We introduce a new concept of rank – relative rank associated to a filtered collection of polynomials. When the filtration is trivial, our relative rank coincides with Schmidt rank (also called strength). We also introduce the notion of relative bias. The main result of the paper is a relation between these two quantities over finite fields (as a special case, we obtain a new proof of the results in [21]). This relation allows us to get an accurate estimate for the number of points on an affine variety given by a collection of polynomials which is of high relative rank (Lemma 3.2). The key advantage of relative rank is that it allows one to perform an efficient regularization procedure which is polynomial in the initial number of polynomials (the regularization process with Schmidt rank is far worse than tower exponential). The main result allows us to replace Schmidt rank with relative rank in many key applications in combinatorics, algebraic geometry, and algebra. For example, we prove that any collection of polynomials $\mathcal P=(P_i)_{i=1}^c$ of degrees $\le d$ in a polynomial ring over an algebraically closed field of characteristic $>d$ is contained in an ideal $\mathcal I({\mathcal Q})$, generated by a collection ${\mathcal Q}$ of polynomials of degrees $\le d$ which form a regular sequence, and ${\mathcal Q}$ is of size $\le A c^{A}$, where $A=A(d)$ is independent of the number of variables.

It is well known that the edge ideal $I(G)$ of a simple graph G has linear quotients if and only if $G^c$ is chordal. We investigate when the property of having linear quotients is inherited by homological shift ideals of an edge ideal. We will see that adding a cluster to the graph $G^c$ when $I(G)$ has homological linear quotients results in a graph with the same property. In particular, $I(G)$ has homological linear quotients when $G^c$ is a block graph. We also show that adding pinnacles to trees preserves the property of having homological linear quotients for the edge ideal of their complements. Furthermore, $I(G)$ has homological linear quotients for every graph G such that $G^c$ is a $\lambda $-minimal chordal graph.

Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl’s book: For the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases, take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with $S^G\subseteq S$ being the natural embedding.

Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring $S^G$ is a pure subring of S, equivalently, $S^G$ is a direct summand of S as an $S^G$-module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$ is pure, then either the invariant ring $S^G$ is regular or the group G is linearly reductive.

The purpose of this paper is to investigate the properties of spectral and tiling subsets of cyclic groups, with an eye towards the spectral set conjecture [9] in one dimension, which states that a bounded measurable subset of $\mathbb {R}$ accepts an orthogonal basis of exponentials if and only if it tiles $\mathbb {R}$ by translations. This conjecture is strongly connected to its discrete counterpart, namely that, in every finite cyclic group, a subset is spectral if and only if it is a tile. The tools presented herein are refinements of recent ones used in the setting of cyclic groups; the structure of vanishing sums of roots of unity [20] is a prevalent notion throughout the text, as well as the structure of tiling subsets of integers [1]. We manage to prove the conjecture for cyclic groups of order $p^{m}q^{n}$ , when one of the exponents is $\leq 6$ or when $p^{m-2}<q^{4}$ , and also prove that a tiling subset of a cyclic group of order $p_{1}^{m}p_{2}\dotsm p_{n}$ is spectral.

Let $f:R\to S$ be a ring homomorphism and J be an ideal of S. Then the subring $R\bowtie ^fJ:=\{(r,f(r)+j)\mid r\in R$ and $j\in J\}$ of $R\times S$ is called the amalgamation of R with S along J with respect to f. In this paper, we characterize when $R\bowtie ^fJ$ is a Hilbert ring. As an application, we provide an example of Hilbert ring with maximal ideals of different heights. We also construct non-Noetherian Hilbert rings whose maximal ideals are all finitely generated (unruly Hilbert rings).

In this paper, we study a family of binomial ideals defining monomial curves in the n-dimensional affine space determined by n hypersurfaces of the form $x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}}$ in $\Bbbk [x_1, \ldots , x_n]$ with $u_{ii} = 0, \ i\in \{ 1, \ldots , n\}$ . We prove that the monomial curves in that family are set-theoretic complete intersections. Moreover, if the monomial curve is irreducible, we compute some invariants such as genus, type and Frobenius number of the corresponding numerical semigroup. We also describe a method to produce set-theoretic complete intersection semigroup ideals of arbitrary large height.

