## Abstract

The ‘deautonomization’ of an integrable mapping of the plane consists in treating the free parameters in the mapping as functions of the independent variable, the precise expressions of which are to be determined with the help of a suitable criterion for integrability. Standard practice is to use the singularity confinement criterion and to require that singularities be confined at the very first opportunity. An algebro-geometrical analysis will show that confinement at a later stage leads to a non-integrable deautonomized system, thus justifying the standard singularity confinement approach. In particular, it will be shown on some selected examples of discrete Painlevé equations, how their regularization through blow-up yields exactly the same conditions on the parameters in the mapping as the singularity confinement criterion. Moreover, for all these examples, it will be shown that the conditions on the parameters are in fact equivalent to a linear transformation on part of the Picard group, obtained from the blow-up.

## 1. Introduction

The monicker ‘deautonomization’ [1] refers to the act of obtaining integrable, non-autonomous, extensions of autonomous mappings through the application of some discrete integrability criterion. It has been of paramount importance in the derivation and discovery of discrete Painlevé equations, the vast majority of which—known today—have in fact been obtained by applying this method to various autonomous, integrable, mappings.

Discrete Painlevé equations are usually derived starting from a mapping of the Quispel–Roberts–Thompson (QRT) family [2]. The reason for this choice lies in the analogy to the continuous case. The continuous Painlevé equations are non-autonomous extensions of equations with solutions that are given in terms of elliptic functions. Because the solution of a QRT mapping, in both its symmetric and asymmetric guises, can be expressed in terms of elliptic functions, it is of course the natural starting point for deriving discrete Painlevé equations. The discrete integrability criterion used for this purpose is either that of singularity confinement [3] or that of zero algebraic entropy [4]. Each of these criteria has its own particular advantages. Requiring the algebraic entropy to be zero is a more stringent criterion, based upon the study of the growth properties of the mapping, while singularity confinement—which is based upon the local study of singularities of the mapping—may turn out to be insufficient in some cases. In practice, however, this deficiency of the singularity confinement criterion can be circumvented if one starts from a QRT mapping, as the growth properties of such a mapping guarantee good behaviour after deautonomization, provided of course the local singularities are taken care of. Singularity confinement, on the other hand, presents a considerable practical advantage over algebraic entropy since one can study each singularity separately and obtain constraints on the parameters for each singularity individually. In the case of algebraic entropy, when one deals with a mapping which has several parameters to be deautonomized, these constraints usually become entangled.

The standard way to apply singularity confinement in the deautonomization process is to require confinement at the very first opportunity, i.e. after a succession of singularities that is the same as that for the underlying autonomous mapping. This last statement, however, needs some clarification and even a caveat. Namely, there exist situations where the same mapping can have more than one singularity pattern, leading to more than one possible deautonomization. Let us illustrate this on the example of the mapping [5]
*a*_{n}=1, but as we are only interested in mappings of infinite order, such a possibility is always discarded in the standard deautonomization approach). The first singularity pattern is *a*_{m+1}*b*_{m+1}=*a*_{m−1}*b*_{m+2}. The second one is *a*_{m+2}*b*_{m−1}/(*a*_{m}*b*_{m})} with the constraint *a*_{m}*b*_{m}=*a*_{m+2}*b*_{m−1}. Combining these two constraints, which are trivially satisfied in the autonomous case, one can integrate for *a*_{n} and *b*_{n}. We find

Another possibility exists however. We can, for example, choose to confine earlier in one of the two patterns (which has as a consequence that the other pattern will be longer). This can be done either by assuming that *a*_{m−1} in the first pattern is equal to 1, i.e. *a*_{n}=1 for all *n*, or by assuming that *a*_{m}*b*_{m}/*b*_{m−1}=*b*_{m+1} in the second pattern. Note however that this second choice is just the dual of the first one. Indeed, introducing *y*_{n}=*b*_{n}/*x*_{n} leads again to (1.1) with *a*_{n} replaced by *b*_{n+1}*b*_{n−1}/(*a*_{n}*b*_{n}). In the *a*_{n}=1 case, the second confinement possibility gives rise to the singularity pattern *b*_{m+5}*b*_{m−1}=*b*_{m+4}*b*_{m} (which is, again, trivially satisfied in the autonomous case). Integration of this constraint leads to

