## 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)

### Keywords:

roots of polynomials; Poincaré series
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### References:

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