Polynomials with high multiplicity at unity and Tarry’s problem
Copyright policy )Polynomials with high multiplicity at unity and Tarry’s problem
Let \(P\) be a non constant polynomial with integer coefficients of degree less than \(N\). The author denotes by \(r(P)\) the multiplicity of \(P\) at the point \(z=1\). In 1932 [Proc. Lond. Math. Soc., II. Ser. 33, 102-114 (1931; Zbl 0003.10501)] \textit{A. Bloch} and \textit{G. Pólya} proved that there exists a nonzero polynomial with coefficients in \(\{0,\pm 1\}\) such that \[ r(P) \geq \left[\sqrt {2 \log {2\log 2{N-1\over \log N}}}\,\right]-1. \] Since then this estimate has been improved several times. Here the author obtains an improvement of these results. His proof is based on an estimate of the number of solutions of Tarry's problem.
- Vilnius University Lithuania
multiplicity of roots, Tarry's problem, Diophantine equations in many variables, Polynomials (irreducibility, etc.)
multiplicity of roots, Tarry's problem, Diophantine equations in many variables, Polynomials (irreducibility, etc.)
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