Discrete Painlev\'e equations from pencils of quadrics in $\mathbb P^3$ with branching generators

Jaume Alonso; Yuri B. Suris

REVIEW 4 major objections 5 minor 21 references

Discrete Painlev\'e equations from pencils of quadrics in $\mathbb P^3$ with branching generators

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A single geometric construction—a pencil of quadrics in $\mathbb P^3$ with a branched double cover—produces every discrete Painlevé equation except the elliptic one.

desk verdict A solid, explicit extension of the pencil-of-quadrics framework that recovers all non-elliptic discrete Painlevé equations; the main open point is a formal proof of the singularity-confinement pattern, and Section 6 needs expanding. read the letter →

arxiv 2506.02275 v1 pith:CVMMULYO submitted 2025-06-02 math-ph math.DSmath.MPnlin.SI

Jaume Alonso , Yuri B. Suris This is my paper

classification math-phmath.DSmath.MPnlin.SI MSC 39A4514J26

keywords discretePainlevéequationspencilsofquadrics3DQRTmapssingularityconfinementdeformationmapbrancheddoublecoverd-Painlevéq-Painlevé

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to show that nearly all discrete Painlevé equations—every case except the elliptic one—can be generated by a single geometric construction: a pencil of quadrics in $\mathbb P^3$ with a branched double cover. The authors define a deformation map that moves from one quadric of the pencil to another by a translation on the universal cover of the Riemann surface of $\sqrt{\Delta(\lambda)}$, then compose it with two 3D QRT involutions. Reduced to pencil-adapted coordinates, the resulting 3D Painlevé maps reproduce the standard $d$-Painlevé and $q$-Painlevé equations of surface types $A_1^{(1)}$, $D_4^{(1)}$, and $A_0^{(1)}$. If the construction is correct, the eight-point-blowup classification of discrete Painlevé equations can be replaced, for all non-elliptic cases, by the classical classification of pencils of quadrics in $\mathbb P^3$.

What carries the argument

The machinery rests on the pencil $\{Q_\lambda\}$ of quadrics in $\mathbb P^3$ and its characteristic polynomial $\Delta(\lambda)=\det(M_0-\lambda M_\infty)$. For the six non-complete-square pencil types, the generators of $Q_\lambda$ are rational in the point and in $\sqrt{\Delta(\lambda)}$, so the construction passes to the branched double cover $\mathcal X$ of $\mathbb P^3$ ramified at the singular quadrics. The Painlevé deformation map $L$ of Theorem 1 is the explicit recipe (9)/(10): if $Q_\infty$ does not depend on $X_3$, then $\widehat X_1=X_1X_4$, $\widehat X_2=X_2X_4$, $\widehat X_3=X_3X_4-(\lambda(\widehat\nu)-\lambda(\nu))Q_\infty(X)$, $\widehat X_4=X_4^2$, and similarly in the other case; it sends $Q_{\lambda(\nu)}$ to $Q_{\lambda(\nu+2\delta)}$ and fixes the base curve. Pencil-adapted coordinates $(x,y,\nu)$ are the parametrization $\phi_\nu:\mathbb P^1\times\mathbb P^1\to Q_{\lambda(\nu)}$ for which the generator families are $x=\text{const}$ and $y=\text{const}$; these coordinates carry the comparison with the literature equations. The 3D Painlevé map $\widetilde F=R_1\circ i_1\circ L_1\circ R_2\circ i_2\circ L_2$ combines the 3D QRT involutions $i_1,i_2$ with half-step factors $L_1,R_2,L_2,R_1$, each shifting $\nu$ by $\delta$. The singularity-confinement mechanism is the pattern (13): under $i_1$ the ruled surface $\Phi_i$ collapses to the base point $S_i$, and under $i_2$ that point is blown up to the ruled surface $\Psi_i$.

