We prove in a broad combinatorial setting, namely for finitely generated semipositive cancellative reduced binoids, that the Hilbert–Kunz multiplicity is a rational number independent of the characteristic.

The Hilbert polynomial of a homogeneous complete intersection is determined by the degrees of the generators of the defining ideal. The degrees of the generators are not, in general, determined by the Hilbert polynomial — but sometimes they are. When? We give some general criteria and completely answer the question up to codimension 6.

We study amoebas of the principal $A$-determinant in relationship with the lopsidedness condition: the (co)amoeba of the principal $A$-determinant stratifies the space of lopsided (co)amoebas according to equivalence of order maps — a refinement of topological equivalence. We extend Nilsson and Passare’s description of the coamoeba of the $A$-discriminant to the case when $A$ defines a projective toric curve. As an application, we compute coefficients of transition matrices necessary when gluing local monodromy groups of $A$-hypergeometric systems.

The resurgence of an ideal of fat points in $\mathbb{𝕂}\left[{\mathbb{P}}^N\right]$ with linear minimal free resolution can be expressed as the quotient of its initial degree and its Waldschmidt constant. This makes it possible to do computations of the resurgence of this type of ideal with less complexity than in general. In this paper, given a fat point subscheme of ${\mathbb{P}}^N$, we construct a new subscheme where its saturated ideal has a linear minimal free resolution. In particular, we show that the saturated ideal of a fat point subscheme $Z=\left(s-2\right)p_1+p_2+\cdots +p_s$, supported on $s\ge 3$ general points of ${\mathbb{P}}^2$, has a linear minimal free resolution, and we compute its resurgence.

We advance an ideal-theoretic analogue of a “finite factorization domain” (FFD), giving such a domain the moniker “finite molecularization domain” (FMD). We characterize FMD’s as those factorable domains (termed “molecular domains” in the paper) for which every nonzero ideal is divisible by only finitely many nonfactorable ideals (termed “molecules” in the paper) and the monoid of nonzero ideals of the domain is unit-cancellative, in the language of Fan, Geroldinger, Kainrath, and Tringali. We develop a number of connections, particularly at the local level, amongst the concepts of “FMD”, “FFD”, and the “finite superideal domains” (FSD’s) of Hetzel and Lawson. Characterizations of when $k\left[X^2,X^3\right]$, where $k$ is a field, and the classical $D+M$ construction are FMD’s are provided. We also demonstrate that if $R$ is a Dedekind domain with the finite norm property, then $R\left[X\right]$ is an FMD.

Let $G$ be a finite simple graph and $I\left(G\right)$ denote the corresponding edge ideal. We prove that if $G$ is a very well-covered graph then for all $s\ge 1$ the regularity of $I{\left(G\right)}^s$ is exactly $2s+\nu \left(G\right)-1$, where $\nu \left(G\right)$ denotes the induced matching number of $G$.

Let $\left(R,\mathfrak{𝔪}\right)$ be a $d$-dimensional Cohen–Macaulay local ring, $I$ be an $\mathfrak{𝔪}$-primary ideal of $R$ and let $J=\left(x_1,\dots ,x_d\right)$ be a minimal reduction of $I$. We show that if, for $i=0$ or $1$, $J_{d-1}=\left(x_1,\dots ,x_{d-1}\right)$ and ${\sum }_{n=1}^{\infty }\lambda \left(I^{n+1}\cap J_{d-1}\right)∕\left(J“I^n\cap J_{d-1}\right)=i$, then $depthG\left(I\right)\ge d-i-1$. Moreover, we prove that if $e_2\left(I\right)={\sum }_{n=2}^{\infty }\left(n-1\right)\lambda \left(I^n∕J“I^{n-1}\right)-2$, or if $e_2\left(I\right)={\sum }_{n=2}^{\infty }\left(n-1\right)\lambda \left(I^n∕J“I^{n-1}\right)-3$ and $I$ is integrally closed, then $e_1\left(I\right)={\sum }_{n=1}^{\infty }\lambda \left(I^n∕J“I^{n-1}\right)-1$, where the integers $e_i$ are the Hilbert coefficients of $I$. In addition, if $J$ is a minimal reduction of $I$ then we prove that the reduction number $r_J\left(I\right)$ is independent of $J$.

Let $\left(A,\mathfrak{𝔪}\right)$ be a local complete intersection ring and let $I$ be an ideal in $A$. Let $M$, $N$ be finitely generated $A$-modules. Then for $l=0,1$, the values $depth{Ext}_A^{2i+l}\left(M,N∕I^{n}N\right)$ become independent of $i$, $n$ for $i,n\gg 0$. We also show that if $\mathfrak{𝔭}$ is a prime ideal in $A$ then the $j$-th Bass numbers ${\mu }_j\left(\mathfrak{𝔭},{Ext}_A^{2i+l}\left(M,N∕I^{n}N\right)\right)$ have polynomial growth in $\left(n,i\right)$ with rational coefficients for all sufficiently large $\left(n,i\right)$.

An extension $R\subseteq S$ of commutative rings with unity is called a root extension if for each element $s\in S$, there exists a positive integer $n$ such that $s^n\in R$. Unlike the integral extension, the root extension is not stable under polynomial ring extension. We characterize when the extension $R\left[X\right]\subseteq S\left[X\right]$ of polynomial rings is a root extension. Using the characterization, we can give a positive answer to the question posed by Anderson, Dumitrescu and Zafrullah (2004), i.e., $R\left[X\right]\subseteq S\left[X\right]$ being a root extension implies that $R\left[X,Y\right]\subseteq S\left[X,Y\right]$ is a root extension. We also characterize when the extension $R\left[\left[X\right]\right]\subseteq S\left[\left[X\right]\right]$ of power series rings is a root extension.

Aghapournahr and Melkersson introduced the notion of Melkersson condition on a Serre subcategory of the category of modules over a commutative Noetherian ring. This paper investigates the structure of a set of prime ideals satisfying the Melkersson condition on a Serre subcategory. We try to calculate members of this set for a subcategory consisting of extension modules in two given Serre subcategories. Moreover, we classify the structure of sets consisting of prime ideals satisfying the Melkersson condition over rings with a small dimension.