A new method to determine the value or the reality of zeros for certain entire functions
- Brugidou, Vincent
DOIAbstract
AbstractWe present a new method to study the zeros of an entire function f(x)=∑n⩾0Anxn by associating a certain holomorphic function of two variables. When An≠0 for all n, we show that each zero appears formally as the inverse of a series, whose terms are rational expressions of the An. With a majorant method we prove the main result: let ρo be the positive root of ∑k=1∞ρk2=1/2, then if supn⩾1|An−1An+1/An2|<ρo2=0.2078… these series converge and determine exactly the zeros. When f(x) is real, the inequality supn⩾1|An−1An+1/An2|⩽ρo2 gives a sufficient criterion so that all zeros are real. The majorant ρo2 is conjectured to be the best possible.The main result required that we extend the composition G∘H of two formal power series to some cases...
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