What would settle it

Take a pencil of type (iv) with generic parameters satisfying the closure condition and a nonzero $q$; compute, in homogeneous coordinates, the image under the undeformed 3D QRT map $i_1\circ i_2$ of a generic point on one of the ruled surfaces $\Phi_i$. If the image is not contained in the corresponding ruled surface $\Psi_i$, the confinement pattern (13) fails and Theorem 2 cannot apply; equivalently, iterate $\widetilde F$ numerically from initial data close to $x=a_i(\nu)$ and check whether the orbit returns to $y=b_i(\nu)$ after two steps.

Watch

Extended reading notes

Core claim

The central discovery is that when the characteristic polynomial $\Delta(\lambda)$ of a pencil of quadrics is not a complete square, the generators through a point are rational in the point and in $\sqrt{\Delta(\lambda)}$, so the natural space for the geometry is a branched double cover $\mathcal X$ of $\mathbb P^3$. On this cover the Painlevé deformation map $L$ acts as a translation $\nu\mapsto\nu+2\delta$ in the uniformizing coordinate, not as a Möbius transformation of the pencil parameter. Theorem 1 gives an explicit coordinate formula for $L$: it preserves the pencil, sends $Q_{\lambda(\nu)}$ to $Q_{\lambda(\nu+2\delta)}$, and fixes the base curve pointwise. Composing $L$ with the two QRT-type involutions $i_1,i_2$ produces a 3D Painlevé map $\widetilde F$; for pencils of types (ii)–(vi) the paper verifies the singularity-confinement pattern (13) needed in Theorem 2, and in Sections 5–9 the reduction of $\widetilde F$ to pencil-adapted coordinates coincides exactly with the $d$-Painlevé and $q$-Painlevé equations of surface types $A_1^{(1)}$, $D_4^{(1)}$, and $A_0^{(1)}$ as given in [10] and [19]. The generic type (i), which would give the elliptic Painlevé equation, is explicitly left to future work.

Load-bearing premise

The whole construction rests on a pattern that the paper checks case by case rather than proves: that after two steps of the basic involution, the singular lines collapse to one of the eight base points and then expand to another line. If this pattern fails for any pencil of types (ii)–(vi), the deformed map may stop being a discrete Painlevé equation.

Editorial extensions

If this is right

The shift $\delta$ (or $q=e^{\delta}$ in the $q$-difference cases) is a free parameter of the deformation map, not a datum of the eight-point configuration, so one pencil generates a family of discrete Painlevé equations with arbitrary time step.
The eight base points $S_i$ are fixed in homogeneous coordinates by $L$; their apparent motion under iteration is an artifact of the pencil-adapted coordinates, so the time evolution of the distinguished points is not intrinsic.
The singular quadrics are exactly the branch points of the cover, and $L$ traverses the pencil by a translation on the universal cover, so the discrete time step is tied to the geometry of the Riemann surface of $\sqrt{\Delta(\lambda)}$.
In the symmetric cases the 3D map squares to a deformed QRT root, yielding scalar second-order non-autonomous equations, so the scheme includes the standard one-field formulation of these discrete Painlevé equations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves the isomonodromic interpretation open; if pursued, the translation $\nu\mapsto\nu+2\delta$ would most naturally be a monodromy-preserving shift on the spectral curve attached to the pencil, with the branch points $\lambda_i$ as the singular loci of monodromy data.
Applied to the remaining generic type (i), where the universal cover is a torus, the same deformation recipe should produce the elliptic Painlevé equation; this would test whether the singularity pattern (13) has a torus analogue.
Because the deformed base points $R_1(S_i)$ do not support a net of quadrics, the construction suggests that singularity confinement here does not force the eight points to evolve within a fixed configuration space; higher-dimensional analogues may exist where no eight-point-blowup description is available.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: this is a real extension of the authors' earlier pencil-of-quadrics framework to the cases where the characteristic polynomial is not a complete square. The new ingredient is a branched double cover of P3 and a deformation map that acts as a translation on the universal cover of the Riemann surface of sqrt(Δ(λ)). Applied to pencils of types (ii)-(vi), the scheme reproduces the d- and q-Painlevé equations of surface types A1(1), D4(1), and A0(1), exactly as in Kajiwara–Noumi–Yamada and Yamada. That is a genuine organizational result: it gives a different geometric route to the same equations, with a free shift parameter δ (or q) rather than a fixed configuration.