The question that can be asked at this point is whether it is imperative to confine at the first opportunity (albeit with the precautions dictated by the example presented above) and, especially, what will happen if one does not do so. Hietarinta & Viallet [6] have addressed this last question through an example based upon the equation known as the discrete Painlevé I. The standard form of this equation is
*z*_{n}=*αn*+*β*+*γ*(−1)^{n}. However, if, for example, one would overlook the first confinement opportunity and proceed further, another opportunity appears three steps later, one that leads to the confinement constraint
*c*_{k} are certain complex numbers, expressible in terms of radicals. Hietarinta & Viallet [6] applied the algebraic entropy criterion to the mapping deautonomized according to constraint (1.4) and found that this deautonomization does not pass this integrability test. In fact, pursuing their analysis they also showed that a confinement opportunity occurs periodically, every (3*N*+1) steps, leading to constraints similar to (1.4) that are however expected to lead, every single time, to a non-integrable deautonomization.

In this paper, we shall address this question of the confinement of singularities through an algebro-geometric approach. More precisely, we shall show in specific examples that if one regularizes a mapping with exactly eight blow-ups, one recovers the confinement constraint obtained at the first confinement opportunity. On the other hand, it will be shown that when more than eight blow-ups are performed, one always obtains confinement conditions that give rise to non-integrable systems. We shall illustrate this in detail in the case of mappings (1.1) and (1.2). Moreover, our analysis leads us to the important finding that the confinement conditions are in fact equivalent to the action of the linear transformation induced by the blow-up on the Picard group, when restricted to a subset of the exceptional lines. Our presentation is intended for a general audience in mathematical physics and we shall therefore not assume more than a general grasp of algebraic geometry and its techniques. For this reason, we shall present all the details of our calculations, with deeper mathematical considerations kept to a minimum and introduced only when necessary.

## 2. A first example: the d-P_{I} case

We first consider equation (1.2), which we shall interpret as a birational mapping on *n* through the function *z*_{n}, which is to be determined (but which we assume to be non-zero). The precise succession of blow-ups (i.e. the base points as well as coordinate charts) that allow for the regularization of the d-P_{I} equation (1.2) on a rational surface were first described by P. Howes in his doctoral thesis [7]. However, in that reference the calculations are carried out for the mapping (2.1) iterated twice, i.e. for *φ*_{n+1}∘*φ*_{n}, with the parameter dependence (1.3) explicitly inserted into the mapping, whereas in our approach the parameter dependence is *a priori* free. Obviously, this does not change the blow-ups that are required, nor does it change the resulting surface on which the mapping becomes regular, but it does change the behaviour of the exceptional lines under the resulting linear action on the Picard group (thereby obscuring the relation to the singularity patterns), the linear action itself being different: our matrix (2.10) for ℓ=1 is in fact, after an appropriate change of basis for the Picard group, the square root of the corresponding result in [7].

As is standard, we introduce the variables *s*_{n}=1/*x*_{n} and *t*_{n}=1/*y*_{n}, in terms of which *φ*_{n} becomes indeterminate at the points (*s*_{n},*y*_{n})=(0,0) and (*s*_{n},*t*_{n})=(0,0) and we shall start by lifting the indeterminacy at (*s*_{n},*y*_{n})=(0,0). This requires a blow-up, performed by introducing two new coordinate charts:
*s*_{n},*y*_{n}/*s*_{n})=(0,*z*_{n}), which requires a new blow-up (the same indeterminacy appears for the other coordinates). The new coordinate charts are
*s*_{n},*t*_{n})=(0,0), can be lifted by the blow-up
*s*_{n},*t*_{n}/*s*_{n})=(0,−1) as a new undefined point. Another blow-up is therefore needed through
*D* in figure 1 have self-intersection −2.

