On the Lorentz degree of a product of polynomials
The smallest integer \(m\) for which a real polynomial \(p\) having no zeros in an interval \((a,b)\) admits a representation of the form \(p(x) = \sum_{k=0}^{m} c_k (x-a)^k (b-x)^{m-k}\) with all \(c_k \geq 0\) or all \(c_k \leq 0,\) is called Lorentz degree of \(p\) over \([a,b]\) and denoted by \(d_{[a,b]}(p)\). \textit{T. Erdélyi} [J. Approx. Theory 67, No. 2, 187--198 (1991; Zbl 0759.41003)] formulated the following two conjectures: Is it true that \(d_{[a,b]}(pq) \geq \min (d_{[a,b]}(p), d_{[a,b]}(q))\) for any polynomials \(p\) and \(q\)? Is it true that \(d_{[a,b]}(pq) \geq | d_{[a,b]}(p) - d_{[a,b]}(q) |\) for any polynomials \(p\) and \(q\)? The author answers negatively the above two questions. The validity of the first one was shown for exact degree two and for a family of other high degree polynomials.
polynomials with nonnegative coefficients
scientific article; zbMATH DE number 6374936
scientific article