Francesco Vaccarino
Let R be a commutative ring and let n,m be two positive integers. Let be the polynomial ring in m x n commuting independent variables R. The symmetric group on n letters acts diagonally on A(n,m). We give generators and relations of the rings of invariants for this action.
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This article studies separating invariants for the ring of multisymmetric polynomials in $m$ sets of $n$ variables over an arbitrary field $\mathbb{K}$. We prove that in order to obtain separating sets it is enough to consider polynomials that depend only on $\lfloor \frac{n}{2} \rfloor + 1$ sets of these variables. This improves a general result by Domokos about separating invariants. In addition, for $n \leq 4$ we explicitly give minimal separating sets (with respect to inclusion) for all $m$ in case $\text{char}(\mathbb{K}) = 0$ or $\text{char}(\mathbb{K}) > n$.
This habilitation (German variant of a PhD on top of a PhD) thesis presents the quintessence of the ideas and experiences with Groebner Bases of Birgit Reinert. She died unexpectedly without providing an abstract. As arXiv requires an abstract, let us quote from the table of contents: History of Groebner Bases. The two definitions of Groebner Bases. Generalizations. Groebner Bases in Function Rings -- A Guide for Introducing Relations to Algebraic Structures. Applications of Groebner Bases to Function Rings. Groebner Bases in Polynomial Rings. Reduction Rings. Quotients of Reduction Rings. Sum…
Generalized Igusa functions and ideal growth in nilpotent Lie rings
本文引入一类新的组合定义有理函数，推导出包含自由2-幂零李环的一大类幂零李环的局部理想zeta函数显式公式，统一并推广了先前大量计算，并证明了自互反性质，进而证实了Grunewald-Segal-Smith关于正规zeta函数一致性的猜想。
We study the ring of polyfunctions over $\mathbb Z/n\mathbb Z$. The ring of polyfunctions over a commutative ring $R$ with unit element is the ring of functions $f:R\to R$ which admit a polynomial representative $p\in R[x]$ in the sense that $f(x)= p(x)$ for all $x\in R$. This allows to define a ring invariant $s$ which associates to a commutative ring $R$ with unit element a value in $\mathbb N\cup\{\infty\}$. The function $s$ generalizes the number theoretic Smarandache function. For the ring $R=\mathbb Z/n\mathbb Z$ we provide a unique representation of polynomials which vanish as a functio…
We study the ring of algebraic functions on the space of persistence barcodes, with applications to pattern recognition.
In this paper, we develop the theory of the necklace ring and the logarithmic function. Regarding the necklace ring, we introduce the necklace ring functor $Nr$ from the category of special $\ld$-rings into the category of special $\ld$-rings and then study the associated Adams operators. As far as the logarithmic function is concerned, we generalize the results in Bryant's paper (J. Algebra. 253 (2002); no.1, 167-188) to the case of graded Lie (super)algebras with a group action by applying the Euler-Poincaré principle.
Regarding polynomial functions on a subset $S$ of a non-commutative ring $R$, that is, functions induced by polynomials in $R[x]$ (whose variable commutes with the coefficients), we show connections between, on one hand, sets $S$ such that the integer-valued polynomials on $S$ form a ring, and, on the other hand, sets $S$ such that the set of polynomials in $R[x]$ that are zero on $S$ is an ideal of $R[x]$.