We prove that a generic homogeneous polynomial of degree $d$ is determined, up to a nonzero constant multiplicative factor, by the vector space spanned by its partial derivatives of order $k$ for $k\leqslant \frac{d}{2}-1$.

We investigate whether the property of having linear quotients is inherited by ideals generated by multigraded shifts of a Borel ideal and a squarefree Borel ideal. We show that the ideal generated by the first multigraded shifts of a Borel ideal has linear quotients, as do the ideal generated by the $k$th multigraded shifts of a principal Borel ideal and an equigenerated squarefree Borel ideal for each $k$. Furthermore, we show that equigenerated squarefree Borel ideals share the property of being squarefree Borel with the ideals generated by multigraded shifts.

A number field K with a ring of integers 𝒪K is called a Pólya field, if the 𝒪K-module of integer-valued polynomials on 𝒪K has a regular basis, or equivalently all its Bhargava factorial ideals are principal [1]. We generalize Leriche's criterion [8] for Pólya-ness of Galois closures of pure cubic fields, to general S3-extensions of ℚ. Also, we prove for a real (resp. imaginary) Pólya S3-extension L of ℚ, at most four (resp. three) primes can be ramified. Moreover, depending on the solvability of unit norm equation over the quadratic subfield of L, we determine when these sharp upper bounds can occur.

A current research theme is to compare symbolic powers of an ideal $I$ with the regular powers of $I$. In this paper, we focus on the case where $I\,=\,{{I}_{X}}$ is an ideal defining an almost complete intersection (ACI) set of points $X$ in ${{\mathbb{P}}^{1}}\,\times \,{{\mathbb{P}}^{1}}$. In particular, we describe a minimal free bigraded resolution of a non-arithmetically Cohen-Macaulay (also non-homogeneous) set $Z$ of fat points whose support is an ACI, generalizing an earlier result of Cooper et al. for homogeneous sets of triple points. We call $Z$ a fat ACI. We also show that its symbolic and ordinary powers are equal, i.e, $I_{Z}^{\left( m \right)}\,=\,I_{Z}^{m}$ for any $m\,\ge \,1$.

Let $S\,=\,K\left[ {{x}_{1}},\,\ldots \,,\,{{x}_{n}} \right]$ be the polynomial ring in $n$-variables over a field $K$ and $I$ a monomial ideal of $S$. According to one standard primary decomposition of $I$, we get a Stanley decomposition of the monomial factor algebra $S/I$. Using this Stanley decomposition, one can estimate the Stanley depth of $S/I$. It is proved that $\text{sdept}{{\text{h}}_{s}}\left( S/I \right)\,\ge \,\text{siz}{{\text{e}}_{S}}\left( I \right)$. When $I$ is squarefree and $\text{bigsiz}{{\text{e}}_{S}}\left( I \right)\,\le \,2$, the Stanley conjecture holds for $S/I$, i.e., $\text{sdept}{{\text{h}}_{S}}\left( S/I \right)\ge \text{dept}{{\text{h}}_{S}}\left( S/I \right)$.

Recent work of Ein–Lazarsfeld–Smith and Hochster–Huneke raised the problem of which symbolic powers of an ideal are contained in a given ordinary power of the ideal. Bocci–Harbourne developed methods to address this problem, which involve asymptotic numerical characters of symbolic powers of the ideals. Most of the work done up to now has been done for ideals defining 0-dimensional subschemes of projective space. Here we focus on certain subschemes given by a union of lines in ${{\mathbb{P}}^{3}}$ that can also be viewed as points in ${{\mathbb{P}}^{1}}\times {{\mathbb{P}}^{1}}$. We also obtain results on the closely related problem, studied by Hochster and by Li and Swanson, of determining situations for which each symbolic power of an ideal is an ordinary power.

