On reality property of Wronski maps (Q1041086)

The B. and M. Shapiro conjecture asserts that if the Wronskian of \(N\) polynomials with complex coefficients has real roots only, then the space spanned by these polynomials has a basis consisting of polynomials with real coefficients. The authors prove that if the Wronskian of \(N\) quasi-exponentials \(e^{\lambda_i x}p_i(x)\),where \(\lambda_i\) are real numbers and \(p_i(x)\) are polynomials with complex coefficients, has real roots only, then the space spanned by these quasi-exponentials has a basis such that all polynomials have real coefficients. The case \(\lambda_1=\cdots=\lambda_N=0\) is the statement of the original B. and M. Shapiro conjecture.