Polynomial Ring Research Articles

The module structure of a group action on a ring

Consider a finite group G acting on a graded Noetherian k-algebra S, for some field k of characteristic p; for example S might be a polynomial ring. Regard S as a kG-module and consider the multiplicity of a particular indecomposable module as a summand in each degree. We show how this can be described in terms of homological algebra and how it is linked to the geometry of the group action on the spectrum of S.

A model for the canonical algebras of bimodules type (1, 4) over truncated polynomial rings

Let C((ε))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}(\\!(\\varepsilon )\\!)$$\\end{document} be the field of complex Laurent series. We use Galois descent techniques to show that the simple regular representations of the species of type (1,4)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1,\\, 4)$$\\end{document} over C((ε))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}(\\!(\\varepsilon )\\!)$$\\end{document} are naturally parametrized by the closed points of Spec(C((ε))[x])∪˙{1,2}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Spec}(\\mathbb {C}(\\!(\\varepsilon )\\!)[x]){\\dot{\\cup }}\\{1,\\,2\\}$$\\end{document}. Moreover, we provide weak normal forms for those representations. We use our representatives of the simple regular representations to describe the canonical algebras associated to the species of type (1, 4) over C((ε))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}(\\!(\\varepsilon )\\!)$$\\end{document}. This suggests a model of those algebras in the sense of the work of Geiss et al. (Invent Math 209(1):61–158, 2017; Math Z 295:1245–1277, 2020).

Towards a theory of Inc-compatible term orders in polynomial rings in infinitely many variables

Towards a theory of Inc-compatible term orders in polynomial rings in infinitely many variables

On products of idempotents and nilpotents

This article studies the ring structure arising from products of idempotents and nilpotents. Thus the argument is concerned essentially with the one-sided IQNN property of rings. We first prove that if the $2$ by $2$ full matrix ring over a principal ideal domain $F$ of characteristic zero is right IQNN then $F$ contains infinitely many non-integer rational numbers; and that the concepts of right IQNN and right quasi-Abelian are independent of each other. We next introduce a ring property, called {\it right IAN}, as a generalization of both right IQNN and right quasi-Abelian; and provide several kinds of methods to construct right IAN rings. In the procedure, we also show that the right IQNN and right IAN do not go up to polynomial rings.

On generalized Star-GCD extensions

– Let ⋆ be a star operation on a ring extension R ⊆ S . The ring extension is said to be a generalized ⋆ -greatest common divisor (G- ⋆ GCD) extension if the intersection of two ⋆ -invertible ideals of R is ⋆ -invertible. We investigate properties of G- ⋆ GCD extensions. We give special attention to polynomial rings and Nagata rings.

Undecidability of indecomposable polynomial rings

Undecidability of indecomposable polynomial rings

Invariant rings of the special orthogonal group have nonunimodal h-vectors

For [Formula: see text] an infinite field of characteristic other than two, consider the action of the special orthogonal group [Formula: see text] on a polynomial ring via copies of the regular representation. When [Formula: see text] has characteristic zero, Boutot’s theorem implies that the invariant ring has rational singularities; when [Formula: see text] has positive characteristic, the invariant ring is [Formula: see text]-regular, as proven by Hashimoto using good filtrations. We give a new proof of this, viewing the invariant ring for [Formula: see text] as a cyclic cover of the invariant ring for the corresponding orthogonal group; this point of view has a number of useful consequences, for example, it readily yields the [Formula: see text]-invariant and information on the Hilbert series. Indeed, we use this to show that the [Formula: see text]-vector of the invariant ring for [Formula: see text] need not be unimodal.

Threshold Changeable Secret Sharing (TCSS) via Skew Polynomials

Secret sharing is a tool to divide a secret into multiple shares, so that to reconstruct the secret, it is necessary to collect several shares under certain conditions. Secret sharing was first introduced by Adi Shamir, the scheme is based on polynomials. The secret is represented as a constant value polynomial, and the points on the polynomial graph serve as shares. Secret reconstruction is performed through Lagrange interpolation at a minimum of k out of n points, known as a threshold scheme (k, n). In 2010, Zhang Y. designed a secret sharing scheme through skew polynomials. Involving the role of the automorphism σ in the skew polynomial ring increases the complexity in share distribution and secret reconstruction. In 2012, Zhang Z. designed a secret sharing scheme with a threshold that can change according to the participants present during the reconstruction process, known as Threshold Changeable Secret Sharing (TCSS). This aims to prevent external parties from pretending to be valid participants in order to learn the secret. In this research, a TCSS scheme will be designed using skew polynomials. The aim is to make the TCSS scheme's calculations more complex, making it harder for adversaries to access the secret.