What the paper does well: the constructions are explicit. Theorems 3–5 and 7 give concrete deformation maps in homogeneous and pencil-adapted coordinates, and the reductions to known equations are direct and checkable. The match with the literature is convincing, and the paper is honest that the generic type (i) (elliptic) is left for future work. The reliance on their earlier classification [2] is acknowledged and appropriate.

Where I have reservations. The first is the one flagged in the stress-test: Theorem 2 rests on the singularity-confinement pattern (13) for the undeformed 3D QRT involutions, and that pattern is asserted from case-by-case computation rather than proved. I don't think this is fatal—the pattern is standard for QRT maps, and the later sections verify it implicitly by producing the correct equations—but a referee should ask for either a general argument or a complete check for each of the five pencil types. The type (vi) case in Section 6 is especially thin: it is presented as a limiting procedure with details deliberately omitted, and the D4(1) identification is asserted rather than demonstrated. That is the weakest section. Second, Theorem 6 contains an obvious typo: the deformation map uses β where the computation requires δ, and the text then uses δ in the same formula. Minor but needs correcting. The paper also has a slightly compressed style in a few places, though not enough to obscure the main thread.

Bottom line: this is a solid paper that deserves a serious referee. It is aimed at specialists in discrete integrable systems and algebraic geometry. I would send it to a journal with a request to expand Section 6 and fix the typo, and to address the singularity-confinement pattern more rigorously. It is not a desk reject.

Referee Report

4 major / 5 minor

Summary. The paper extends the authors' earlier program for constructing discrete Painlev\'e equations from pencils of quadrics in P^3 to the technically harder case where the characteristic polynomial Delta(lambda) is not a complete square. The construction starts from a pencil {Q_lambda} of quadrics and a second pencil {P_mu} sharing one quadric, lifts the 2D QRT construction to a 3D setting on the branched double cover of P^3, and defines a Painlev\'e deformation map L that moves between fibers Q_{lambda(nu)} and Q_{lambda(nu+2delta)}, where nu is the uniformizing variable on the Riemann surface of sqrt(Delta(lambda)). A 3D Painlev\'e map eF is then formed by composing L with the two 3D QRT involutions. Sections 5--9 work out the explicit coordinate formulas for pencil types (v), (vi), (iv), (iii), and (ii), and identify the resulting non-autonomous systems with the d-Painlev\'e and q-Painlev\'e equations of surface types A_1^(1), D_4^(1), and A_0^(1) as given in references [10] and [19]. The generic pencil type (i), which is claimed to correspond to the elliptic Painlev\'e equation, is deferred to a separate publication.

Significance. If the construction is fully justified, this gives a uniform geometric explanation of discrete Painlev\'e equations in terms of pencils of quadrics, complementing Sakai's classification by generalized Halphen surfaces. The paper contains explicit, checkable coordinate computations for several nontrivial cases, and the treatment of the branched double cover is a natural and original step beyond the preceding work [2]. The main conceptual contribution is the deformation map L: it fixes the base curve pointwise and implements the non-autonomous shift by a translation on the universal cover of the Riemann surface. The significance is currently conditional: the central singularity-confinement hypothesis is not proved, the type (vi) derivation is omitted, and one theorem contains an inconsistent uniformization formula. These are fixable gaps rather than fundamental flaws, so a major revision is appropriate.

major comments (4)

