Related papers: Bounds for test exponents

Let $(S, \mathfrak n) $ be a regular local ring and let $I \subseteq \mathfrak n^2 $ be a perfect ideal of $S. $ Sharp upper bounds on the minimal number of generators of $I$ are known in terms of the Hilbert function of $R=S/I. $ Starting…

Given $N$ positive integers $a_1, ..., a_N$ with $\gcd(a_1, ..., a_N)=1$, let $f_N$ denote the largest natural number which is not a positive integer combination of $a_1, ..., a_N$. This paper gives an optimal lower bound for $f_N$ in terms…

We improve certain degree bounds for Grobner bases of polynomial ideals in generic position. We work exclusively in deterministically verifiable and achievable generic positions of a combinatorial nature, namely either strongly stable…

Motivated by recent result of P\'erez and R.G. on equality of test ideal of module closure operation and trace ideal, and the well-known result by Smith that parameter test ideal cannot be contained in parameter ideals, we study the…

We study different form of boundness for ideals of almost Dedekind domains, generalizing the notions of critical ideals, radical factorization, and SP-domains. We show that every almost Dedekind domain has at least one noncritical maximal…

Motivated by some recent results of F. P\'erez and R. R.G connecting test ideal of module closure operations and trace ideals, we investigate the test ideal restricted to principal ideals corresponding to a module closure operation of a…

Let $\mathbf{A}$ be a finite nilpotent algebra in a congruence modular variety with finitely many fundamental operations. If $\mathbf{A}$ is of prime power order, then it is known that there is a polynomial $p$ such that for every $n \in…

When $I$ is the radical homogeneous ideal of a finite set of points in projective $N$-space, ${\bf P}^N$, over a field $K$, it has been conjectured that $I^{(rN-N+1)}$ should be contained in $I^r$ for all $r\geq 1$. Recent counterexamples…

In this paper, we introduce and study the numerical semigroups generated by $\{a_1, a_2, \ldots \} \subset \mathbb{N}$ such that $a_1$ is the repunit number in base $b > 1$ of length $n > 1$ and $a_i - a_{i-1} = a\, b^{i-2},$ for every $i…

In this paper we study various equivalent conditions for tight closure to commute with localization. If N is a submodule of a finitely generated module M over a Noetherian commutative ring of characteristic p, then a test exponent for c,N,M…

We show that for any finite-dimensional algebra $\Lambda$ of infinite representation type, over a perfect field, there is a bounded principal ideal domain $\Gamma$ and a representation embedding from $\Gamma -$mod into $\Lambda -$mod. As an…

We generalize the notion of Hilbert-Kunz multiplicity of a graded triple $(M,R,I)$ in characteristic $p>0$ by proving that for any complex number $y$, the limit $$\underset{n \to \infty}{\lim}(\frac{1}{p^n})^{\text{dim}(M)}\sum \limits_{j=…

Given an ideal $a \subseteq R$ in a (log) $Q$-Gorenstein $F$-finite ring of characteristic $p > 0$, we study and provide a new perspective on the test ideal $\tau(R, a^t)$ for a real number $t > 0$. Generalizing a number of known results…

Many Hilbert modules over the polynomial ring in m variables are essentially reductive, that is, have commutators which are compact. Arveson has raised the question of whether the closure of homogeneous ideals inherit this property and…

Let $G$ be a Frobenius group with an abelian Frobenius kernel $F$ and let $k$ be a finite extension of $\mathbb{Q}$. We obtain an upper bound for the number of degree $|F|$ algebraic extensions $K/k$ with Galois group $G$ with the norm of…

We find the number of compositions over finite abelian groups under two types of restrictions: (i) each part belongs to a given subset and (ii) small runs of consecutive parts must have given properties. Waring's problem over finite fields…

A group element is called generalized torsion if a finite product of its conjugates is equal to the identity. We show that in a finitely generated abelian-by-finite group, an element is generalized torsion if and only if its image in the…

The test ideal $\tau(R)$ of a ring $R$ of prime characteristic is an important object in the theory of tight closure. In this paper, we study a generalization of the test ideal, which is the ideal $\tau(\a^t)$ associated to a given ideal…

We present a new effective Nullstellensatz with bounds for the degrees which depend not only on the number of variables and on the degrees of the input polynomials but also on an additional parameter called the {\it geometric degree of the…

For a prime number $\ell$ and an extension of number fields $K/F$, we prove new lower bounds on the $\ell$-rank of the ideal class group of $K$ based on prime ramification in $K/F$. Unlike related results from the literature, our bound is…