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Motivated by the search for methods to establish strong minimality of certain low order algebraic differential equations, a measure of how far a finite rank stationary type is from being minimal is introduced and studied: Thedegree of nonminimalityis the minimum number of realisations of the type required to witness a nonalgebraic forking extension. Conditional on the truth of a conjecture of Borovik and Cherlin on the generic multiple-transitivity of homogeneous spaces definable in the stable theory being considered, it is shown that the nonminimality degree is bounded by theU-rank plus 2. The Borovik–Cherlin conjecture itself is verified for algebraic and meromorphic group actions, and a bound ofU-rank plus 1 is then deduced unconditionally for differentially closed fields and compact complex manifolds. An application is given regarding transcendence of solutions to algebraic differential equations.more » « lessFree, publicly-accessible full text available February 5, 2026
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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
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Kráľovič, Rastislav; Kučera, Antonín (Ed.)In this paper we give several applications of Littlestone dimension. The first is to the model of [Angluin and Dohrn, 2017], where we extend their results for learning by equivalence queries with random counterexamples. Second, we extend that model to infinite concept classes with an additional source of randomness. Third, we give improved results on the relationship of Littlestone dimension to classes with extended d-compression schemes, proving the analog of a conjecture of [Floyd and Warmuth, 1995] for Littlestone dimension.more » « less
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In this paper, it is shown that if p is a complete type of Lascar rank at least 2, in the theory of differentially closed fields of characteristic zero, then there exists a pair of realisations a, b such that p has a nonalgebraic forking extension over a, b. Moreover, if A is contained in the field of constants then p already has a nonalgebraic forking extension over a. The results are also formulated in a more general setting.more » « less
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In this paper we develop a new technique for showing that a nonlinear algebraic differential equation is strongly minimal based on the recently developed notion of the degree of non-minimality of Freitag and Moosa. Our techniques are sufficient to show that generic order $$h$$ differential equations with non-constant coefficients are strongly minimal, answering a question of Poizat (1980).more » « less
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Abstract We study the structure of the solution sets in universal differential fields of certain differential equations of order two, the Poizat equations, which are particular cases of Liénard equations. We give a necessary and sufficient condition for strong minimality for equations in this class and a complete classification of the algebraic relations for solutions of strongly minimal Poizat equations. We also give an analysis of the non-strongly minimal cases as well as applications concerning the Liouvillian and Pfaffian solutions of some Liénard equations.more » « less
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Bahoo, Yeganeh; Georgiou, Konstantinos (Ed.)In this work we consider the Steiner tree problem under Bilu-Linial stability. We give strong geometric struc- tural properties that need to be satisfied by stable in- stances. We then make use of, and strengthen, these geometric properties to show that 1.59-stable instances of Euclidean Steiner trees are polynomial-time solvable by showing it reduces to the minimum spanning tree problem. We also provide a connection between certain approximation algorithms and Bilu-Linial stability for Steiner trees.more » « less
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