Some remarks on the ring of arithmetical functions
本文研究算术函数环的结构，证明其非诺特、非阿廷，具有无限克鲁尔维数，并构造多种素理想。
Skew-convex function rings and evaluation of skew rational functions
本文利用斜多项式乘积的赋值公式构造了一类环，并基于此提出对斜多项式商进行赋值的方法。
In characteristic zero, Zinovy Reichstein and the author generalized the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where "varieties" carry a PGL_n-action, regular and rational "functions" on them are matrix-valued, "coordinate rings" are prime polynomial identity algebras, and "function fields" are central simple algebras of degree n. In the present paper, much of this is extended to prime characteristic. In addition, a mistake in the earlier paper is corrected. …
On structures of the ring of arithmetical functions: prime ideals and beyond
本文研究算术函数环的结构性质，证明其既非诺特环也非阿廷环，构造多种素理想并给出非素的半素理想例子，同时证明该环具有无穷克鲁尔维数。
Given a ring $R$, the notion of Sylvester rank function was conceived within the context of Cohn's classification theory of epic division $R$-rings. In this paper we study and describe the space of Sylvester rank functions on certain families of rings, including Dedekind domains, simple left noetherian rings and skew Laurent polynomial rings $\mathcal{D}[t^{\pm 1};τ]$ for any division ring $\mathcal{D}$ and any automorphism $τ$ of $\mathcal{D}$.
In calculus, an indefinite integral of a function $f$ is a differentiable function $F$ whose derivative is equal to $f$. In present paper, we generalize this notion of the indefinite integral from the ring of real functions to any ring. The main goal of the paper is to focus on the properties of such generalized integrals that are inherited from the well-known basic properties of indefinite integrals of real functions.
In this paper we introduce new modules over the ring of ponderation functions, so we recover old results in harmonic analysis from the side of ring theory. Moreover, we prove that Laplace transform, Fourier transform and Hankel transform generate some kind of modules over the ring of ponderation functions.
We define the notions of finite-state and functionally recursive matrices and their growth. We also introduce two rings generated by functionally recursive matrices. The first is isomorphic to the 2-generated free ring. The second is a 2-generated monomial ring such that the multiplicative semigroup of monomials in the generators is nil of degree 5 and the ring has Gelfand Kirillov dimension 1 + log(2)/log(a) where a=1/2(1+sqrt(5)).
Exploring Functional Identities: From Division Rings to Matrix Algebras
本文通过严格分析与定理证明，解决了Ferreira等人提出的未决问题，深入探究了除环上矩阵代数中加性函数的复杂行为，并揭示了其对代数结构的影响。
Rings of Quotients of Rings of Functions
将Johnson和Utumi引入的商环应用于完全正则空间X上的连续实值函数环C(X)，研究了最大商环Q(X)的实现及其与经典商环的关系。
The ring of dual numbers over a ring $R$ is $R[α] = R[x]/(x^2)$, where $α$ denotes $x+(x^2)$. For any finite commutative ring $R$, we characterize null polynomials and permutation polynomials on $R[α]$ in terms of the functions induced by their coordinate polynomials ($f_1,f_2\in R[x]$, where $f=f_1+αf_2$) and their formal derivatives on $R$. We derive explicit formulas for the number of polynomial functions and the number of polynomial permutations on $\mathbb{Z}_{p^n}[α]$ for $n\le p$ ($p$ prime).
We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where "varieties" carry a PGL_n-action, regular and rational "functions" on them are matrix-valued, "coordinate rings" are prime polynomial identity algebras, and "function fields" are central simple algebras of degree n. In particular, a prime polynomial identity algebra of degree n is finitely generated if and only if it arises as the "coordinate ring" of a "variety" in this setting. For n = 1 our defi…
Centralizers in non-associative rings with a pseudo-degree function
本文研究非结合代数中元素的中心化子，证明其为有限秩自由模。
A detailed proof is given of the well-known facts that greatest common divisors exist in rings of non-Archimedean entire functions of several variables and that these rings of entire functions are almost factorial, in the sense that an entire function can be uniquely written as a countable product of irreducible entire functions.
Zeta functions for table algebras and fusion rings with irrational-valued characters
本文针对具有无理值不可约特征标的秩3标准整表代数与整融合环，通过显式计算zeta积分，求出了相应序的zeta函数。
We produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class-$2$-nilpotent Lie rings, introduced in [1], in terms of Igusa functions. As corollaries we obtain information about analytic properties of global ideal zeta functions, local functional equations, topological, reduced, and graded ideal zeta functions, as well as representation zeta functions for the unipotent group schemes associated to the Lie rings in question.