Structure of zero-divisors to go up to related ideals

There are many ways for zero-dividing polynomials to go up to zero-dividing ideals when the base rings are IFP. The importance of these in ring theory leads us to consider the following ring conditions and study new useful roles of matrices for ring theory. Let R be a ring and a , b ∈ R \\ { 0 } . The first is the condition (*) that if ab = 0 then Ib = 0 for some nonzero ideal I ⊆ RaR of R or aJ = 0 for some nonzero ideal J ⊆ RbR of R. It is shown that from given any IFP ring, there can be constructed a non-IFP ring with the condition (*). We prove that a semiprime ring R with the condition (*) is both right and left nonsingular. The second is the condition (**) that if ab = 0 then IJ = 0 for some nonzero ideals I ⊆ RaR and J ⊆ RbR of R. We prove that every ring can be a subring of rings with the condition (**), that if R is an irreducible ring with the condition (**) then R is either a domain or non-semiprime, and that the condition (**) passes to polynomial rings when the base ring is semiprime. Various sorts of examples are given to illustrate and delimit the results obtained.

Equivariant Schubert calculus and geometric Satake

The main classical result of Schubert calculus is that multiplication rules for the basis of Schubert cycles inside the cohomology ring of the Grassmannian G(n,m) are the same as multiplication rules for the basis of Schur polynomials in the ring of symmetric polynomials. In this paper, we explain how to recover this somewhat mysterious connection by using the geometric Satake correspondence to put the structure of a representation of GLm on H•(G(n,m)) and comparing it to the Fock space representation on symmetric polynomials. This proof also extends to equivariant Schubert calculus, and gives an explanation of the relationship between torus-equivariant cohomology of Grassmannians and double Schur polynomials.

Computing Gröbner bases on the Weyl algebras over fields with valuations

The computational aspect of tropical Gröbner bases for a polynomial ring K[x] with respect to tropical term orders studied by Chan and Maclagan in 2019 is extended to the Weyl algebra Dn(K), where K is a field with a valuation. The term order in this paper is not only an extension of the tropical term order on K[x] by Chan and Maclagan, but also of the tropical term order on K[x] studied by Vaccon et al. (2021). Due to the involvement of the valuations of term coefficients, this term order is not well-ordering. Therefore, a suitable division algorithm with respect to this term order is needed. This algorithm holds only for homogeneous operators, so utilizing the homogenized Weyl algebra is required. A computation example and an implementation in Risa/Asir Computer Algebra System are also presented in this paper.

Factorization in the monoid of integrally closed ideals

Given a Noetherian ring A, the collection of integrally closed ideals in A which contain a nonzerodivisor forms a cancellative monoid under the operation I * J = IJ ¯ , the integral closure of the product. The monoid is torsion-free and atomic. Restricting to monomial ideals in a polynomial ring, there is a surjective homomorphism from the Integral Polytope Group onto the Grothendieck group of integrally closed monomial ideals under translation invariance of their Newton Polyhedra. The Integral Polytope Group, the Grothendieck group of polytopes with integer vertices under Minkowski addition and translation invariance, has an explicit basis, allowing for explicit factoring in the monoid.

Products and powers of principal symmetric ideals

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.

On strongly quasi S-primary ideals

This paper centers around one of several generalizations of primary ideals. Which is an intermediate class between S-primary ideals and quasi S-primary ideals. Let R be a commutative ring with identity and S be a multiplicative closed subset of R. A proper ideal I of R disjoint from S is called strongly quasi S-primary if there exists an s ∈ S such that whenever x , y ∈ R and xy ∈ I , then either s x 2 ∈ I or sy ∈ I . Many basic properties of strongly quasi S-primary ideals are given, and examples are presented to distinguish the last concept from other classical ideals. Moreover, forms of strongly quasi S-primary ideals in polynomial rings, power series rings and idealization of a module are investigated.