It is easily verified that the indeterminacy of the mapping *φ*_{n} at the point (0,0) is fully resolved after the blow-up (2.3) but that the one at (0,−1) still persists, even in the new coordinate charts (2.4). However, instead of continuing with trying to lift this indeterminacy, it pays to first analyse the singularities of the inverse mapping
*x*_{n+1},*t*_{n+1})=(0,0), requiring coordinates
*x*_{n+1}/*t*_{n+1},*t*_{n+1})=(*z*_{n},0):
*s*_{n+1},*t*_{n+1})=(0,0), requires a first blow-up
*s*_{n+1},*t*_{n+1}/*s*_{n+1})=(0,−1), with coordinate charts
*P*_{n}:(*x*_{n},*t*_{n})=(0,0) is obviously singular for the mapping *x*_{n}/*t*_{n},*t*_{n}) in (2.5) at *n*−1. Moreover, its image _{n}=*φ*_{n}(*P*_{n}) coincides exactly with the point (0,−1) in the coordinate charts of (2.4) at *n*+1:
*φ*_{n+1} is indeterminate at _{n}, it is clear that the indeterminacy at the point _{n−1} (for *φ*_{n}) requires yet another blow-up which, however, as is easily verified, does not yet resolve the singularity entirely. On the other hand, it is important to note that _{n−1} is in fact the pre-image of *Q*:(*s*_{n+1},*y*_{n+1})=(0,0),
*φ*_{n+1} and thus also requires a blow-up. This entire sequence of blow-ups is presented in figure 2. As mentioned above, the blow-up at *P*_{n} does not fully resolve the indeterminacy in *R*_{n}: (*x*_{n}/*t*_{n},*t*_{n})=(*z*_{n−1},0) using the coordinate chart (2.6) at *n*−1.

Similarly, we define _{n}=*φ*_{n}(*R*_{n}), which, in the coordinate chart used for the blow-up of _{n}, is easily found to be
_{n−1}. Moreover, the indeterminacy left in *φ*_{n+1} after the blow-up at the point *Q* can be resolved by blowing up at *S*_{n}: (*s*_{n+1},*y*_{n+1}/*s*_{n+1})=(0,*z*_{n+1}). This can of course be done using the coordinate charts (2.3) for *n*+1, after which the singularity of *φ*_{n+1} at (*s*_{n+1},*y*_{n+1})=(0,0) is fully resolved. Furthermore, as *S*_{n} is the base point for the blow-up of *Q*, it is interesting to look at the consequence of this blow-up on the relation expressed in (2.7),
_{n−1}=(0,*z*_{n−2}−*z*_{n−1}−1). This of course implies a constraint on *z*_{n}, which turns out to be exactly the integrability condition (1.3): *z*_{n+1}−*z*_{n}−*z*_{n−1}+*z*_{n−2}=0. Note that this is exactly the same stage as where the corresponding autonomous mapping becomes fully regularized. Indeed, implementing the above constraint results in a fully regularized non-autonomous mapping, which can be obtained by glueing together all the different coordinate charts introduced in the blow-ups. This was first shown in [7]. The curves in figure 4 encode the positions of these coordinate charts.

The curves labelled *D*_{1},…,*D*_{7} in figure 4 are different in nature from the others as they all have self-intersection −2 in the surface obtained after eight blow-ups. The other curves all have self-intersection −1, but *C*_{1}, *C*_{2} and *C*_{3} are distinguished because they are the exceptional curves obtained in the three last blow-ups. The curves represented in this figure are of fundamental importance in the description of the properties of the surface obtained in the regularization of *φ*_{n}. They can be thought of as part of a finitely generated free Abelian group, the so-called Picard (Pic) group, the rank of which is equal to the number of blow-ups +2 (when blowing-up *rank*(*Pic*)=10, and we can take (*D*_{1},…,*D*_{7},*C*_{1},*C*_{2},*C*_{3}) as a basis generating the whole group. The intersection patterns of the curves *D*_{1},…,*D*_{7} can be thought of as forming a Dynkin diagram, in this case for the affine algebra

Another crucial feature of the curves in figure 4 is that, having regularized the mapping for all *n*, their arrangement is essentially independent of *n*, i.e. although the exact position of each curve will depend on *z*_{n}, their mutual intersections will be the same for all *n*. The diagram only represents each curve up to linear equivalence and will therefore be the same for all *n*.

The evolution under *φ*_{n} induces the following map between the curves in figure 4:
*D*_{7} is left invariant. The fact that the basis (*D*_{1},…,*C*_{3}) is closed under this map can be used to great effect in calculating the algebraic entropy, as was shown by Takenawa in [11]. We shall come back to this point at the end of this section. Another important consequence of the existence of a well-defined map on the Picard group is that we immediately obtain the singularity pattern for *φ*_{n}. From {*y*=0}→*C*_{1}→*C*_{2}→*C*_{3}→{*x*=0}, we find for *y*: *φ*_{n}, for arbitrary initial conditions, as shown in [12].

