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1. Frobenius powers

   https://doi.org/10.1007/s00209-019-02442-2  

   Hernández, Daniel J.; Teixeira, Pedro; Witt, Emily E. (January 2020, Mathematische Zeitschrift)

   null (Ed.)

   This article extends the notion of a Frobenius power of an ideal in prime characteristic to allow arbitrary nonnegative real exponents. These generalized Frobenius powers are closely related to test ideals in prime characteristic, and multiplier ideals over fields of characteristic zero. For instance, like these well-known families of ideals, Frobenius powers also give rise to jumping exponents that we call critical Frobenius exponents. In fact, the Frobenius powers of a principal ideal coincide with its test ideals, but Frobenius powers appear to be a more refined measure of singularities than test ideals in general. Herein, we develop the theory of Frobenius powers in regular domains, and apply it to study singularities, especially those of generic hypersurfaces. These applications illustrate one way in which multiplier ideals behave more like Frobenius powers than like test ideals. 

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2. Orthogonal Dirichlet Polynomials

   https://doi.org/10.1007/978-3-030-84122-5_30  

   Lubinsky, D. (January 2022, Approximation and Computation in Science and Engineering)

   Daras, N.; Rassias, T. (Ed.)

   Abstract. Let fj g1 j=1 be a sequence of distinct positive numbers. Let w be a nonnegative function, integrable on the real line. One can form orthogonal Dirichlet polynomials fng from linear combinations of n 

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3. Products and powers of principal symmetric ideals

   https://doi.org/10.1142/S0219498825503207  

   Dannetun, Eric; Fang, Bruce; Formenti, Riccardo; Gao, Bo Y; Geraci, Juliann; Kogel, Ross; Li, Yuelin; Mandal, Shreya; Rupasinghe, Vinuge; Seceleanu, Alexandra; et al (July 2024, Journal of Algebra and Its Applications)

   Principal symmetric ideals were recently introduced by Harada et al. in [The minimal free resolution of a general principal symmetric ideal, preprint (2023), arXiv:2308.03141], where their homological properties are elucidated. They are ideals generated by the orbit of a single polynomial under permutations of variables in a polynomial ring. In this paper, we determine when a product of two principal symmetric ideals is principal symmetric and when the powers of a principal symmetric ideal are again principal symmetric ideals. We characterize the ideals that have the latter property as being generated by polynomials invariant up to a scalar multiple under permutation of variables. Recognizing principal symmetric ideals is an open question for the purpose of which we produce certain obstructions. We also demonstrate that the Hilbert functions of symmetric monomial ideals are not all given by symmetric monomial ideals, in contrast to the non-symmetric case. 

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4. F‐purity of binomial edge ideals

   https://doi.org/10.1112/jlms.70474  

   LaClair, Adam; McCullough, Jason (February 2026, Journal of the London Mathematical Society)

   Abstract In 2012, Matsuda introduced the class of weakly closed graphs and investigated when binomial edge ideals are F‐pure. He proved that weakly closed binomial edge ideals are F‐pure whenever the base field has positive characteristic. He conjectured that: (i) when the base field has characteristic 2, every F‐pure binomial edge ideal comes from a weakly closed graph; and (ii) that every binomial edge ideal is F‐pure provided that the characteristic of the residue field is sufficiently large. In this paper, we resolve both of Matsuda's conjectures. We confirm Matsuda's first conjecture, showing that the binomial edge ideal of a graph defines an F‐pure quotient in characteristic 2 if and only if the graph is weakly closed. We also show that Matsuda's second conjecture is false in a very strong way by showing that graphs containing asteroidal triples, such as the net, define non‐F‐pure binomial edge ideals in any positive characteristic. Our results yield a complete classification of F‐pure binomial edge ideals of chordal graphs as well as large families of standard graded algebras that are F‐injective but neither F‐pure nor F‐rational in all characteristics. 

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5. Algorithms for computing greatest common divisors of parametric multivariate polynomials

   Kapur, Deepak; Lu, Dong; Monagan, Micheal; Sun, Yao; Wang, Dingkang (January 2021, Journal of symbolic computation)

   Two new efficient algorithms for computing greatest common divisors (gcds) of parametric multivariate polynomials over k[U][X]are presented. The key idea of the first algorithm is that the gcd of two non-parametric multivariate polynomials can be obtained by dividing their product by the generator of the intersection of two principal ideals generated by the polynomials. The second algorithm is based on another simple insight that the gcd can be extracted using the generator of the ideal quotient of a polynomial with respect to the second polynomial. Since the ideal intersection and ideal quotient in these cases are also principal ideals, their generators can be obtained by computing minimal Gröbner bases of the ideal intersection and ideal quotient, respectively. To avoid introducing new variables which can adversely affect the efficiency, minimal Gröbner bases computations are performed on modules. Both of these constructions generalize to the parametric case as shown in the paper. Comprehensive Gröbner system constructions are used for the parametric ideal intersection and ideal quotient using the Kapur-Sun-Wang’s algorithm. It is proved that whether in a minimal comprehensive Gröbner system of a parametric ideal 20intersection or in that of a parametric ideal quotient, each branch of the specializations corresponds to a principal parametric ideal with a single generator. Using this generator, the parametric gcd of that branch is obtained by division. For the case of more than two parametric polynomials, we can use the above two algorithms to compute gcds recursively, and get an extended algorithm by generalizing the idea of the second algorithm. Algorithms do not suffer from having to apply expensive steps such as ensuring whether parametric polynomials are primitive w.r.t. the main variable as used in both the algorithms proposed by Nagasaka (ISSAC, 2017). The resulting algorithms are not only conceptually simple to understand but are more efficient in practice. The proposed algorithms and both of Nagasaka’s algorithms have been implemented in Singular, and their performance is compared on a number of examples. 

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