Given a function from $\mathbb{Z}_n$ to itself one can determine its polynomial representability by using Kempner function. In this paper we present an alternative characterization of polynomial functions over $\mathbb{Z}_n$ by constructing a generating set for the $\mathbb{Z}_{n}$-module of polynomial functions. This characterization results in an algorithm that is faster on average in deciding polynomial representability. We also extend the characterization to functions in several variables.
The aim of this paper is to give an algebraic characterization of the rings $C(X,\mathbb{Q}_p)$ of all continuous $\mathbb{Q}_p$-valued functions on a compact space $X$. The characterization is similar to that of M. Stone from 1940 for the case of $\mathbb{R}$-valued functions.
In 1969 Höchster proved that for every quasi-compact T1-space $X$ we can find a commutative ring $R$ such that $X$ is homeomorphic to the maximal spectrum $\mathrm{Specm}(R)$ of $R$. This result implies the existence of a commutative ring $R$ that admits a non-Hausdorff and totally disconnected maximal spectrum $\mathrm{Specm}(R)$. However, there has not been an example of such a commutative ring yet. The aim of this paper is to investigate rings that almost admit a non-Hausdorff and totally disconnected maximal spectrum.
The homogeneous weights and the Möbius functions and Euler phi-functions on finite rings are discussed; some computational formulas for these functions on finite principal ideal rings are characterized; for the residue rings of integers, they are reduced to the classical number-theoretical Möbius functions and the classical number-theoretical Euler phi-functions.
In this paper we prove relationships between two generalizations of commutative valuation theory for noncommutative central simple algebras: (1) Dubrovin valuation rings; and (2) the value functions called gauges introduced by Tignol and Wadsworth in [TW1] and [TW2]. We show that if v is a valuation on a field F with associated valuation ring V and v is defectless in a central simple F-algebra A, and C is a subring of A, then the following are equivalent: (a) C is the gauge ring of some minimal v-gauge on A, i.e., a gauge with the minimal number of simple components of C/J(C); (b) C is integra…
We compute the zeta functions enumerating graded ideals in the graded Lie rings associated with the free $d$-generator Lie rings $\mathfrak{f}_{c,d}$ of nilpotency class $c$ for all $c\leq2$ and for $(c,d)\in\{(3,3),(3,2),(4,2)\}$. We apply our computations to obtain information about $\mathfrak{p}$-adic, reduced, and topological zeta functions, in particular pertaining to their degrees and some special values.
Pseudocomplementation in rings of continuous functions
本文研究连续函数环的伪补全条件，通过素谱上的各种格来分析，全面 characterize 伪补全并在相对伪补全方面有几乎完整的 characterization。
The Whitney group $K_1(A)$ is isomorphic to $A^\times \times \operatorname{SK}_1(A)$ for some subgroup $\operatorname{SK}_1(A)$, where $A$ is a commutative ring and $A^\times$ denotes the set of units in $A$. Consider an o-minimal expansion of a real closed field $\mathcal R=(R,0,1,+,\cdot,\ldots)$. Let $M$ be an affine definable $C^r$ manifold, where $r$ is a nonnegative integer. We demonstrate its homotopy theorem and that the group $\operatorname{SK}_1(C_{\text{df}}^r)$ is isomorphic to $\operatorname{SK}_1(C_{\text{df}}^0(M))$, where $C_{\text{df}}^r(M)$ denotes the ring of definable $C^r$…
Rings of almost everywhere defined functions
本文证明偏序交换环可表示为紧 Hausdorff 空间上几乎处处定义的连续实值函数环子环的充要条件是其兼具 Archimede 性与可局部化性，并给出多类代数应用。
We study the asymptotics at zero of continuous functions on (0, 1] by means of their asymptotic ideals, i.e., ideals in the ring of continuous functions on (0, 1] satisfying a polynomial growth condition at 0 modulo rapidly decreasing functions at 0. As our main result, we characterize maximal and prime ideals in terms of maximal and prime filters.
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