### (a) The case of late confinement

The singularity pattern resulting from (2.8) is the shortest one possible. It was obtained by requiring that _{n−1} (or, equivalently, by requiring that *S*_{n+1}=*φ*_{n+1}*φ*_{n}(*R*_{n})), which at the level of the blow-ups was also the first opportunity to regularize the mapping. However, we may well decide to postpone regularization (or, equivalently, confinement) until another opportunity appears. Let us analyse this scenario in detail. It is clear that the only troublesome indeterminacies in the mapping *φ*_{n} arise on the chain of curves *D*_{1}→*D*_{2}→*D*_{3}→*D*_{1}. Therefore, starting from the point *R*_{n}:(*x*_{n}/*t*_{n},*t*_{n})=(*z*_{n−1},0) on *D*_{1} (the blow-up of which gives the exceptional curve *C*_{1}), iteration of the mapping yields
*D*_{2} and *C*_{2}, and subsequently
*D*_{3} and *C*_{3}. As explained above, requiring this point to coincide with *S*_{n+1}=(0,*z*_{n+2}) offers a first opportunity to regularize the mapping. If one chooses not to seize this opportunity, one has to iterate the mapping further, obtaining the points
*D*_{1}, *D*_{2} and *D*_{3}, respectively. It is easily verified that the only way to escape another loop through the same chain and to regularize the mapping at this stage is to require that the point *φ*_{n+4}*φ*_{n+3}*φ*_{n+2}*φ*_{n+1}*φ*_{n}(*R*_{n}) coincide with *S*_{n+4}=(0,*z*_{n+5}), all other possible recombinations of points leading to contradictions. We obtain thus precisely condition (1.4). It goes without saying that these three new points necessitate blowing-up, yielding the three new exceptional curves, *C*_{4}, *C*_{5} and *C*_{6}, depicted in figure 5. The map induced on the curves *D*_{1},…,*C*_{6}, under the evolution of *φ*_{n}, differs from (2.8) only in the part of the exceptional curves: {*y*=0}→*C*_{1}→⋯→*C*_{6}→{*x*=0}. This chain corresponds to the singularity pattern

In general, we can choose to regularize *φ*_{n} after an arbitrary number of loops around the curves *D*_{1},*D*_{2} and *D*_{3}. Let us first define points *D*_{j} (*j*=1,2,3):
*φ*_{n} gives
*D*_{3} allows us to regularize the mapping after ℓ loops through *D*_{1},*D*_{2} and *D*_{3}. This leads to the (late confinement) condition
*D*_{j} (*j*=1,2,3) at exactly ℓ base-points, generating on each such curve ℓ exceptional curves, as shown in figure 6. The map induced on the Picard group is the same as that given in (2.8) for the curves *D*_{1},…,*D*_{7}, while for the exceptional curves we have {*y*=0}→*C*_{1}→⋯→*C*_{3ℓ}→{*x*=0}, which leads to the singularity pattern:

### (b) Computing the algebraic entropy

The surface *X*_{n} on which *φ*_{n} is regular, constructed by blowing-up *X*_{n} does offer important information on the behaviour of the mapping *φ*_{n} even in the non-integrable case, as pointed out by Takenawa [11].

As we are dealing here with a non-autonomous system, the surface *X*_{n} depends explicitly on *n* and, strictly speaking, we do not have a single surface but rather a family of surfaces. However, as is clear from the construction of *X*_{n}, although the exact positions of the base-points in the blow-ups depend on *n*, the intersection pattern of curves depicted in figure 6, which are only defined up to linear equivalence, is independent of *n*. The same applies to the map *φ*_{*}: *Pic*(*X*_{n})→*Pic*(*X*_{n+1}) induced by *φ*_{n} on the Picard group *Pic*(*X*_{n}) of the surface *X*_{n}. As *Pic*(*X*_{n}) is of rank 7+3ℓ, choosing (*D*_{1},…,*D*_{7},*C*_{1},…,*C*_{3ℓ}) as a basis, we have
*φ*_{*} on this basis can be expressed through the matrix
2.10
where *A* is the 3ℓ×3ℓ submatrix given by
*x*=0} is linearly equivalent to
*φ*_{n} can be obtained from the largest eigenvalue *λ*_{0} of the induced map *φ*_{*}:
*φ*, *degφ*, is the maximum of the degrees of its numerator and denominator. Given the block structure of the matrix (2.10) and the fact that the upper-left block is unitary, it is clear that |*λ*_{0}| cannot be less than 1. To check whether |*λ*_{0}| is greater than 1, it suffices to compute the eigenvalues of the submatrix *A*. Because this matrix is in (a particularly simple) Frobenius normal form, its characteristic polynomial can be read off from the last column:
*f*(*λ*)=(*λ*−1)^{2}(*λ*+1). Thus, |*λ*_{0}|=1 and the algebraic entropy ℰ is 0. This is a well-known fact, the growth of the iterates of the mapping *φ*_{n} subject to condition (1.3) being quadratic. (This information can be gleaned from the Jordan normal form of the full matrix (2.10).) On the other hand, when ℓ>1, we remark readily that *f*(1)=1−ℓ<0 while *φ*_{n} is positive for all regularizations with ℓ>1 (i.e. with more than eight blow-ups), proving the non-integrability of all late confinements conjectured in [6].

