We present a novel way to apply the singularity confinement property as a discrete integrability criterion.We shall use what we call a full deautonomisation approach, which consists in treating the free parameters in the mapping as functions of the independent variable, applied to a mapping complemented with terms that are absent in the original mapping but which do not change the singularity structure. We shall show, on a host of examples including the well-known mapping of Hietarinta-Viallet, that our approach offers a way to compute the algebraic entropy for these mappings exactly, thereby allowing one to distinguish between the integrable and non-integrable cases even when both have confined singularities.

We study the Laurent property, the irreducibility and co-primeness of discrete integrable and non-integrable equations. First we study a discrete integrable equation related to the Somos-4 sequence, and also a non-integrable equation as a comparison. We prove that the conditions of irreducibility and co-primeness hold only in the integrable case. Next, we generalize our previous results on the singularities of the discrete Korteweg-de Vries (dKdV) equation. In our previous paper [1] we described the singularity confinement test (one of the integrability criteria) using the Laurent property, and the irreducibility, and co-primeness of the terms in the bilinear dKdV equation, in which we only considered simplified boundary conditions. This restriction was needed to obtain simple (monomial) relations between the bilinear form and the nonlinear form of the dKdV equation. In this paper, we prove the co-primeness of the terms in the nonlinear dKdV equation for general initial conditions and boundary conditions, by using the localization of Laurent rings and the interchange of the axes. We assert that co-primeness of the terms can be used as a new integrability criterion, which is a mathematical reinterpretation of the confinement of singularities in the case of discrete equations.

We introduce a series of discrete mappings, which is considered to be an extension of the Hietarinta-Viallet mapping with one parameter. We obtain the algebraic entropy for this mapping by obtaining the recurrence relation for the degrees of the iterated mapping. For some parameter values the mapping has a confined singularity, in which case the mapping is equivalent to a recurrence relation between six irreducible polynomials. For other parameter values, the mapping does not pass the singularity confinement test. The properties of irreducibility and co-primeness of the terms play crucial roles in the discussion.MSC: 37K10, 39A20

We consider a two dimensional extension of the so-called linearizable mappings. In particular, we start from the Heideman-Hogan recurrence, which is known as one of the linearizable Somoslike recurrences, and introduce one of its two dimensional extensions. The two dimensional lattice equation we present is linearizable in both directions, and has the Laurent and the coprimeness properties. Moreover, its reduction produces a generalized family of the Heideman-Hogan recurrence.Higher order examples of two dimensional linearizable lattice equations related to the Dana-Scott recurrence are also discussed.

We investigate some of the discrete Painlevé equations (dP II , qP I and qP II ) and the discrete KdV equation over finite fields 1 . The first part concerns the discrete Painlevé equations. We review some of the ideas introduced in our previous papers [1,2] and give some detailed discussions. We first show that they are well defined by extending the domain according to the theory of the space of initial conditions. We then extend them to the field of p-adic numbers and observe that they have a property that is called an 'almost good reduction' of dynamical systems over finite fields. We can use this property, which can be interpreted as an arithmetic analogue of singularity confinement, to avoid the indeterminacy of the equations over finite fields and to obtain special solutions from those defined originally over fields of characteristic zero. In the second part we study the discrete KdV equation. We review the discussions in [3] and present a way to resolve the indeterminacy of the equation by treating it over a field of rational functions instead of the finite field itself. Explicit forms of soliton solutions and their periods over finite fields are obtained.