[Section 4, Eq. (13)] The singularity confinement pattern (13) is the key hypothesis of Theorem 2 and therefore underpins the entire construction of the 3D Painlev\'e maps, but it is only asserted as the result of 'case-by-case computations' with no computations shown. Since the QRT involutions for each of the pencil types are written explicitly in Sections 5--9, the authors should provide a verification of (13) for each type, or a general argument deriving it from those formulas. Without this, the reader cannot confirm that the deformed maps ei1 and ei2 have the confinement structure (15)--(16) and hence that eF is a discrete Painlev\'e map.
[Section 8, Eq. (105) and Theorem 6] The uniformization is stated as lambda(nu) = nu^2 - 1/4 with sqrt(Delta) = nu, but this is inconsistent: if lambda = nu^2 - 1/4, then Delta = 1 + 4 lambda = 4 nu^2, whose square root is not nu. The parametrization in (108) and the quadric equation (103) require lambda(nu) = (nu^2 - 1)/4. With the printed formula, the map L given in Theorem 6 does not preserve the pencil; with the corrected formula it does, and the factor beta in the expression for bX3 should be delta (or should be defined explicitly). This is not a purely cosmetic typo, since the deformation map is the central object of the section.
[Section 6] The derivation of the D_4^(1) system (61)--(62) is not actually given. The text says 'By a simple limiting procedure, the results of the previous section lead to similar results' and later 'Computing the 3D Painlev\'e map ... we come to the following non-autonomous system', but the limiting procedure and the intermediate steps are omitted. Since Section 6 is one of the five main derivations and is needed for the claim of recovering D_4^(1), the authors should supply the computation or at least a detailed outline of the limiting argument.
[Section 4, Theorem 2] The proof of the singularity confinement pattern (15)--(16) for the deformed maps is only sketched. In particular, the assertion that ei2 maps R1(Si) to R2(Psi_i) requires checking that L2(R1(Si)) = Si, and the assertion about the inverse image L1^{-1}(Phi_i) requires a short argument that L1 is invertible on the relevant locus. Please include these steps so that Theorem 2 is self-contained.

minor comments (5)

[Theorem 1 proof] The word 'Futher' should be 'Further'.
[Section 7, Eq. (78)] The second expression for y has a suspicious denominator: it reads kappa(1 - kappa w) X4 - (kappa - w) X4, which simplifies to w(1 - kappa^2) X4 and would give y = X2/X4. This is likely a typo; the intended denominator may be kappa(1 - kappa w) X4 - (kappa - w) X3.
[Section 8, Theorem 6] The symbol beta is used without prior definition. In the corrected version with lambda(nu) = (nu^2 - 1)/4, the factor beta should be delta; please state this explicitly.
[Sections 5--9] The identifications of the derived systems with the standard equations in [10] and [19] are asserted without displaying the corresponding standard forms. A short table or explicit matching of variables and parameters for each of the five cases would make the paper considerably easier to verify.
[Introduction] The sentence 'We recover, within our novel framework, all discrete Painlev\'e equations except for the elliptic one' is stronger than what is demonstrated in this paper alone, since the complete-square cases from [2] are referenced rather than re-derived here. Please clarify that the statement refers to the combined program of [2] and the present paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction derives known discrete Painlevé equations from a new pencil-of-quadrics scheme, with known equations entering only as a final comparison.

full rationale

The paper's derivation chain is self-contained rather than circular. The 3D Painlevé map eF is built from explicit geometric ingredients: a pencil of quadrics {Qλ}, the 3D QRT involutions i1,i2, and the deformation map L of Theorem 1, with the shift δ (or q) left free. The known d-Painlevé and q-Painlevé equations appear only at the end of Sections 5, 7, 8, and 9, where the derived non-autonomous systems are identified with equations 'as given in [10]' or '[19], [10]'. These target equations are not used as inputs to fix the construction. Similarly, the singularity confinement pattern (13) is stated as an unproved case-by-case observation; although this is the paper's weakest assumption and a possible correctness gap, it is not equivalent to the target equations and does not make the derivation circular. The citations to the authors' prior work [1] and [2] supply the 3D QRT construction and the classification of pencils of quadrics, both independent background results rather than the paper's conclusions; no load-bearing premise reduces to a self-citation. There is no fitted parameter renamed as a prediction, no self-definitional identification, and no importation of a uniqueness theorem from the authors' earlier work. Hence the paper is not circular.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The construction introduces no new entities. It uses the classical classification of pencils of quadrics, uniformization, and the singularity-confinement structure of QRT maps as inputs. The only free numbers are the shift parameter and the coordinates of the eight base points.