An important consequence of the regularization procedure is that the confinement condition (2.9) can be expressed solely in terms of the submatrix *A*:

## 3. A second example: a *q*-difference equation

We now perform a similar analysis for equation (1.1), which we shall again think of as a birational mapping on *a*_{n}≠0 and *b*_{n}≠0,1. As before, we cover *ϕ*_{n} becomes indeterminate at (*x*_{n},*y*_{n})=(0,*b*_{n}) and (*s*_{n},*y*_{n})=(0,1), and its inverse *x*_{n+1},*y*_{n+1})=(*b*_{n},0) and (*x*_{n+1},*t*_{n+1})=(1,0). For convenience, we introduce the notation *P*_{n},*Q*_{n},*R*_{n} and *S*_{n} for the points
*ϕ*_{n+1}.

In analogy to the detailed calculations in the previous section, we first perform blow-ups at the indeterminate points *S*_{n−1} and *R*_{n−1} for the mapping *ϕ*_{n} and at *P*_{n+1} and *Q*_{n+1} (indeterminate points for *P*_{n} and *Q*_{n} (and *ϕ*_{n+1} at *R*_{n} and *S*_{n}) we must perform blow-ups at these points as well. The resulting exceptional lines are depicted in figure 7.

Next, we calculate the images of *P*_{n} and *Q*_{n} under the mapping *ϕ*_{n}
*ϕ*_{n}(*P*_{n}) to coincide with *R*_{n} and *ϕ*_{n}(*Q*_{n}) with *S*_{n}, in which case the mapping *ϕ*_{n} requires no further blow-ups. The condition on the parameters in this case is *a*_{n}=1 and *b*_{n+1}*b*_{n−1}=*b*_{n}, which means that *b*_{n} is periodic, with period 6. Moreover, it is easily verified that the mapping *ϕ*_{n} itself is periodic, with period 12 (for arbitrary initial conditions), which implies that the growth of the degree of its iterates is bounded. Note that as the mapping is periodic, it does not fall into the class of mappings that is the object of proposition 2.1 in [15], which states that a mapping with bounded growth (but of infinite order) is conjugate to an automorphism on *n*≠1. Indeed, the present mapping is regularized on a family of del Pezzo surfaces of degree 4 (the arrangement of −1 curves in the top left-most diagram in figure 7 can be thought of as depicting such a surface). However, as mentioned in the Introduction, the purpose of deautonomizing an integrable mapping is to obtain mappings of infinite order, and as such this first possible regularization does not count as a true ‘first confinement’ and it should therefore be discarded.

Another possible way to regularize the mapping *ϕ*_{n} would be to keep the constraint *a*_{n}=1 (i.e. *ϕ*_{n}(*P*_{n})=*R*_{n}) but to wait for the next opportunity for one of the iterates *ϕ*_{m}*ϕ*_{m−1}⋯*ϕ*_{n}(*Q*_{n}) of *Q*_{n} to coincide with some *S*_{m} (*m*>*n*). Calculating the next few iterates (under the condition *a*_{n}=1),
*ϕ*_{n+4}*ϕ*_{n+3}⋯*ϕ*_{n}(*Q*_{n}) coincide with *S*_{n+4}. The resulting constraint is identical to the confinement condition described in the Introduction: *b*_{n+5}*b*_{n−1}=*b*_{n}*b*_{n+4}. The indeterminacies that arise at the intermediate points *ϕ*_{n}(*Q*_{n}),…,*ϕ*_{n+3}*ϕ*_{n+2}*ϕ*_{n+1}*ϕ*_{n}(*Q*_{n}) in the above chain of iterates must now also be eliminated by blow-up, yielding the four extra exceptional curves *C*′_{2},*C*′_{3},*C*′_{4} and *C*′_{5}, respectively, that are depicted in figure 8. The curve *C*′_{6} in this figure corresponds to the blow-up at the point *S*_{n}.

