More Like this

1. 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. 

   more » « less
2. 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. 

   more » « less
3. 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. 

   more » « less
4. Finiteness Properties of Affine Difference Algebraic Groups

   https://doi.org/10.1093/imrn/rnaa177  

   Wibmer, Michael (July 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We establish several finiteness properties of groups defined by algebraic difference equations. One of our main results is that a subgroup of the general linear group defined by possibly infinitely many algebraic difference equations in the matrix entries can indeed be defined by finitely many such equations. As an application, we show that the difference ideal of all difference algebraic relations among the solutions of a linear differential equation is finitely generated. 

   more » « less
5. Exploring AC Power Flow Resultants through the Lens of Multi-homogeneous Algebraic Geometry

   https://doi.org/10.1109/KPEC58008.2023.10215389  

   Barati, Masoud (April 2023, IEEE)

   The computational aspects of power systems have been exploring the steady states of AC power flow for several decades. In this paper, we propose a novel approach to AC power flow calculation using resultants and discriminants for polynomials, which are primarily compiled for quadratic power flow equations. In the case of AC power flow nonlinear systems, it is not possible to determine the number of isolated solutions. However, for polynomial systems, the theorem of Bézout is the primary theorem of algebraic geometry. This study considers a certain multi-homogeneous structure in an algebraic geometry system to demonstrate that the theorem of Bézout is indeed a generalization of the fundamental theorem, among other results. 

   more » « less