free parameters (3)

delta (or q=e^delta) = free complex parameter, q != +/-1
Shift parameter for the translation on the universal cover; explicitly stated in Section 1 and Theorems 3-7 to be a free parameter of the construction, not determined by the configuration.
base point parameters a_i (or c_i, z_i) = arbitrary complex numbers subject to one constraint (e.g., sum a_i = 2 for type (v))
Positions of the eight distinguished points supporting the QRT pencil; chosen by hand to realize each surface type, inherited from the 2D QRT map.
kappa (Sections 7-9) = kappa != 0,1 (or kappa1, kappa2 with constraints)
Parameter distinguishing the (1,1)-curves C_infty and the quadric Q_infty; free choice within the construction.

assumptions (4)

standard math Classification of pencils of quadrics in P3 into thirteen Segre types, including types (i)-(vi) used here.
Invoked in Section 2 to enumerate the non-square cases; relies on classical results (Reid, Casas-Alvero) and the authors' previous paper [2].
standard math Uniformization theorem: the universal cover of the Riemann surface of sqrt(Delta(lambda)) is C (or C^* for q-cases).
Used in Section 2 to justify the translation nu -> nu+2delta on the universal cover.
domain assumption The 3D QRT involutions i1, i2 restricted to each quadric Q_{lambda(nu)} act in pencil-adapted coordinates by the same formulas as the original 2D QRT involutions with points s_i replaced by s_i(nu).
Verified case-by-case in Sections 5,7,8,9; it is a computational precondition for deriving the final equations.
domain assumption Singularity confinement pattern (13) holds for the undeformed involutions.
Stated as an observation from case-by-case computations in Section 4; used as the premise of Theorem 2.

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Cite this review

Pith. "Pith review of Discrete Painlev\'e equations from pencils of quadrics in $\mathbb P^3$ with branching generators." pith.science (2026). https://pith.science/paper/CVMMULYO

@misc{pith2026250602275,
  author       = {Pith},
  title        = {Pith review of: Discrete Painlev\'e equations from pencils of quadrics in $\mathbb P^3$ with branching generators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CVMMULYO}},
  note         = {Machine review of arXiv:2506.02275}
}

abstract

In this paper we extend the novel approach to discrete Painlev\'e equations initiated in our previous work [2]. A classification scheme for discrete Painlev\'e equations proposed by Sakai interprets them as birational isomorphisms between generalized Halphen surfaces (surfaces obtained from $\mathbb P^1\times\mathbb P^1$ by blowing up at eight points). Sakai's classification is thus based on the classification of generalized Halphen surfaces. In our scheme, the family of generalized Halphen surfaces is replaced by a pencil of quadrics in $\mathbb P^3$. A discrete Painlev\'e equation is viewed as an autonomous transformation of $\mathbb P^3$ that preserves the pencil and maps each quadric of the pencil to a different one. Thus, our scheme is based on the classification of pencils of quadrics in $\mathbb P^3$. Compared to our previous work, here we consider a technically more demanding case where the characteristic polynomial $\Delta(\lambda)$ of the pencil of quadrics is not a complete square. As a consequence, traversing the pencil via a 3D Painlev\'e map corresponds to a translation on the universal cover of the Riemann surface of $\sqrt{\Delta(\lambda)}$, rather than to a M\"obius transformation of the pencil parameter $\lambda$ as in [2].

Figures

Figures reproduced from arXiv: 2506.02275 by the authors.

