Bad pairs of polynomial zeros (Q2764925)

Let a polynomial \(p\) be a linear combination \(ap_1 + bp_2\) of two polynomials \(p_1\) and \(p_2\), or their Hadamard product \(p_1 \otimes p_2\) or any other of their combinations. The classical question is to determine the locus of the zeros of \(p\) given by the loci of the zeros of \(p_1\) and \(p_2\) - see, e.g., \textit{M. Marden's} Geometry of polynomials (1966; Zbl 0162.37101) or \textit{N. Obreschkoff's}, Verteilung und Berechnung der Nullstellen reeller Polynome, Berlin (1963; Zbl 0156.28002).NEWLINENEWLINENEWLINEThe present author introduces the following definition. Let \(U\) and \(V\) be two disjoint \(n-\)sets of real numbers, \(T=U \cup V = \{t_1,t_2, \ldots , t_{2n} \}\) be their union written in the increasing order and \(T_1 = \{ \{t_1, t_2 \}, \{t_3, t_4 \}, \ldots , \{t_{2n-1}, t_{2n} \}\) be the split of \(T\) in \(n\) consecutive pairs. A pair \(\{t_i, t_{i+1} \} \in T_1\) is called \(U-\)\textit{bad} iff both \(t_i, t_{i+1} \in U\). Let \(P_U\) (\(P_V\))be a monic polynomial with the zero set \(U\) (respectively \(V\)). It is known [see, e.g., \textit{H. Fell}, Pac. J. Math. 89, 43-50 (1980; Zbl 0441.30014)] that if \(T\) has no bad pair, then \(P_U \cup P_V\) also has only real roots. The author considers some examples and draws certain relationships between the \(n\), the number of bad pairs and the number of non-real roots of these polynomials.