Moreover, from the above calculations, it is clear that the curves *D*_{1},*D*_{2},*D*_{3},*D*_{4} will always be mapped cyclically into each other as
*ϕ*_{n} on this surface, the eight exceptional curves *C*_{1},*C*_{2},*C*′_{1},…,*C*′_{6} form two separate chains, {*y*=1}→*C*_{1}→*C*_{2}→{*x*=1}, which corresponds to the singularity pattern *y*=*b*}→*C*′_{1}→⋯→*C*′_{6}→{*x*=*b*}, which corresponds to the pattern

Of course, another possibility would be to relax the condition *a*_{n}=1, by requiring that some iterate of *P*_{n} beyond *ϕ*_{n}(*P*_{n}) coincides with some *R*_{m}, which will then also require further blow-ups. In order to describe this general set-up, we define the points (for general *ϕ*_{n} take the simple form
*P*_{n} can be obtained as
*Q*_{n} as
*ϕ*_{n} can be regularized after exactly 4+4(ℓ+ℓ′) blow-ups (for arbitrary non-negative integers ℓ and ℓ′) by requiring that the chain of iterates for *P*_{n} terminate at *Q*_{n} at *a*_{n} and *b*_{n}. The family of surfaces *X*_{n} obtained after 4+4(ℓ+ℓ′) blow-ups is depicted in figure 9 (figure 8 corresponds to the special case ℓ=0,ℓ′=1). Besides the fundamental chain (3.2) for the −(ℓ+ℓ′+1) curves *D*_{1},*D*_{2},*D*_{3} and *D*_{4}, the mapping *ϕ*_{n} induces the following behaviour for the exceptional curves on the surface depicted in figure 9:

In general, the Picard group *Pic*(*X*_{n}) for this surface will have rank 6+4(ℓ+ℓ′). Choosing (*D*_{1},*D*_{2},*D*_{3},*D*_{4},*C*_{2},…,*C*_{4ℓ+1},*C*′_{1},…,*C*′_{4ℓ′+2}) as a basis for *Pic*(*X*_{n}), the action *ϕ*_{*} induced by *ϕ*_{n} on *Pic*(*X*_{n}) can be expressed by means of the matrix
3.7
where the size 4(ℓ+ℓ′)+2 (square) submatrix *Φ* is zero everywhere, except on the main sub-diagonal, where it is 1, and along its 4ℓth column, which is
*Φ*). Here we have used the fact that *C*_{4ℓ+2} and the strict transform of {*x*=*b*} are, respectively, linearly equivalent to the curves

As before, because of the block structure of matrix (3.7) and in particular because of the unitarity of its upper left-most block, the integrability of the corresponding deautonomizations of *ϕ*_{n} is decided by the largest eigenvalue of the matrix *Φ*. The characteristic polynomial of *Φ* is
*f*_{Φ}(1)=1−(ℓ′+ℓ). Hence, for any choice of ℓ and ℓ′ such that ℓ′+ℓ>1, there is a real root that is greater than 1 and the algebraic entropy of the corresponding deautonomized mapping is necessarily positive. The only integrable deautonomizations of (1.1) obtained from the above regularization are therefore the trivial, periodic, case ℓ=ℓ′=0 (for which the matrix (3.7) is in fact a 12th root of the identity matrix) and the cases ℓ=0, ℓ′=1 and ℓ=1, ℓ′=0 (which, as explained in the Introduction, are in fact dual to each other). In the latter two cases *f*_{Φ}(*λ*)=(*λ*−1)^{2}(*λ*^{4}+*λ*^{3}+*λ*^{2}+*λ*+1), which implies that all eigenvalues have modulus 1 and that the algebraic entropy of the corresponding mappings is 0. Discarding the case ℓ=ℓ′=0 for which the mapping is of finite order, we see that the only integrable cases indeed correspond to the first confinement opportunities for two—distinct but dual—singularity patterns. This result implies, in particular, that any number of blow-ups greater than 8 for mapping (3.1) necessarily leads to a non-integrable mapping.

