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1. Polynomial Parametrization for SL2 over Quadratic Number Rings

   https://doi.org/10.1093/imrn/rnz046  

   Larsen, Michael; Nguyen, Dong Quan (March 2019, International Mathematics Research Notices)

   null (Ed.)

   Abstract If $$R$$ is the ring of integers of a number field, then there exists a polynomial parametrization of the set $$\operatorname{SL}_2(R)$$, that is, an element $$A\in{\textrm{SL}}_2(\mathbb{Z}[x_1,\ldots ,x_n])$$ such that every element of $$\operatorname{SL}_2(R)$$ is obtained by specializing $$A$$ via some homomorphism $$\mathbb{Z}[x_1,\ldots ,x_n]\to R$$. 

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2. Differential polynomial rings in several variables over locally nilpotent rings

   https://doi.org/10.1142/S0218196719500668  

   Chen, Fei Yu; Hagan, Hannah; Wang, Allison (February 2020, International Journal of Algebra and Computation)

   We show that a differential polynomial ring over a locally nilpotent ring in several commuting variables is Behrens radical, extending a result by Chebotar. 

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3. Outlier robust multivariate polynomial regression

   Arora, Vipul; Bhattacharyya, Arnab; Boban, Mathews; Guruswami, Venkatesan; Kelman, Esty (September 0024, Proceedings of the European Symposium on Algorithms (ESA),)

   We study the problem of robust multivariate polynomial regression: let p\colon\mathbb{R}^n\to\mathbb{R} be an unknown n-variate polynomial of degree at most d in each variable. We are given as input a set of random samples (\mathbf{x}_i,y_i) \in [-1,1]^n \times \mathbb{R} that are noisy versions of (\mathbf{x}_i,p(\mathbf{x}_i)). More precisely, each \mathbf{x}_i is sampled independently from some distribution \chi on [-1,1]^n, and for each i independently, y_i is arbitrary (i.e., an outlier) with probability at most \rho < 1/2, and otherwise satisfies |y_i-p(\mathbf{x}_i)|\leq\sigma. The goal is to output a polynomial \hat{p}, of degree at most d in each variable, within an \ell_\infty-distance of at most O(\sigma) from p. Kane, Karmalkar, and Price [FOCS'17] solved this problem for n=1. We generalize their results to the n-variate setting, showing an algorithm that achieves a sample complexity of O_n(d^n\log d), where the hidden constant depends on n, if \chi is the n-dimensional Chebyshev distribution. The sample complexity is O_n(d^{2n}\log d), if the samples are drawn from the uniform distribution instead. The approximation error is guaranteed to be at most O(\sigma), and the run-time depends on \log(1/\sigma). In the setting where each \mathbf{x}_i and y_i are known up to N bits of precision, the run-time's dependence on N is linear. We also show that our sample complexities are optimal in terms of d^n. Furthermore, we show that it is possible to have the run-time be independent of 1/\sigma, at the cost of a higher sample complexity. 

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4. Generalized Splines on Graphs with Two Labels and Polynomial Splines on Cycles

   https://doi.org/10.37236/11155  

   Anderson, Portia; Matherne, Jacob P; Tymoczko, Julianna (January 2024, The Electronic Journal of Combinatorics)

   Generalized splines are an algebraic combinatorial framework that generalizes and unifies various established concepts across different fields, most notably the classical notion of splines and the topological notion of GKM theory. The former consists of piecewise polynomials on a combinatorial geometric object like a polytope, whose polynomial pieces agree to a specified degree of differentiability. The latter is a graph-theoretic construction of torus-equivariant cohomology that Shareshian and Wachs used to reformulate the well-known Stanley-Stembridge conjecture, a reformulation that was recently proven to hold by Brosnan and Chow and independently Guay-Paquet. This paper focuses on the theory of generalized splines. A generalized spline on a graph $$G$$ with each edge labeled by an ideal in a ring $$R$$ consists of a vertex-labeling by elements of $$R$$ so that the labels on adjacent vertices $u, v$ differ by an element of the ideal associated to the edge $uv$. We study the $$R$$-module of generalized splines and produce minimum generating sets for several families of graphs and edge-labelings: $1)$ for all graphs when the set of possible edge-labelings consists of at most two finitely-generated ideals, and $2)$ for cycles when the set of possible edge-labelings consists of principal ideals generated by elements of the form $(ax+by)^2$ in the polynomial ring $$\mathbb{C}[x,y]$$. We obtain the generators using a constructive algorithm that is suitable for computer implementation and give several applications, including contextualizing several results in the theory of classical (analytic) splines. 

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5. Symmetric group fixed quotients of polynomial rings

   https://doi.org/10.1016/j.jpaa.2023.107537  

   Pevzner, Alexandra (April 2024, Journal of Pure and Applied Algebra)

   Given a representation of a finite group G over some commutative base ring k, the cofixed space is the largest quotient of the representation on which the group acts trivially. If G acts by k-algebra automorphisms, then the cofixed space is a module over the ring of G-invariants. When the order of G is not invertible in the base ring, little is known about this module structure. We study the cofixed space in the case that G is the symmetric group on n letters acting on a polynomial ring by permuting its variables. When k has characteristic 0, the cofixed space is isomorphic to an ideal of the ring of symmetric polynomials in n variables. Localizing k at a prime integer p while letting n vary reveals striking behavior in these ideals. As n grows, the ideals stay stable in a sense, then jump in complexity each time n reaches a multiple of p. 

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