  The famous Descartes' rule of signs from 1637 giving an upper bound on the
number of positive roots of a real univariate polynomials in terms of the
number of sign changes of its coefficients, has been an indispensable source of
inspiration for generations of mathematicians. Trying to extend and sharpen
this rule, we consider below the set of all real univariate polynomials of a
given degree, a given collection of signs of their coefficients, and a given
number of positive and negative roots. In spite of the elementary definition of
the main object of our study, it is a non-trivial question for which sign
patterns and numbers of positive and negative roots the corresponding set is
non-empty. The main result of the present paper is a discovery of a new
infinite family of non-realizable combinations of sign patterns and the numbers
of positive and negative roots.
