Rational Functions, Diagonals, Automata and Arithmetic

Lipshitz, Leonard; Poorten, Alfred J. van der; Mollin, R. A.

doi:10.1515/9783110848632-029

Cited by 20 publications

(32 citation statements)

References 14 publications

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“…Let us first recall the fundamental, and deep, mathematical relation between Hadamard product of series, multiple integration, and diagonal reduction [20,21]. Christol's conjecture amounts to saying that any algebraic power series in n variables is the "diagonal"of a rational power series of 2 n variables [20,22,23,24].…”

Section: The Diagonal Modelmentioning

confidence: 99%

“…Christol's conjecture amounts to saying that any algebraic power series in n variables is the "diagonal"of a rational power series of 2 n variables [20,22,23,24]. These kind of quite remarkable results [24,25,26,27] gave us the idea to build our second "toy model" by considering the "diagonal" of the integrand (seen as a function of (n − 1) angles φ j ) occurring in the multiple integrals corresponding to the χ (n) 's, and thus perform, instead of n − 1 integrals, only one integration.…”

Section: The Diagonal Modelmentioning

confidence: 99%

“…On the other hand, in view of the previous non-trivial results [20,22,23,24,25,26,27], one can, a contrario, imagine that such a "diagonal reduction" procedure keeps "some" relevant analytical informations, especially on the singularities. Replacing the quite involved multiple integral by an integration on only one angle, and since the integrand is an algebraic function of trigonometric functions of this angle and of the variable w = s/(1 + s 2 )/2, the result will be holonomic.…”

Section: The Diagonal Modelmentioning

confidence: 99%

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Landau singularities and singularities of holonomic integrals of the Ising class

Boukraa¹,

Hassani²,

Maillard³

et al. 2007

J. Phys. A: Math. Theor.

159

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Abstract.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 ODE's 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 behavior of these integrals is obtained in the self-dual variable w = s/2/(1 + s 2 ), 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 behavior (J. Phys. A 38 (2005) 9439-9474) of the third contribution to the magnetic susceptibility of Ising model χ (3) at the points 1+3w +4w 2 = 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 + s 2 = 0 oustside the unit circle (|s| > 1).

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Section: The Diagonal Modelmentioning

confidence: 99%

Section: The Diagonal Modelmentioning

confidence: 99%

Section: The Diagonal Modelmentioning

confidence: 99%

See 1 more Smart Citation

Landau singularities and singularities of holonomic integrals of the Ising class

Boukraa¹,

Hassani²,

Maillard³

et al. 2007

J. Phys. A: Math. Theor.

159

View full text Add to dashboard Cite

show abstract

“…There are a great many familiar sequences satisfying recurrence equations of the present kind (see, say, [6], [11], [17]). Indeed, (A h ) is always the sequence of Taylor coefficients of a formal power series satisfying a linear differential equation with coefficients which are rational functions; a particular case is that of a power series representing an algebraic function (note [17, Theorem 2.1]).…”

Section: S 0 (H)a H+n = S 1 (H)a H+n _ L + + S N (H)a H mentioning

confidence: 99%

On linear recurrence sequences with polynomial coefficients

Poorten

Shparlinski

1996

Glasgow Math. J.

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We consider sequences (Ah)defined over the field ℚ of rational numbers and satisfying a linear homogeneous recurrence relationwith polynomial coefficients sj;. We shall assume without loss of generality, as we may, that the sj, are defined over ℤ and the initial values A0A]…, An−1 are integer numbers. Also, without loss of generality we may assume that S0 and Sn have no non-negative integer zero. Indeed, any other case can be reduced to this one by making a shift h → h – l – 1 where l is an upper bound for zeros of the corresponding polynomials (and which can be effectively estimated in terms of their heights)

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“…In the last part of this paper we come back to the algebraic independence of certain formal power series which have been previously studied in [2]. 1 [4] et [5], de Denef et Lipshitz [9], et au récent survol de Lipshitz et van der Poorten [18].…”

mentioning

confidence: 99%

Note sur un article de Sharif et Woodcock

Allouche¹

1989

Journal De Théorie Des Nombres De Bordeaux

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Rational Functions, Diagonals, Automata and Arithmetic

Cited by 20 publications

References 14 publications

Landau singularities and singularities of holonomic integrals of the Ising class

Landau singularities and singularities of holonomic integrals of the Ising class

On linear recurrence sequences with polynomial coefficients

Note sur un article de Sharif et Woodcock

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