Related papers: Completeness of the ring of polynomials

We study the algebraic $K$-theory of rings of the form $R[x]/x^e$. We do this via trace methods and filtrations on topological Hochschild homology and related theories by quasisyntomic sheaves. We produce computations for $R$ a perfectoid…

Let $p(Y_1, \dots, Y_d, Z_1, \dots, Z_e)$ be a self-adjoint noncommutative polynomial, with coefficients from $\mathbb{C}^{r \times r}$, in the indeterminates $Y_1, \dots, Y_d$ (considered to be self-adjoint), the indeterminates $Z_1,…

In this paper, we give a detailed proof to a result of Gabber (unpublished) on the lifting problem of quasi-excellent rings, extending the previous work on Nishimura-Nishimura. As a corollary, we establish that an ideal-adic completion of…

In this paper we prove that the gradient ideal of a Morse polynomial is radical. This gives a generic class of polynomials whose gradient ideals are radical. As a consequence we reclaim a previous result that the unconstrained polynomial…

We consider ideals in the ring $\mathbb{Z}_2[x_1,\ldots, x_n]$ that contain the polynomials $x_i^2 - x_i$ for $i = 1, \ldots, n$ and give various results related to the one-to-one correspondence between these ideals and the subsets of…

We prove the following;Theorem:Let R be a prime noetherian ring with k.dimR = n, n a finite non-negative integer. We refer the reader to the definitions (1.1) of this paper.For a fixed non-negative integer m, m<n let Xm be the full set of…

For a commutative ring R we investigate the property that the sets of minimal primes of finitely generated ideals of R is always finite. We prove this property passes to polynomial ring extensions (in an arbitrary number of variables) over…

We introduce a new basis for the polynomial ring which lifts the complete homogeneous symmetric polynomials while retaining representation theoretic significance. Using a specialized RSK algorithm we give an explicit nonnegative expansion…

Let $I$ be a proper graded ideal in a positively graded polynomial ring $S$ over a field of characteristic 0. In this note it is shown that $S/I^k$ is Golod for all $k\geq 2$.

Let $R$ be a commutative ring and $M$ be an $R$-module, and let $I(R)^*$ be the set of all non-trivial ideals of $R$. The $M$-intersection graph of ideals of $R$, denoted by $G_M(R)$, is a graph with the vertex set $I(R)^*$, and two…

Motivated by classic theorems of Thompson and Berger on the Fitting height of finite groups with a fixed-point-free automorphism of coprime order, we conjecture that, for every non-zero polynomial $f(x) = a_0 + a_1 x + \cdots + a_d x^d \in…

A \textit{symmetric ideal} $I \subseteq R = K[x_1,x_2,...]$ is an ideal that is invariant under the natural action of the infinite symmetric group. We give an explicit algorithm to find Gr\"obner bases for symmetric ideals in the infinite…

In this paper, we present a criterion for $(K,v)$ to be henselian and defectless in terms of finite complete sequences of key polynomials. For this, we use the theory of Mac Lane-Vaqui\'e chains and abstract key polynomials. We then prove…

In a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free $R$-modules to finitely…

We show that, on a complete and possibly non-compact Riemannian manifold of dimension at least 2 without close conjugate points at infinity, the existence of a closed geodesic with local homology in maximal degree and maximal index growth…

Let I be a finitely supported complete m-primary ideal of a regular local ring (R, m). A theorem of Lipman implies that I has a unique factorization as a *-product of special *-simple complete ideals with possibly negative exponents for…

We study perfect multiple coverings in translation invariant graphs with vertex set $\mathbb{Z}^2$ using an algebraic approach. In this approach we consider any such covering as a two-dimensional binary configuration which we then express…

In this paper, we study conditions guaranteeing that a product of ideals defines a Golod ring. We show that for a $3$-dimensional regular local ring (or $3$-variable polynomial ring) $(R , \m)$, the ideal $I \m$ always defines a Golod ring…

In 1971, Ruzsa conjectured that if $f:\ \mathbb{N}\rightarrow\mathbb{Z}$ with $f(n+k)\equiv f(n)$ mod $k$ for every $n,k\in\mathbb{N}$ and $f(n)=O(\theta^n)$ with $\theta<e$ then $f$ is a polynomial. In this paper, we investigate the…

Let R be a commutative noetherian ring. Lindo and Pande have recently posed the question asking when every ideal of R is isomorphic to some trace ideal of R. This paper studies this question and gives several answers. In particular, a…