By Dr. M. J. Bertin, Dr. A. Decomps-Guilloux, Prof. M. Grandet-Hugot, Dr. M. Pathiaux-Delefosse, Prof. J. P. Schreiber (auth.)

the recognition of The ebook of Charles Pisot's thesis in 1938 delivered to the mathematical neighborhood these wonderful numbers referred to now because the Pisot numbers (or the Pisot-Vijayaraghavan numbers). even though those numbers were found previous via A. Thue after which through G. H. Hardy, it used to be Pisot's lead to that paper of 1938 that supplied the hyperlink to harmonic research, as stumbled on by means of Raphael Salem and defined in a sequence of papers within the Nineteen Forties. in a single of those papers, Salem brought the comparable classification of numbers, now universally often called the Salem numbers. those units of algebraic numbers are uncommon via a few impressive arith metic houses that account for his or her visual appeal in lots of varied parts of arithmetic: harmonic research, ergodic idea, dynamical structures and alge braic teams. previously, the easiest recognized and such a lot available creation to those num bers has been the gorgeous little monograph of Salem, Algebraic Numbers and Fourier research, first released in 1963. because the book of Salem's ebook, besides the fact that, there was a lot development within the research of those numbers. Pisot had lengthy expressed the will to submit an updated account of this paintings, yet his loss of life in 1984 left this job unfulfilled.

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Possesses k zeros in D(O, 1). Letting >.. tend to 1 gives us that h1 = f - 9 has at most k zeros in D(O, 1). Hence n ~ k. b) Let a1, a2, ... , as be the zeros of 9 of modulus 1. Set: s P(z) = II(z - ai). i=l h = f! P and gl = 9 / P are analytic on = P~~) zn +... Vz E D(O, 1) and gl has k From inequality i) we deduce that D(0,1). Moreover h(z) - gl(Z) zeros in D(O, 1). By a) we have n ~ k. 1. If f = A/Q is a limit point of the set F(q, k, 8), then A is different from =fQ*, where Q* is the reciprocal polynomial of Q.

Proof. 3 implies that there exists a polynomial Pin C[z] such that P(O) i- 0, dO(P)= n - 2r with r E N, and ord(F P - P*)2': n + 1- r. Assume that n i- 0 and r 2': 1, and set Q = (1 + z) P, then dO (Q) = n + 1 - 2r = n - 1 - 2(r - 1) and ord(FQ - Q*) 2': n - (r - 1) with r - 1 E N. 3, On-l = O. So r = O. ii) Assume that n i- 0, and set R = Pgcd (P, P*). As P(O) i- 0, R(O) i- O. We can suppose that R = R* because if P = RQ, P* = RS then P* = R*Q*, P = R* S* . Then R* divides P and P* and there exists A E C* such that .

Generalized Schur algorithm Equations (1) and (2) imply that • We end this chapter with Smyth's theorem. 5. Let P be a polynomial contained in Z[z], different from ±P*, monic, and with P(O) "# O. Denote (It'(h, ... , Os the roots of P. 3247 ... is the real root of 03 - 0 -1 number). = O. (0 0 is the smallest Pisot Before embarking on the proof of the theorem, we prove two lemmas about the functions of M. 1. Let J be a function belonging to M, and such that J(z) EnEN anz n Vz E D(O, 1) where an are real Vn E N.