The Automorphisms of Differential Extensions of Characteristic p

Nonassociative differential extensions are generalizations of associative differential extensions, either of a purely inseparable field extension K of exponent one of a field F, F of characteristic p, or of a central division algebra over a purely inseparable field extension of F. Associative differential extensions are well known central simple algebras first defined by Amitsur and Jacobson. We explicitly compute the automorphisms of nonassociative differential extensions. These are canonically obtained by restricting automorphisms of the differential polynomial ring used in the construction of the algebra. In particular, we obtain descriptions for the automorphisms of associative differential extensions of D and K, which are known to be inner.

Isomorphism between skew polynomial rings

Isomorphism between skew polynomial rings

Action of derivations on polynomials and on Jacobian derivations

Let $\mathbb K$ be a field of characteristic zero, $A := \mathbb K[x_{1}, x_{2}]$ the polynomial ring and $W_2(\mathbb K)$ the Lie algebra of all $\mathbb K$-derivations on $A$. Every polynomial $f \in A$ defines a Jacobian derivation $D_f\in W_2(\mathbb K)$ by the rule $D_f(h)=\det J(f, h)$ for any $h\in A$, where $J(f, h)$ is the Jacobi matrix for $f, h$. The Lie algebra $W_2(\mathbb K)$ acts naturally on $A$ and on itself (by multiplication). We study relations between such actions from the viewpoint of Darboux polynomials of derivations from $W_2(\mathbb K)$. It is proved that for a Jordan chain $T(f_1)=\lambda f_1+f_2$, ..., $T(f_{k-1})=\lambda f_{k-1}+f_k$, $T(f_k)=\lambda f_k$ for a derivation $T\in W_2(\mathbb K)$ on $A$ there exists an analogous chain $[T,D_{f_1}]=(\lambda -\mathop{\mathrm{div}} T)D_{f_1} + D_{f_2}$, ..., $[T,D_{f_{k}}]=(\lambda -\mathop{\mathrm{div}} T)D_{f_{k}}$ in $W_2(\mathbb K)$. In case $A:=\mathbb K[x_1, \ldots , x_n]$, the action of normalizers of elements $f$ from $A$ in $W_n(\mathbb K)$ on the principal ideals $(f)$ is considered.

A characterization of differential operators in the ring of complex polynomials

The paper aims to provide a full characterization of all operators T:P(ℂ)→P(ℂ) acting on the space of all complex polynomials that satisfy the Leibniz rule T(f⋅g)=T(f)⋅g+f⋅T(g)for all f,g∈P(ℂ). We do not assume the linearity of T. As we will see, contrary to the well-known theorems for function spaces there are many other solutions here, not only differential operators. From our main result, we also derive two corollaries, showing that in some special cases operators that satisfy the Leibniz rule have some particular form.

On covering systems of polynomial rings over finite fields

In 1950, Erdős posed a question known as the minimum modulus problem on covering systems for Z \mathbb {Z} , which asked whether the minimum modulus of a covering system with distinct moduli is bounded. This long-standing problem was finally resolved by Hough [Ann. of Math. (2) 181 (2015), no. 1, pp. 361–382] in 2015, as he proved that the minimum modulus of any covering system with distinct moduli does not exceed 10 16 10^{16} . Recently, Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [Invent. Math. 228 (2022), pp. 377–414] developed a versatile method called the distortion method and significantly reduced Hough’s bound to 616 , 000 616,000 . In this paper, we apply this method to present a proof that the smallest degree of the moduli in any covering system for F q [ x ] \mathbb {F}_q[x] of multiplicity s s is bounded by a constant depending only on s s and q q . Consequently, we successfully resolve the minimum modulus problem for F q [ x ] \mathbb {F}_q[x] and disprove a conjecture by Azlin [Covering Systems of Polynomial Rings Over Finite Fields, University of Mississippi, Electronic Theses and Dissertations. 39, 2011].

On polynomial invariant rings in modular invariant theory

Let k be a field of characteristic p>0, V a finite-dimensional k-vector-space, and G a finite p-group acting k-linearly on V. Let S=SymV⁎. Confirming a conjecture of Shank-Wehlau-Broer, we show that if SG is a direct summand of S, then SG is a polynomial ring, in the following cases:(a)k=Fp and dimk⁡V=4; or(b)|G|=p3. In order to prove the above result, we also show that if dimk⁡VG≥dimk⁡V−2, then the Hilbert ideal hG,S is a complete intersection.