Geodesic
Reprint: A remark on a paper of mine on polynomials (1964)
Documenta mathematica, Mahler Selecta (2019), pp. 625-629.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

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 $\lambda1.91$. 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)].
Classification : 11-03 
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     author = {Mahler, Kurt},
     title = {Reprint: {A} remark on a paper of mine on polynomials (1964)},
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     url = {http://geodesic.mathdoc.fr/item/DOCMA_2019__S1__a36/}
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Mahler, Kurt. Reprint: A remark on a paper of mine on polynomials (1964). Documenta mathematica, Mahler Selecta (2019), pp. 625-629. http://geodesic.mathdoc.fr/item/DOCMA_2019__S1__a36/