Let /? be a real number > 1. Addition and multiplication by a fixed positive integer of real numbers represented in bsise /3 cire shown to be computable by an on-line algorithm, and thus are continuous functions. When /? is a Pisot number, these functions cire computable by an on-line finite automaton.

Mathematical Foundations of Computer Science 1999

We consider families of multiple and simple integrals of the 'Ising class' and the linear ordinary differential equations with polynomial coefficients they are solutions of. We compare the full set of singularities given by the roots of the head polynomial of these linear ODEs and the subset of singularities occurring in the integrals, with the singularities obtained from the Landau conditions. For these Ising class integrals, we show that the Landau conditions can be worked out, either to give the singularities of the corresponding linear differential equation or the singularities occurring in the integral. The singular behaviour of these integrals is obtained in the self-dual variable w = s/2/(1 + s2), with s = sinh(2K), where K = J/kT is the usual Ising model coupling constant. Switching to the variable s, we show that the singularities of the analytic continuation of series expansions of these integrals actually break the Kramers–Wannier duality. We revisit the singular behaviour (Zenine et al 2005 J. Phys. A: Math. Gen. 38 9439–74) of the third contribution to the magnetic susceptibility of Ising model χ(3) at the points 1 + 3w + 4w2 = 0 and show that χ(3)(s) is not singular at the corresponding points inside the unit circle |s| = 1, while its analytical continuation in the variable s is actually singular at the corresponding points 2 + s + s2 = 0 outside the unit circle (|s| > 1).

Landau singularities and singularities of holonomic integrals of the Ising class

We prove a quantitative version of a result of Furstenberg and Deligne stating that the the diagonal of a multivariate algebraic power series with coefficients in a field of positive characteristic is algebraic. As a consequence, we obtain that for every prime $p$ the reduction modulo $p$ of the diagonal of a multivariate algebraic power series $f$ with integer coefficients is an algebraic power series of degree at most $p^{A}$ and height at most $A^2p^{A+1}$, where $A$ is an effective constant that only depends on the number of variables, the degree of $f$ and the height of $f$. This answers a question raised by Deligne.

Diagonalization and Rationalization of algebraic Laurent series

We give a new proof of Fatou's theorem: if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function. This result is applied to show that for any non-trivial completely multiplicative function from N to {-1,1}, the series Σ ∞ n=1 f(n)z n is transcendental over Z(z); in particular, Σ ∞ n=1 λ(n)z n is transcendental, where λ is Liouville's function. The transcendence of Σ ∞ n=1 μ(n)z n is also proved.

/pdf/transcendence-of-power-series-for-some-number-theoretic-338impk2xt.pdf

Transcendence of power series for some number theoretic functions

Helleseth, Kim, and No (2003) described the linear complexity over F/sub p/ of Sidel'nikov sequences of length p/sup m/-1 for p=3, 5, and 7. In this correspondence, the result is generalized to all odd primes.

Linear complexity over F/sub p/ of Sidel'nikov sequences

Transcendence of formal power series with rational coefficients

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Transcendence of formal power series with rational coefficients