By Descartes’ rule of signs, a real degree $d$ polynomial $P$ with all nonvanishing coefficients with $c$ sign changes and $p$ sign preservations in the sequence of its coefficients ($c+p=d$) has $\mathrm{pos}\le c$ positive and $\neg \le p$ negative roots, where $\mathrm{pos}\equiv c(mod2)$ and $\neg \equiv p(mod2)$. For $1\le d\le 3$, for every possible choice of the sequence of signs of coefficients of $P$ (called sign pattern) and for every pair $(\mathrm{pos},\mathrm{neg})$ satisfying these conditions there exists a polynomial $P$ with exactly $\mathrm{pos}$ positive and exactly $\neg$ negative roots (all of them simple). For $d\ge 4$ this is not...

Let f(x) be a complex rational function. We study conditions under which f(x) cannot be written as the composition of two rational functions which are not units under the operation of function composition. In this case, we say that f(x) is prime. We give sufficient conditions for complex rational functions to be prime in terms of their degrees and their critical values, and we also derive some conditions for the case of complex polynomials.

The purpose of this paper is (1) to highlight some recent and heretofore unpublished results in the theory of multiplier sequences and (2) to survey some open problems in this area of research. For the sake of clarity of exposition, we have grouped the problems in three subsections, although several of the problems are interrelated. For the reader’s convenience, we have included the pertinent definitions, cited references and related results, and in several instances, elucidated the problems by...