The Complement of the Bifurcation Diagram of Trigonometric Polynomials

Abstract

1. We consider the space F = {f(t) = cosnt+n−1 j=1 (Aj cos jt+Bj sin jt) | Aj , Bj ∈ R} of real trigonometric polynomials defined on the circle (f : S1 = R/2πZ → R). The bifurcation diagram Σ1 ⊂ F is the set of non-Morse polynomials. The complex form of a polynomial f ∈ F is the meromorphic function g(z) : C → C obtained from f by the substitution z = eit . Let Σ2 be the set of polynomials f ∈ F such that their complex forms have nonsimple real critical values. Obviously, Σ1 and Σ2 are hypersurfaces in F and Σ1 ⊂ Σ2 . In this paper, we describe the complements F \ Σ1 and F \ Σ2 . Earlier, similar questions were considered in [1] for ordinary polynomials and in [2, 3] for trigonometric polynomials with 2n critical points on the circle. 2. Two functions f1, f2 : S1 → R are said to be topologically equivalent if there exist orientationpreserving homeomorphisms α : S1 → S1 and β : R → R such that f1α = βf2 . If f, f ′ ∈ F are topologically nonequivalent Morse polynomials, then they obviously belong to distinct components of the complement F \ Σ1 . Consider a Morse function f0 : S1 → R. Let [f0] ⊂ F be the set of Morse polynomials topologically equivalent to f0 . Suppose that the number of critical points of f0 on the circle is 2k. Obviously, [f0] is empty for k > n. Hence in the following we assume that k n. Theorem 1. The set [f0] is not empty for all Morse functions f0 such that k n. Moreover, if k < n, then [f0] is a connected submanifold of F . If k = n, then [f0] is a submanifold of F homotopically equivalent to n points. This theorem describes all components of the complement F \ Σ1 . 3. A bracket symbol is a syntactically correct sequence of 2k successive pairs of square brackets and n − k pairs of parentheses nested in square brackets. By way of example, we write out all bracket symbols for n = 3 and k = 1: [(( ))][ ], [( )( )][ ], [( )][( )], [ ][(( ))], [ ][( )( )]. It was proved in [1] that if n and k are given integers, then the number of all bracket symbols is equal to ( 2n−1 n−k ) − ( 2n−1 n−k−1 ) . Now to each polynomial f ∈ [f0] \ Σ2 we assign a bracket symbol. Consider the complex form g of the polynomial f ∈ F . Obviously, g(z) = 12(z + z−1) + ∑n−1 j=1 (Cjz j + Cjz−j). Let us describe the subset of the unit disk on which g takes real values. It is the union of the unit circle, 2k lines λi connecting zero with the critical points on the unit circle, and n− k circles ωm containing zero. There are no other points of intersection of these lines except for the following: λj ∩ λi = λi ∩ ωm = ωm ∩ ωl = {0}, 1 i, j 2k, 1 m, l n− k. Let us go around zero in the positive direction starting from the line λ1 connecting zero with the global minimum of g on the unit circle. We write out a bracket symbol by the following rules: 1) at the beginning, we write “[”, and at the end, we write “]”; 2) as we cross λi , 2 i 2k, we write “][”; 3) the first time we cross ωm , 1 m n − k, we write “(”; the second time, we write “)”. Needless to say, it is assumed that we cross each λi exactly once and each ωm exactly two times along the path.