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1. Algebraic relations between solutions of Painlevé equations

   https://doi.org/10.1112/jlms.70190  

   Freitag, James; Nagloo, Joel (June 2025, Journal of the London Mathematical Society)

   Abstract In this manuscript, we make major progress classifying algebraic relations between solutions of Painlevé equations. Our main contribution is to establish the algebraic independence of solutions of various pairs of equations in the Painlevé families; for generic coefficients, we show that all algebraic relations between solutions of equations in the same Painlevé family come from classically studied Bäcklund transformations. We also apply our analysis of ranks to establish some transcendence results for pairs of Painlevé equations from different families. In that area, we answer several open questions of Nagloo, and in the process answer a question of Boalch. We calculate model‐theoretic ranks of all Painlevé equations in this article, extending results of Nagloo and Pillay. We show that the type of the generic solution of any equation in the second Painlevé family is geometrically trivial, extending a result of Nagloo. We give the first model‐theoretic analysis of several special families of the third Painlevé equation, proving results analogous to Nagloo and Pillay. We also give a novel new proof of the irreducibility of the third, fifth, and sixth Painlevé equations using recent work of Freitag, Jaoui, and Moosa. Our proof is fundamentally different from the existing transcendence proofs by Watanabe, Cantat and Loray, or Casale and Weil. 

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2. Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III $$({\rm D}_7)$$ Equation

   https://doi.org/10.3842/SIGMA.2024.008  

   Buckingham, Robert J; Miller, Peter D (January 2024, Symmetry, Integrability and Geometry: Methods and Applications)

   It is well known that the Painlevé equations can formally degenerate to autonomous differential equations with elliptic function solutions in suitable scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou steepest-descent techniques to a Riemann-Hilbert representation of a family of solutions. This method leads to an explicit approximation formula in terms of theta functions and related algebro-geometric ingredients that is difficult to directly link to the expected limiting differential equation. However, the approximation arises from an outer parametrix that satisfies relatively simple conditions. By applying a method that we learned from Alexander Its, it is possible to use these simple conditions to directly obtain the limiting differential equation, bypassing the details of the algebro-geometric solution of the outer parametrix problem. In this paper, we illustrate the use of this method to relate an approximation of the algebraic solutions of the Painlevé-III (D$$_7$$) equation valid in the part of the complex plane where the poles and zeros of the solutions asymptotically reside to a form of the Weierstraß equation. 

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3. Stabbing Planes

   https://doi.org/10.48550/arXiv.1710.03219  

   Beame, Paul; Fleming, Noah; Impagliazzo, Russell; Pankratov, Denis; Pitassi, Toniann; Robere, Robert (May 2022, ArXivorg)

   We develop a new semi-algebraic proof system called Stabbing Planes which formalizes modern branch-and-cut algorithms for integer programming and is in the style of DPLL-based modern SAT solvers. As with DPLL there is only a single rule: the current polytope can be subdivided by branching on an inequality and its “integer negation.” That is, we can (non-deterministically choose) a hyperplane ax ≥ b with integer coefficients, which partitions the polytope into three pieces: the points in the polytope satisfying ax ≥ b, the points satisfying ax ≤ b, and the middle slab b − 1 < ax < b. Since the middle slab contains no integer points it can be safely discarded, and the algorithm proceeds recursively on the other two branches. Each path terminates when the current polytope is empty, which is polynomial-time checkable. Among our results, we show that Stabbing Planes can efficiently simulate the Cutting Planes proof system, and is equivalent to a tree-like variant of the R(CP) system of Krajicek [54]. As well, we show that it possesses short proofs of the canonical family of systems of F_2-linear equations known as the Tseitin formulas. Finally, we prove linear lower bounds on the rank of Stabbing Planes refutations by adapting lower bounds in communication complexity and use these bounds in order to show that Stabbing Planes proofs cannot be balanced. 

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4. Functional Transcendence of Periods and the Geometric André–Grothendieck Period Conjecture

   https://doi.org/10.1017/fms.2025.10036  

   Bakker, Benjamin; Tsimerman, Jacob (January 2025, Forum of Mathematics, Sigma)

   Abstract We prove a functional transcendence theorem for the integrals of algebraic forms in families of algebraic varieties. This allows us to prove a geometric version of André’s generalization of the Grothendieck period conjecture, which we state using the formalism of Nori motives. More precisely, we prove a version of the Ax–Schanuel conjecture for the comparison between the flat and algebraic coordinates of an arbitrary admissible graded polarizable variation of integral mixed Hodge structures. This can be seen as a generalization of the recent Ax–Schanuel theorems of [13, 18] for mixed period maps. 

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5. Bertini theorems for differential algebraic geometry

   https://doi.org/10.1090/proc/13588  

   Freitag, James (January 2024, Proceedings of the American Mathematical Society)

   We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the differential analogue of Bertini’s theorem, namely that for an arbitrary geometrically irreducible differential algebraic variety which is not an algebraic curve, generic hypersurface sections are geometrically irreducible and codimension one. Surprisingly, we prove a stronger result in the case that the order of the differential hypersurface is at least one; namely that the generic differential hypersurface sections of an irreducible differential algebraic variety are irreducible and codimension one. We also calculate the Kolchin polynomials of the intersections and prove several other results regarding intersections of differential algebraic varieties. 

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