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A COMPLEX ANALOGUE OF THE ROLLE THEOREM AND POLYNOMIAL ENVELOPES OF IRREDUCIBLE DIFFERENTIAL EQUATIONS IN THE COMPLEX DOMAIN

Published online by Cambridge University Press:  01 October 1997

D. NOVIKOV  and
S. YAKOVENKO
D. NOVIKOV
Affiliation:
Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel. E-mail address: dmitri@math.weizmann.ac.il, yakov@math.weizmann.ac.il
S. YAKOVENKO
Affiliation:
Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel. E-mail address: dmitri@math.weizmann.ac.il, yakov@math.weizmann.ac.il
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Abstract

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.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 56 , Issue 2 , October 1997 , pp. 305 - 319
Copyright
The London Mathematical Society 1997

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