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On the zeros of certain polynomials. (English) Zbl 0992.12001

It is well known that if \(U\) is a complex polynomial of degree \(e\), not vanishing at \(1\) and \(d>0\) is an integer, then \(U(t)/(1-t)^d=\sum_{n=0}^\infty h_nt^n\) and there exists a complex polynomial \(H\) such that for \(n\geq\max\{0,e-d+1\}\) one has \(h_n=H(n)\). The author proves that if all roots of \(U\) lie on the unit circle then every root of \(H\) either lies on the vertical line \(\operatorname{Re} z= (e-d)/2\) or, which happens only in the case \(d-e\geq 2\), is one of the numbers \(-1,-2,\dots,e+1-d\). This result is then applied to Poincaré series of finitely generated graded algebras. In the final section the author speculates about possible connections with \(L\)-functions and the Riemann zeta-function.

MSC:

12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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