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

Something You Always Wanted to Know About Real Polynomials (But Were Afraid to Ask)