Spectra and eigenvectors of the Segre transformation

Given two sequences a=(an)a=(an) and b=(bn)b=(bn) of complex numbers such that their generating series can be written as rational functions where the denominator is a power of 1−t1−t, we consider their Segre product a⁎b=(anbn)a⁎b=(anbn). We are interested in the bilinear transformations that compute the coefficient sequence of the numerator polynomial of the generating series of a⁎ba⁎b from those of the generating series of aa and bb. The motivation to study this problem comes from commutative algebra as the Hilbert series of the Segre product of two standard graded algebras equals the Segre product of the two individual Hilbert series. We provide an explicit description of these transformations and compute their spectra. In particular, we show that the transformation matrices are diagonalizable with integral eigenvalues. We also provide explicit formulae for the eigenvectors of the transformation matrices. Finally, we present a conjecture concerning the real-rootedness of the numerator polynomial of the r  -th Segre product of the sequence aa if r   is large enough, under the assumption that the coefficients of the numerator polynomial of the generating series of aa are non-negative.