Location of Zeros Part II: Ordered fields

Abstract

In this sequel to [6], we continue the investigation into the behavior of multiplier sequences and n-sequences (of the first kind) begun in Part I. The definitions and notation used in Part I for studying the real numbers will be carried over to arbitrary fields in this part. In particular, a multiplier sequence for a fieldF is an infinite sequence F (y0, Yl, y2, ...) of elements ofF with the property that iff(x) Eakx k is a polynomial which splits in F, then F[f] Eykax also splits in F. An n-sequence is a finite sequence r (v0, v, v with the above property for polynomials of degree at most n. A multiplier sequence F will be called an exponential sequence if Yk Cr, k O, 1, 2, for some elements c and r in the field. These sequences, together with those such that y 0 for k 4: n, n + 1 (some fixed n), will be called trivial multiplier sequences; they are precisely the multiplier sequences which work for all fields [4]. If F {’0, yl, Y2, ...} and s is a positive integer, the sequence {y, y+, Ys+2, } will be called a shift of F; it is again a multiplier sequence [4, Proposition 2.2]. For further definitions and notation see Part I [6]. We shall refer to results in Part I by using the form Theorem 1.2.3 to mean Theorem 2.3 of Part I. Recall that a field is said to be formally real if it can be ordered [2], [11]. In the next section, we explore the extent to which multiplier sequences and n-sequences can be characterized over arbitrary formally real fields in ways similar to that for the real numbers as first proved by P61ya and Schur [4, Theorem 3.1]. This leads us to study certain special classes of formally real fields. In the third section, we take one of the main results of Part I for the real numbers and try to extend it to arbitrary real closed fields. In Part I we showed that a multiplier sequence F for the real numbers can be applied to an arbitrary polynomial with real coefficients, giving a new polynomial