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We give a new elementary proof of the following theorem: if all critical points of a rational function g belong to the real line then there exists a fractional linear transformation φ such that φ ◦ g is a real rational function. One of the many equivalent formulations of the Shapiro conjecture recently proved by Mukhin, Tarasov and Varchenko is the following. Let f = (f1,...,fp) be a vector of polynomials in one complex variable, and assume that the Wronski determinant W(f) = W(f1,...,fp) has only real roots. Then there exists a matrix A ∈ GL(p,C) such that fA is a vector of real polynomials. This result plays an important role in real enumerative geometry [16, 17], theory of real algebraic curves [8] and has applications to control theory [12, 4]. We refer to a survey [18] and the book [19] for a comprehensive discussion of this result and related problems. In [1] we proved the Shapiro conjecture in the first non-trivial case p = 2. The main idea was a coding of the elements of the preimage of the Wronski map with certain combinatorial objects. The proof in [1] was quite complicated, and its main drawback was the non-constructive character, and especially the use of the Uniformization theorem. Later we found a simpler proof [5] for p = 2 which preserves the main idea of [1] but employs only elementary arguments. Supported by NSF grants DMS-0555279 and by the Humboldt Foundation

Year: 2005

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