Some remarks about Descartes' rule of signs

Alain Albouy, Yanning Fu

Published 2013-07-23, updated 2014-10-29Version 2

What can we deduce about the roots of a real polynomial in one variable by simply considering the signs of its coefficients? On one hand, we give a complete answer concerning the positive roots, by proposing a statement of Descartes' rule of signs which strengthens the available ones while remaining as elementary and concise as the original. On the other hand, we provide new kinds of restrictions on the combined numbers of positive and negative roots.