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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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.)
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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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Abstract We set up a general context in which one can prove Sauer–Shelah type lemmas. We apply our general results to answer a question of Bhaskar [1] and give a slight improvement to a result of Malliaris and Terry [7]. We also prove a new Sauer–Shelah type lemma in the context of $ \operatorname {\textrm{op}}$ -rank, a notion of Guingona and Hill [4].more » « less
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null (Ed.)We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory, monodromy of linear differential equations, the study of algebraic and Liouvillian solutions, differential algebraic work of Nishioka towards the Painlevé irreducibility of certain Schwarzian equations, and considerable machinery from the model theory of differentially closed fields. Our techniques allow for certain generalizations of the Ax-Lindemann-Weierstrass theorem that have interesting consequences. In particular, we apply our results to give a complete proof of an assertion of Painlevé (1895). We also answer certain cases of the André-Pink conjecture, namely, in the case of orbits of commensurators of Fuchsian groups.more » « less
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