On a conjecture of Posner and Rumsey
Let \(m_ 1 > m_ 2 > \cdots > m_{i-1} > 0\), \(a_ 1, \dots, a_ i\) be nonzero integers. An expression of the form \(\sum^{i-1}_{j=1} a_ j x^{m_ j} + a_ i\) is called standard \(i\)-nomial. \textit{E. C. Posner} and \textit{H. Rumsey} jun. [Mich. Math. J. 12, 339--348 (1965; Zbl 0144.03504)] conjectured, that if a polynomial with rational coefficients divides infinitely many standard \(i\)-nomials over \(\mathbb Q\), it divides a nonzero polynomial of degree less than \(i\) in \(x^ r\) over \(\mathbb Q\), for some \(r \geq 1\). Posner and Rumsey proved the conjecture in the case \(i = 3\). Let \(p\) be a nonzero polynomial with rational coefficients and \(k\) distinct roots. Denote by \(K\) the splitting field of \(p\), and \(d = [K : \mathbb Q]\). Let \(S\) be the set of places of \(K\) consisting of all infinite places and all valuations induced by the prime ideal factors of the nonzero roots of \(p(x)\) and set \(s = \text{card }S\). The authors prove the conjecture of Posner and Rumsey for \(i = 3\) in the following stronger quantitative sense: If \(p(x)\) divides more than \((4sd)^{s^ 6 \cdot 2^{180d} + 8sk}\) standard trinomials over \(\mathbb Q\), then it divides a linear or quadratic polynomial in \(x^ r\) over \(\mathbb Q\) for some integer \(r \geq 1\). Moreover, the authors show, that the conjecture is false for \(i \geq 4\). The main tool of the proof is the application of quantitative results on \(S\)-unit equations.
scientific article; zbMATH DE number 584045
scientific article