Q(\boldsymbol{f}). In this paper, Mahler shows that Cmn2(n2n)λm,C_{mn}\le 2(n^2-n)\lambda^m, where λ<1.91\lambda<1.91. This is a significant improvement over the trivial bound of Cmn2m+1(n1)C_{mn}\le 2^{m+1}(n-1).

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