The vanishing ideal of an arrangement of linear subspaces in a vector space is considered, and the paper investigates when this ideal can be generated by products of linear forms. A combinatorial construction (blocker duality) is introduced which yields such generators in cases with a great deal of combinatorial structure, and examples are presented that inspired the work. A construction is given which produces all elements of this type in the vanishing ideal of the arrangement. This leads to an algorithm for deciding if the ideal is generated by products of linear forms. Generic arrangements of points in ${\bf P}^2$ and lines in ${\bf P}^3$ are also considered.

Let $R$ be a one-dimensional locally analytically irreducible Noetherian domain with finite residue fields. In this note it is shown that if $I$ is a finitely generated ideal of the ring $\text{Int(}R)$ of integer-valued polynomials such that for each $\text{x}\,\in \,R$ the ideal $I\text{(}x\text{)}=\{f(x)|f\in I\}$ is strongly $\text{n}$-generated, $n\,\ge \,2$, then $I$ is $\text{n}$-generated, and some variations of this result.

We consider the rings of invariants RG, where R is the symmetric algebra of a tensor product between two vector spaces over the field Fp andG = Un × Um. A polynomial algebra is constructed and these invariants provide Chern classes for the modular cohomology of Un+m.

Let k: be a perfect field such that is solvable over k. We show that a smooth, affine, factorial surface birationally dominated by affine 2-space is geometrically factorial and hence isomorphic to . The result is useful in the study of subalgebras of polynomial algebras. The condition of solvability would be unnecessary if a question we pose on integral representations of finite groups has a positive answer.

Soit A un anneau intègre de corps des fractions K et soit A[X]sub = {P(X) ∈ K[X] ; P(ℤ) ⊂ A} l'anneau des polynômes à valeurs entières sur A. Lorsque la caractéristique de A est nulle, le A-module A[X]sub est contenu dans le A-module {P(X) ∊ K[X] ; P(Z) ⊂ A] engendré par les polynômes binomiaux Bn(X) – X(X – 1) (X – n + 1)/n′. Nous caractérisons ici les anneaux de Dedekind A pour lesquels ces A-modules sont égaux.

Puis nous étudions la situation plus générale dans laquelle A[X]sub = {P(X) ∈ K[X] ; P(A0) ⊂ A} OÙ A0 désigne un anneau de Dedekind contenu dans A. Ce sont alors des polynômes généralisant les binômes de Fermat FP(X) – (Xq – X)/p qui jouent le rôle central.

Let V be a vector space of dimension n over a field k of characteristic p. Let G ⊆ Gl(V) be a finite group with p-Sylow subgroup P. G and P act on the symmetric algebra R of V. Denote the respective rings of invariants by RG and Rp. We show that if Rp is Cohen-Macaulay (CM) so also is RG, generalizing a result of M. Hochster and J. A. Eagon. If P is normal in G and G is generated by P and pseudo-reflections, we show that if RG is CM so also is Rp. However, in general, RG may even be polynomial with Rp not CM. Finally, we give a procedure for determining a set of generators for RG given a set of generators for Rp.

Let M be a free module of rank n over a commutative ring R with unit and let Σn denote the symmetric group acting on a fixed basis of M in the usual way. Let Mm denote the direct sum of m copies of M and let S be the symmetric ring of Mm over R. Then each element of Σn acts diagonally on Mm and consequently on S; denote by Xn the subgroup of Gl(Mm) so defined. The ring of invariants, SΣn, is called the ring of vector invariants by H. Weyl [ 3, Chapter II, p. 27] when R = Q. In this paper a set of generators valid over any ring R is given. This set of generators is somewhat larger than Weyl's. It is interesting to note that, over the integers, his algebra and SΣn have the same Hilbert-Poincaré series, are equal after tensoring with the rationals, and have the same fraction fields, although they are not equal.

We give a simple construction of non-Noetherian Hilbert domains whose maximal ideals are all finitely generated. Such domains we call unruly Hilbert domains.