Metric properties of polynomials

Monogenic A_4 quartic fields

“…Remark. We note that a theorem of Erdös guarantees that t(t 2 + 108) is squarefree for infinitely many positive integers t, (see [2]) so that our theorem will imply that there exist infinitely many monogenic A 4 quartic fields.…”

A remark on polynomials and the transfinite diameter

“…If D has transfinite diameter l, then perhaps the area of E(f) can be made < E for every e > 0 if n > no(E) (here the connectedness of D will not be needed) . That this is so when D is the unit circle or the interval (-2, + 2) is proved in [1] ; the general case is open .…”

“…If D is the unit circle, it is proved in [1 ;Th. 7] that the maximum number is n -1, and if D is the interval ( -2, + 2), it is easy to see that E(f) can have n components .…”

Length functions of lemniscates

“…In 1958 Erdös, Herzog and Piranian [13] posed a number of problems concentrated around the metric properties of lemniscates (see also the later paper [12]). Among them is the following Erdös Conjecture (Problem 12, [13]; Problem VI, [12]). For fixed degree n of P , is the length of the lemniscate |P (z)| = 1 greatest in the case where P (z) = Q n (z) := z n − 1?…”