### (a) A second blow-up pattern

From the above analysis, it should be clear that there exists yet another pattern of blow-ups through which mapping (3.1) can be regularized. Indeed, instead of requiring the iterates of *P*_{n} to coincide, at some stage, with a future point *R*_{m} (and similarly for the chain of iterates of *Q*_{n} to link up with some *S*_{m}), it is of course entirely possible to switch these two requirements and to ask that *P*_{n}, at some iterate, ends up at *S*_{m′} and *Q*_{n} at some *R*_{m}. Using notation (3.3) and the general result (3.4), the requirement on the chain of iterates for *P*_{n} then becomes
*Q*_{n}:
*ϕ*_{n} becomes regular after 8+4(ℓ′+ℓ) blow-ups. The resulting family of surfaces *a*_{n} and *b*_{n} can be summarized as follows:

The fundamental behaviour (3.2) of the curves *D*_{1},*D*_{2},*D*_{3} and *D*_{4} induced by the mapping *ϕ*_{n} remaining unchanged, that of the exceptional curves on *D*_{1},*D*_{2},*D*_{3},*D*_{4},*C*_{1},…,*C*_{4ℓ+3}, *C*′_{2},…,*C*′_{4ℓ′+4}) as a basis to represent the action induced on

Here we have used the fact that *C*_{4ℓ+4} and the strict transform of {*x*=*b*} are, respectively, linearly equivalent to the curves
*λ*=1, but its derivative *a*_{n}*b*_{n+3}=*a*_{n+2}*b*_{n+2},*b*_{n−1}*a*_{n+2}=*a*_{n}*b*_{n} (cf. (3.8) at ℓ=ℓ′=0). It is obtained after eight blow-ups and it corresponds, again, to the first opportunity to regularize the mapping for this specific blow-up pattern. Hence, the notion of first confinement is indeed equivalent to regularizing the mapping at the first opportunity, for a specific blow-up pattern.

### (b) The conditions on the parameters

Examining the different blow-up patterns, or, equivalently, conditions (3.5) and (3.8) on *a*_{n} and *b*_{n} for each blow-up pattern, it is clear that the effect of the duality (*x*_{n},*y*_{n})→(*b*_{n}/*x*_{n},*b*_{n}/*y*_{n}), (*a*_{n},*b*_{n})→(*b*_{n+1}*b*_{n−1}/(*a*_{n}*b*_{n}),*b*_{n}) on mapping (3.1) is to interchange the roles played by ℓ and ℓ′. Hence, in the following, we can without loss of generality suppose that ℓ′≥ℓ≥0.

For the first blow-up pattern, rewriting condition (3.5) on *a*_{n} and *b*_{n} in terms of their logarithms *A*_{n}≡0 (i.e. *a*_{n}=1 for all *n*) and the difference equation for the *B*'s corresponds exactly to the matrix *Φ* in (3.7) (which in this case only contains one special column—the last one—and is in Frobenius normal form):
*Φ*. Using relation (3.11), one must therefore systematically reduce the number of variables involved in equation (3.12), which ultimately yields:
*q* and *r* are the quotient and remainder when dividing 2ℓ′+1 by 2ℓ+1: 2ℓ′+1=*q*(2ℓ+1)+*r*,*r*<2ℓ+1. Systems (3.11) and (3.13) can be expressed as
*M* of size 4ℓ′+4ℓ+2, i.e. of the same size as *Φ*. Brute force calculations of the first 400 or so cases (up to ℓ=20,ℓ′=20) show that, in all cases, the matrices *M* have the same Frobenius normal form as *Φ* and are therefore similar to *Φ*.