**Figure 1.** (a) Base set of the surface type A (1) 1 : two quadruples of points on two touching (1,1)-curves in P 1 × P 1 . (b) Pencil of quadrics through two touching non-coplanar conics The vertical involution i1 for this pencil can be described by the following equation: i1(x, y) = (x, ye), (ye+ x)(x + y) (ye+ x − 1)(x + y − 1) = Q4 i=1(x − ai) Q8 i=5(x − ai) . (20) Similarly, the horizontal involution i2 can be described by… view at source ↗

**Figure 2.** (a) Base set of the surface type D (1) 4 : four double points on a double (1,1)-curve in P 1 × P 1 . (b) Pencil of quadrics touching along a conic The characteristic polynomial of the pencil {Qλ} is: ∆(λ) = det(Mλ) = −1 + 4λ, the same as in Section 5. The normalizing transformation of Qλ(ν) to the canonical form Q0 reads:     X1 X2 X3 X4     = Aν     Y1 Y2 Y3 Y4     , (53) where Aν =   1 2ν (1 … view at source ↗

**Figure 3.** (a) Base set of the surface type A (1) 1 : two quadruples of points on two (1,1)-curves (hyperbolas) in P 1 × P 1 intersecting at two points (∞, 0) and (0,∞). (b) Pencil of quadrics through two non-coplanar conics intersecting at two points This pencil contains a reducible curve consisting of two (1,1)-curves: C∞ = (xy − 1)(xy − κ 2 ) = 0 . (67) The vertical involution i1 can be described by the following equation… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: (a) Base set of the surface type A (1) 0 : eight points on a cuspidal (2,2)-curve in P 1 × P 1 . (b) Pencil of quadrics through cuspidal spatial quartic in P 3 The vertical involution i1 can be described by the following equation: i1(x, y) = (x, ye), [PITH_FULL_IMAGE:…

**Figure 5.** Figure 5: (a) Base set of the surface type A (1) 0 : eight points on a nodal (2,2)-curve in P 1×P 1 . (b) Pencil of quadrics through a nodal spatial quartic in P 3 The vertical involution i1 can be described by the following equation: i1(x, y) = (x, ye), ye− 1 ξ − ξ κ2 y − …

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Works this paper leans on

21 extracted references · 21 canonical work pages

[2]

Discrete Painlev\'e equations and pencils of quadrics in $\mathbb P^3$

J. Alonso, Yu. B. Suris, K. Wei. Discrete Painlev´ e equations and pencils of quadrics in P3. arXiv:2403.11349 [nlin.SI]

work page Pith review arXiv
[10]

Kajiwara, M

K. Kajiwara, M. Noumi, Y. Yamada. Geometric aspects of Painlev´ e equations. J. Phys. A. Math. Theor. 50 (2017), 073001

work page 2017
[19]

Y. Yamada. A simple expression for discrete Painlev´ e equations. RIMS Kˆ okyˆ uroku Bessatsu B47 (2014), pp. 87–95

work page 2014
[1]

Alonso, Yu

J. Alonso, Yu. B. Suris, K. Wei. A three-dimensional generalization of QRT maps. J. Nonlinear Sci. 33 (2023), 117

work page 2023
[3]

M. P. Bellon, C.-M. Viallet. Algebraic entropy. Commun. Math. Phys. 204 (1999), 425– 437

work page 1999
[4]

Carstea, A

A.S. Carstea, A. Dzhamay, T. Takenawa. Fiber-dependent deautonomization of integ- rable 2D mappings and discrete Painlev´ e equations.J. Phys. A. Math. Theor. 50 (2017), 405202

work page 2017
[5]

Casas-Alvero

E. Casas-Alvero. Analytic projective geometry. EMS Textbooks in Mathematics. Z¨ urich: European Mathematical Society (2014)

work page 2014
[6]

Duistermaat

J.J. Duistermaat. Discrete Integrable Systems. QRT Maps and Elliptic Surfaces. Springer Monographs in Mathematics. Springer, New York (2010)

work page 2010

Show all 21 references

[7]

Grammaticos, A

B. Grammaticos, A. Ramani. Discrete Painlev´ e equations: a review. In: Grammaticos, B. et al. (eds.), Discrete integrable systems , Proceedings of the international CIMPA 31 school, Pondicherry, India, February 2–14, 2003. Lecture Notes in Physics 644 (2004), 245–321

2004
[8]

Grammaticos, A

B. Grammaticos, A. Ramani, V. Papageorgiou. Do integrable mappings have the Pain- lev´ e property?Phys. Rev. Lett. 67, No. 14 (1991) 1825–1828

