Algèbres de Hadamard

Arithmetic properties of certain recurrence sequences

“…If / E A + and k E N we denote by (f{m)) x/k the real fcth root if k is odd and the positive real root if k is even. Let B -{g: N -R, g(m) = llLiCiWm)) 1 "", C, E Z, k t E N, / E A+ }; obviously B is a ring.…”

“…Moreover in the paper of Benzaghou [1] there is a reference to an apparently unpublished proof by Pisot of the particular case of Theorem 2 in which the coefficient of the leading term of the exponential polynomial (1) is constant, and the P-sequence coincides with N .…”

Endomorphismes d’algèbres de suites

“…À l'exemple de [1], nous noterons r(K) l'ensemble des suites récurrentes linéaires à valeurs dans K.…”

A Note On the Hadamard Kth Root of a Rational Function

“…Let x x be the largest element of X under the lexicographic ordering, and y x the largest element of Y. Then there is a unique term in the product u(h) k b(h) which has root Xyy v Similarly if x 2 ^ X and _y 2 G Y are smallest, there is a unique term with root x\y 2 . Unless X and Y are both one-element sets, these would be two distinct roots of b(h) of maximal absolute value, contrary to the hypothesis.…”

“…Consider the Taylor expansion [2 ] A note on the Hadamard /cth root of a rational function 315 and suppose that there is a sequence (a' h ) of e lements of a finitely generated extension field F of Q so that a' h k = b h , h = 0,1,2,..., for some given positive integer k. Then it is (a generalisation of) a conjecture of Pisot, see [2], page 249 and also [3], [4], that there is a sequence (a h ) so that a\ = b h , h = 0,1,2,..., and T. h>o with the roots /?, distinct non-zero complex numbers and the multiplicities n i positive integers. Except perhaps if r(X), s(X) have common factors, the /}, are just the reciprocals of the poles of the given rational function.…”