Related papers: Uniformity in Higher class Free Lie algebras

Let $L$ be a nilpotent algebra of class two over a compact discrete valuation ring $A$ of characteristic zero or of sufficiently large positive characteristic. Let $q$ be the residue cardinality of $A$. The ideal zeta function of $L$ is a…

In the present paper, we characterize all possible Hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to $d$, where $d$ is a positive integer. In addition, we prove the following result which…

We determine all F,G in C[X] of degree at least 2 for which the semigroup generated by F and G under composition is not the free semigroup on the letters F and G. We also solve the same problem for F,G in X^2 C[[X]], and prove partial…

We consider the problem of characterizing all functions $f$ defined on the set of integers modulo $n$ with the property that an average of some $n$th roots of unity determined by $f$ is always an algebraic integer. Examples of such…

Let $k$ be a unital commutative ring. In this paper, we study polynomial functors from the category of finitely generated free nilpotent groups to the category of $k$-modules, focusing on comparisons across different nilpotency classes and…

Over a field of characteristic zero, we prove that for each r, there exists a constant C(r) so that the prime ideal of the rth secant variety of any Veronese embedding of any projective space is generated by polynomials of degree at most…

Let $(\mathcal{O}_n, \mathfrak{m})$ denote the ring of germs of holomorphic functions $\mathbb{C}^n\to \mathbb{C}$, and let $I\subseteq \mathcal{O}_n$ be an $\mathfrak{m}$-primary ideal. Demailly and Pham showed that $\mathrm{lct}(I) \geq…

In this article we investigate when a homogeneous ideal in a graded ring is normal, that is, when all positive powers of the ideal are integrally closed. We are particularly interested in homogeneous ideals in an N-graded ring generated by…

A square-free monomial ideal $I$ is called an {\it $f$-ideal}, if both $\delta_{\mathcal{F}}(I)$ and $\delta_{\mathcal{N}}(I)$ have the same $f$-vector, where $\delta_{\mathcal{F}}(I)$ ($\delta_{\mathcal{N}}(I)$, respectively) is the facet…

Let $R$ be an algebra over a commutative ring $k$. Suppose that $R$ is endowed with a descending filtration indexed on an ordered group $(G,<)$ such that the restriction to $k$ is positive. We show that the existence of free algebras on a…

In this paper, motivated by studying special values of zeta functions attached to finite type F_p-schemes, we introduce a category of ``arithmetic C(S^1,R)-modules'' attached to any Dedekind ring R, and compute the 0th K-group of this…

We associate to a semisimple complex Lie algebra $\mathfrak{g}$ a sequence of polynomials $P_{\ell,\mathfrak{g}}(x)\in\mathbb{Q}[x]$ in $r$ variables, where $r$ is the rank of $\mathfrak{g}$ and $\ell=0,1,2,\ldots $. The polynomials…

We study uniform stability of discrete groups, Lie groups and Lie algebras in the rank metric, and the connections between uniform stability of these objects. We prove that semisimple Lie algebras are far from being flexibly…

We obtain order-exact estimates for uniform approximations by using Zygmund sums $Z^{s}_{n}$ of classes $C^{\psi}_{\beta,p}$ of $2\pi$-periodic continuous functions $f$ representable by convolutions of functions from unit balls of the space…

Let $k$ be a field of positive characteristic and $R = k[x_0,\dots, x_n]$. We consider ideals $I\subseteq R$ generated by homogeneous polynomials of degree $d$. Takagi and Watanabe proved that $\mathrm{fpt}(I)\geq \mathrm{height}(I)/d$; we…

We consider standard graded toric rings $R_{\Delta}$ whose generators correspond to the faces of a simplicial complex $\Delta$. When $R_{\Delta}$ is normal, it is shown that its divisor class group is free. For a flag complex $\Delta$ which…

As an extension of previous ungraded work, we define a graded $p$-polar ring to be an analog of a graded commutative ring where multiplication is only allowed on $p$-tuples (instead of pairs) of elements of equal degree. We show that the…

In this work, we extend the definition of the graded prime ideals from those in commutative graded rings to the ideals over graded Lie algebras. We prove some facts about graded prime Lie ideals in arbitrary Lie algebras that are similar to…

Let $UT_n(F)$ be the algebra of the $n\times n$ upper triangular matrices and denote $UT_n(F)^{(-)}$ the Lie algebra on the vector space of $UT_n(F)$ with respect to the usual bracket (commutator), over an infinite field $F$. In this paper,…

We study the sequence of zeta functions $Z(C_p,T)$ of a generic Picard curve $C:y^3=f(x)$ defined over $\mathbb{Q}$ at primes $p$ of good reduction for $C$. We define a degree 9 polynomial $\psi_f\in \mathbb{Q}[x]$ such that the splitting…