A Problem of P olya Concerning Polynomials Which Obey Descartes ' Rule of Signs
نویسنده
چکیده
We formulate and duscuss two open problems. The rst one is due to PP olya. It states that the sequence of polynomials formed by a polynomial p with only real zeros and all its derivatives, obeys Descartes' rule of signs for any x, greater than the largest zero of p. The other problem is due to Karlin and states that certain Hankel determinants associated with an entire function in the Laguerre-Polya class do not change their signs. We prove that the statement of Karlin's problem is a consequence of that of PP olya's problem. The interest in these problems is motivated by the fact that once Karlin's problem is solved, it would yield necessary conditions that the Riemann hypothesis holds.
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