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1. Bounding cohomology classes over semiglobal fields

   https://doi.org/10.1007/s11856-023-2549-x  

   Harbater, David; Hartmann, Julia; Krashen, Daniel (November 2023, Israel Journal of Mathematics)

   We provide a uniform bound for the index of cohomology classes over semiglobal fields (i.e., over one-variable function fields over a complete discretely valued field K). The bound is given in terms of the analogous data for the residue field of K and its finitely generated extensions of transcendence degree at most one. We also obtain analogous bounds for collections of cohomology classes. Our results provide recursive formulas for function fields over higher rank complete discretely valued fields, and explicit bounds in some cases when the information on the residue field is known. In the process, we prove a splitting result for cohomology classes of degree 3 in the context of surfaces over finite fields. ∗ 

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2. LARGE FIELDS IN DIFFERENTIAL GALOIS THEORY

   https://doi.org/10.1017/S1474748020000018  

   Bachmayr, Annette; Harbater, David; Hartmann, Julia; Pop, Florian (January 2020, Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   We solve the inverse differential Galois problem over differential fields with a large field of constants of infinite transcendence degree over  $$\mathbb{Q}$$ . More generally, we show that over such a field, every split differential embedding problem can be solved. In particular, we solve the inverse differential Galois problem and all split differential embedding problems over $$\mathbb{Q}_{p}(x)$$ . 

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3. Rank varieties and 𝜋-points for elementary supergroup schemes

   https://doi.org/10.1090/btran/74  

   Benson, Dave; Iyengar, Srikanth; Krause, Henning; Pevtsova, Julia (November 2021, Transactions of the American Mathematical Society, Series B)

   We develop a support theory for elementary supergroup schemes, over a field of positive characteristic p ⩾ 3 p\geqslant 3 , starting with a definition of a π \pi -point generalising cyclic shifted subgroups of Carlson for elementary abelian groups and π \pi -points of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra k [ t , τ ] / ( t p − τ 2 ) k[t,\tau ]/(t^p-\tau ^2) , where t t has even degree and τ \tau has odd degree. The strength of the theory is demonstrated by classifying the parity change invariant localising subcategories of the stable module category of an elementary supergroup scheme. 

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4. Generalized characteristic sets and new multivariate difference dimension polynomials

   https://doi.org/10.1007/s00200-023-00628-0  

   Levin, Alexander (October 2023, Applicable Algebra in Engineering, Communication and Computing)

   We introduce a new type of characteristic sets of difference polynomials using a generalization of the concept of effective order to the case of partial difference polynomials and a partition of the basic set of translations σ. Using properties of these characteristic sets, we prove the existence and outline a method of computation of a multivariate dimension polynomial of a finitely generated difference field extension that describes the transcendence degrees of intermediate fields obtained by adjoining transforms of the generators whose orders with respect to the components of the partition of σ are bounded by two sequences of natural numbers. We show that such dimension polynomials carry essentially more invariants (that is, characteristics of the extension that do not depend on the set of its difference generators) than previously known difference dimension polynomials. In particular, a dimension polynomial of the new type associated with a system of algebraic difference equations gives more information about the system than the classical univariate difference dimension polynomial. 

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5. Generalized Gröbner Bases and New Properties of Multivariate Difference Dimension Polynomials

   https://doi.org/10.1145/3452143.3465544  

   Levin, Alexander (July 2021, ISSAC '21: Proceedings of the 46th International Symposium on Symbolic and Algebraic Computation)

   null (Ed.)

   We present a method of Gröbner bases with respect to several term orderings and use it to obtain new results on multivariate dimension polynomials of inversive difference modules. Then we use the difference structure of the module of Kahler differentials associated with a finitely generated inversive difference field extension of a given difference transcendence degree to describe the form of a multivariate difference dimension polynomial of the extension. 

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