1991
[9]

N. Joshi. Discrete Painlev´ e equations. CBMS Regional Conference Series in Mathematics
[11]

T. Mase, R. Willox, A. Ramani, and B. Grammaticos. Singularity confinement as an integrability criterion, J. Phys. A 52 (2019), 205201

2019
[12]

Noumi, S

M. Noumi, S. Tsujimoto, Y. Yamada. Pad´ e interpolation for elliptic Painlev´ e equation. In: Symmetries, Integrable Systems and Representations , Ed. K. Iohara et al, Springer Proceedings in Mathematics and Statistics 40 (2013), pp. 463–482

2013
[13]

Quispel, J.A.G

G.R.W. Quispel, J.A.G. Roberts, C.J.Thompson. Integrable mappings and soliton equa- tions. Phys. Lett. A 126, no. 7 (1988) 419–421

1988
[14]

Quispel, J.A.G

G.R.W. Quispel, J.A.G. Roberts, C.J.Thompson. Integrable mappings and soliton equa- tions II. Phys. D 34, no. 1–2 (1989) 183–192

1989
[15]

Ramani, B

A. Ramani, B. Grammaticos, J. Hietarinta. Discrete versions of the Painlev´ e equations. Phys. Rev. Lett. 67, No. 14 (1991) 1829–1832

1991
[16]

M. Reid. The complete intersection of two or more quadrics . Thesis, Cambridge, 1972. http://homepages.warwick.ac.uk/∼masda/3folds/qu.pdf

1972
[17]

H. Sakai. Rational surfaces associated with affine root systems and geometry of the Painlev´ e equations. Commun. Math. Phys. 220 (2001), 165–229

2001
[18]

Takenawa

T. Takenawa. Discrete dynamical systems associated with the configuration space of 8 points in P3(C). Commun. Math. Phys. 246, No. 1 (2004) 19–42

2004
[20]

Y. Yamada. Theory and applications of the elliptic Painlev´ e equation . In: Partition Functions and Automorphic Forms , Eds. V.A. Gritsenko and V.P. Spiridonov, Moscow Lectures 5 (2020) pp. 369–415. 32

2020
[131]

Providence, RI: American Mathematical Society (2019)

2019

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [2]

Discrete Painlev\'e equations and pencils of quadrics in $\mathbb P^3$

J. Alonso, Yu. B. Suris, K. Wei. Discrete Painlev´ e equations and pencils of quadrics in P3. arXiv:2403.11349 [nlin.SI]

work page Pith review arXiv

[2] [10]

Kajiwara, M

K. Kajiwara, M. Noumi, Y. Yamada. Geometric aspects of Painlev´ e equations. J. Phys. A. Math. Theor. 50 (2017), 073001

work page 2017

[3] [19]

Y. Yamada. A simple expression for discrete Painlev´ e equations. RIMS Kˆ okyˆ uroku Bessatsu B47 (2014), pp. 87–95

work page 2014

[4] [1]

Alonso, Yu

J. Alonso, Yu. B. Suris, K. Wei. A three-dimensional generalization of QRT maps. J. Nonlinear Sci. 33 (2023), 117

work page 2023

[5] [3]

M. P. Bellon, C.-M. Viallet. Algebraic entropy. Commun. Math. Phys. 204 (1999), 425– 437

work page 1999

[6] [4]

Carstea, A

A.S. Carstea, A. Dzhamay, T. Takenawa. Fiber-dependent deautonomization of integ- rable 2D mappings and discrete Painlev´ e equations.J. Phys. A. Math. Theor. 50 (2017), 405202

work page 2017

[7] [5]

Casas-Alvero

E. Casas-Alvero. Analytic projective geometry. EMS Textbooks in Mathematics. Z¨ urich: European Mathematical Society (2014)

work page 2014

[8] [6]

Duistermaat

J.J. Duistermaat. Discrete Integrable Systems. QRT Maps and Elliptic Surfaces. Springer Monographs in Mathematics. Springer, New York (2010)

work page 2010