A class of quadrinomial Garsia numbers (Q2855519)

From the text: An algebraic integer is called a Garsia number if it is real and larger than one, if all conjugates have modulus larger than one and if its norm has modulus two. Examples of Garsia numbers are given in [\textit{A. M. Garsia}, Trans. Am. Math. Soc. 102, 409--432 (1962; Zbl 0103.36502) and \textit{De-Jun Feng}, Bernoulli convolutions associated with some algebraic numbers, Journées deNEWLINENumération Graz 2007, manuscript], and exhaustive calculations were performed by A. Kovács and by P. Burcsi and A. Kovács. A characterizationNEWLINEof totally real and trinomial Garsia numbers and a list of Garsia numbers of degrees up to four can be found in the author's earlier paper [Acta Math. Acad. Paedagog. Nyházi. (N.S.) 25, No. 1, 9--16 (2009; Zbl 1224.11089)] (the title stated Garcia instead of Garsia). NEWLINENEWLINEHere we deal with quadrinomial Garsia numbers, and in view of the aforementioned results we restrict to degrees at least five. More specifically, we classify monic quadrinomials with symmetric exponent sequence which are minimal polynomials of Garsia numbers (Theorem 1), and we specify their signatures. It turns out that these Garsia numbers are rather small, and we exploit this fact to generate a dense subset of the reals (see Proposition 2). Further we show that certain other polynomials cannot have Garsia numbers among its roots (see Theorem 3 and Lemma 7). The main theorem proved here is as follows:NEWLINENEWLINETheorem 1. Let \(n, k\in\mathbb N\), \(n\geq 5\), \(0<2k<n\) and \(b, c, d\in\mathbb Z \backslash\{0\}\). Then the following statements hold.NEWLINENEWLINE(i) The polynomial (1) \(X^n + bX^{n-k} + cX^k + d\) is the minimal polynomial of a Garsia number if and only if \(bc = -1\) and \(d = -2\).NEWLINENEWLINE(ii) Let \(\gamma\) be a Garsia number with minimal polynomial (2)NEWLINE\(X^n -X^{n-k} + X^k - 2\) or \(X^n + X^{n-k} - X^k - 2\). ThenNEWLINE\(\gamma < 2^{1/ \max\{2,k\}}\) and \(\house{\gamma} <\sqrt 2\).NEWLINEMoreover, if \(n\) is odd then all other conjugates of \(\gamma\) are nonreal, and if \(n\) is even then \(\gamma\) has exactly one other real conjugate, and this conjugate is negative.