The case of the second blow-up pattern, giving rise to the map *B*′_{n}=*B*_{n}−*A*_{n−1} the two matrices do coincide. Moreover, this linear transformation on the parameters leads to a conservation law expressing the invariance of the quantity *B*′_{n+1}−*A*_{n−1} under shifts in *n*, accompanied by the equation *A*_{n+1}=*A*_{n−1}+*A*_{n−2}−*A*_{n−4}. In general, however, the sizes of the relevant matrices do differ and the number of variables needs to be reduced. This can be done using relation (3.14), leading to (ℓ′≥ℓ≥0)
*q* and *r* are now the quotient and remainder when dividing ℓ′+1 by ℓ+1: ℓ′+1=*q*(ℓ+1)+*r*,*r*<ℓ+1. Systems (3.14) and (3.16) can be expressed as
*Φ*, at least up to ℓ=ℓ′=20. We therefore conjecture that in all cases, confounding all blow-up patterns for all values of ℓ and ℓ′, the matrix dictating the behaviour of the parameters in the mapping will always be similar to that for the linear map that governs the behaviour of the −1 curves in the basis of the Picard group for the blown-up surface. For the moment, however, a proof of this remarkable property still eludes us.

## 4. Conclusion

In this paper, we set out to study the validity of a deautonomization scheme that relies on the singularity confinement integrability criterion, as a way of deriving non-autonomous integrable mappings (and discrete Painlevé equations in particular). Our analysis was based on the regularization of non-autonomous extensions of integrable mappings through blow-up. The use of such algebro-geometric techniques allowed us to draw conclusions of fairly general validity.

The standard practice in deautonomization through singularity confinement has always been to confine at the very first opportunity. This ansatz is corroborated by the present study. As every possible ‘late’ confinement inevitably leads to a non-integrable system, if one aims at obtaining an integrable deautonomization, it is now clear that the singularity pattern of the non-autonomous system must be identical to that of the autonomous one. This, furthermore, corroborates another standard practice when deautonomizing integrable mappings—that of requiring that for integrable deautonomizations the degree growth of the deautonomized mapping be the same as that of its autonomous integrable counterpart. Indeed, as the degree growth is completely determined by the intersections of the curves on the blown-up (family of) surfaces [11], it is clear that since the regularized autonomous and non-autonomous mappings necessarily correspond to the same types of surfaces, their respective degree growths must also coincide.

The non-integrability of deautonomizations obtained from late confinement was illustrated on some selected examples from the family of discrete Painlevé equations (and was, in fact, confirmed by many more examples that could not be presented here). As explained in §3, there exist situations where the choice of a special value for one of the parameters in the mapping drastically modifies the singularity pattern. We showed that this is reflected in the blow-up pattern, as the parameters of the mapping invariably appear in the base points for the blow-ups. The interesting result here is that there exists a perfect parallel between the regularization procedure and singularity confinement: the map on the exceptional curves resulting from the blow-ups furnishes the confinement condition, even for non-integrable systems as was shown in the case of late confinement. We conjecture that this correspondence will be true in general, which then leads us to an intriguing possibility. If the eigenvalues of the matrix that governs the confinement constraints (on the parameters) are indeed identical to those of the map on the exceptional part of the basis of the Picard group (obtained through blow-up) for a specific mapping, then the integrability or non-integrability of the deautonomization at hand can be read off from the confinement constraints directly. Of course, the examples treated here are quite special in that they did not contain any superfluous parameters (e.g. parameters that might be gauged out), but even the presence of such parameters would not fundamentally change our conclusion. The constraints, obtained from singularity confinement, on a more general set of parameters could, conceivably, correspond to a linear transformation on a larger part of the Picard group than that given just by the exceptional curves. However, as the only additional eigenvalues that can appear necessarily belong to the unitary part of the transformation on the Picard group, the correspondence between the respective largest eigenvalues remains unchanged. We claim that this correspondence is even true for deautonomizations of non-QRT mappings, which will be the subject of a forthcoming paper.

## Authors' contributions

T.M., R.W., B.G. and A.R. jointly conceived the idea for this paper and derived and interpreted the mathematical results contained in it.

## Competing interests

The authors declare they have no competing interests.

## Funding

T.M. acknowledges support from the Japan Society for the Promotion of Science (JSPS) through the Grant-in-Aid for Scientific Research 25-3088. R.W. also acknowledges support from JSPS, through grant no. KAKENHI 24540204.

## Acknowledgements

T.M. and A.R. would like to thank the Graduate School of Mathematical Sciences of the University of Tokyo for support extended through its Program for Leading Graduate Schools, MEXT, Japan.

- Received December 11, 2014.
- Accepted October 15, 2015.

- © 2015 The Author(s)