Counting Real Zeros of Analytic Functions Satisfying Linear Ordinary Differential Equations

✍ Scribed by Yulii Il'yashenko; Sergei Yakovenko

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 663 KB
• Volume: 126
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1996.0045

✦ Synopsis

We suggest an explicit procedure to establish upper bounds for the number of real zeros of analytic functions satisfying linear ordinary differential equations with meromorphic coefficients. If the equation

has no singular points in a small neighborhood U of a real segment K, all the coefficients a j (t) have absolute value A on U and a 0 (t)#1, then any solution of this equation may have no more than ;(A+&) zeros on K, where ;=;(U, K ) is a geometric constant depending only on K and U. If the principal coefficient a 0 (t) is nonconstant, but its modulus is at least a>0 somewhere on K, then the number of real zeros on K of any solution analytic in U, does not exceed (AÂa+&) + with some +=+(U, K ).