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A Complex Analogue of the Rolle Theorem and Polynomial Envelopes of Irreducible Differential Equations in the Complex Domain
Author(s) -
Novikov D.,
Yakovenko S.
Publication year - 1997
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461079700536x
Subject(s) - mathematics , monodromy , envelope (radar) , polynomial , ordinary differential equation , pure mathematics , linear differential equation , differential equation , irreducible polynomial , entire function , function (biology) , domain (mathematical analysis) , mathematical analysis , matrix polynomial , telecommunications , radar , evolutionary biology , computer science , biology
We prove a complex analytic analogue of the classical Rolle theorem asserting that the number of zeros of a real smooth function can exceed that of its derivative by at most 1. This result is used then to obtain upper bounds for the number of complex isolated zeros of: (1) functions defined by linear ordinary differential equations (in terms of the magnitude of the coefficients of the equations); (2) elements from the polynomial envelope of a linear differential equation with an irreducible monodromy group (in terms of the degree of the envelope); (3) successive derivatives of a function defined by a linear irreducible equation (in terms of the order of the derivative). These results generalize the bounds from [ 2, 5, 6 ] that were previously obtained for the number of real isolated zeros.