The Analogue of the Shapiro - Tater Conjecture for the Painlevé IV Equation

In this thesis we study an intrinsic connection relating the fourth Painlevé equation and the even sextic anharmonic oscillator with a pole. We compare 1. zeros of generalized Hermite polynomials and Okamoto polynomials, coming as building blocks for rational solutions of the Painlevé equation, 2. zeros of certain resultants, coming as the condition of degeneration of algebraic spectrum of the sextic oscillator. Similar comparison for Painlevé II and the quartic oscillator is know as Shapiro-Tater conjecture. In this work we state the analog of the conjecture and prove presize matching between generalized Hermite polynomials and resultants; asymptotic matching between lattices of zeros of Okamoto polynomials and resultants under certain conditions.