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 Gröbner bases for symmetric ideals in the infinite dimensional polynomial ring $R$. This allows for symbolic computation in a new class of rings. In particular, we solve the ideal membership problem for symmetric ideals of $R$.
Let $K$ be a field equipped with a valuation. Tropical varieties over $K$ can be defined with a theory of Gr{ö}bner bases taking into account the valuation of $K$.Because of the use of the valuation, the theory of tropical Gr{ö}bner bases has proved to provide settings for computations over polynomial rings over a $p$-adic field that are more stable than that of classical Gr{ö}bner bases.Beforehand, these strategies were only available for homogeneous polynomials. In this article, we extend the F5 strategy to a new definition of tropical Gr{ö}bner bases in an affine setting.We provide numerica…
Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{ö}bner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Gr{ö}bner bases has proved to provide settings for computations over polynomial rings over a p-adic field that are more stable than that of classical Gr{ö}bner bases. In this article, we investigate how the FGLM change of ordering algorithm can be adapted to the tropical setting. As the valuations of the polynomial coefficients are taken into account, the classical FGLM algorithm…
Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{ö}bner bases taking into account the valuation of K. While generalizing the classical theory of Gr{ö}bner bases, it is not clear how modern algorithms for computing Gr{ö}bner bases can be adapted to the tropical case. Among them, one of the most efficient is the celebrated F5 Algorithm of Faug{è}re. In this article, we prove that, for homogeneous ideals, it can be adapted to the tropical case. We prove termination and correctness. Because of the use of the valuation, the theory of tropical …
Let $(f\_1,\dots, f\_s) \in \mathbb{Q}\_p [X\_1,\dots, X\_n]^s$ be a sequence of homogeneous polynomials with $p$-adic coefficients. Such system may happen, for example, in arithmetic geometry. Yet, since $\mathbb{Q}\_p$ is not an effective field, classical algorithm does not apply.We provide a definition for an approximate Gr{ö}bner basis with respect to a monomial order $w.$ We design a strategy to compute such a basis, when precision is enough and under the assumption that the input sequence is regular and the ideals $\langle f\_1,\dots,f\_i \rangle$ are weakly-$w$-ideals. The conjecture of…
We consider a finite dimensional representation of the dihedral group $D_{2p}$ over a field of characteristic two where $p$ is an odd prime and study the corresponding Hilbert ideal $I_H$. We show that $I_H$ has a universal Gr\" {o}bner basis consisting of invariants and monomials only. We provide sharp bounds for the degree of an element in this basis and in a minimal generating set for $I_H$. We also compute the top degree of coinvariants.
The discrete polymatroids and their base rings are studied recently in many papers (see \cite{HH}, \cite{HHV}, \cite{V1}, \cite{V2}). It is important to give conditions when the base ring associated to a transversal polymatroid is Gorenstein (see \cite{HH}). In \cite{SA} we introduced a class of such base rings. In this paper we note that an intersection of such base rings (introduced in \cite{SA}) is Gorenstein and give necessary and sufficient conditions for the intersection of two base rings from \cite{SA} to be still a base ring of a transversal polymatroid. Also, we compute the $a$-invari…
In this paper we determine the facets of the polyhedral cone generated by the exponent set of the monomials defining the base ring associated to some transversal polymatroid. We need the description of these facets to find the canonical module of the base ring which is expressed in terms of the relative interior of the cone. This would allow us to compute the $a$-invariant of those base rings. The results presented were discovered by extensive computer algebra experiments performed with {\it{Normaliz}} \cite{BK}.
In this thesis, we focus on the study of the base rings associated to some transversal polymatroids. A transversal polymatroid is a special kind of discrete polymatroid. Discrete polymatroids were introduced by Herzog and Hibi \cite{HH} in 2002.
In this note we consider monoidal complexes and their associated algebras, called toric face rings. These rings generalize Stanley-Reisner rings and affine monoid algebras. We compute initial ideals of the presentation ideal of a toric face ring, and determine its graded Betti numbers. Our results generalize celebrated theorems of Hochster in combinatorial commutative algebra.
Let $k$ be a field. We determine the ideals $I$ in a finitely generated graded $k$-algebra $A$, whose associated graded rings are isomorphic to $A$. Also we compute the graded local cohomologies of the Rees rings $A[I t]$ and give the condition for $A[I t]$ to be generalized Cohen-Macaulay.
The theory of standard bases in polynomial rings with coefficients in a ring R with respect to local orderings is developed. R is a commutative Noetherian ring with 1 and we assume that linear equations are solvable in R.
In this paper we give versions of Hilbert's syzygy theorem for finitely generated modules over polynomial rings over direct product of principal ideal rings.
In this paper, the principal tool to describe transversal polymatroids with Gorenstein base ring is polyhedral geometry, especially the $Danilov-Stanley$ theorem for the characterization of canonical module. Also, we compute the $a-invariant$ and the Hilbert series of base ring associated to this class of transversal polymatroids.
In this paper we will define analogs of Gröbner bases for $R$-subalgebras and their ideals in a polynomial ring $R[x_1,\ldots,x_n]$ where $R$ is a noetherian integral domain with multiplicative identity and in which we can determine ideal membership and compute syzygies. The main goal is to present and verify algorithms for constructing these Gröbner basis counterparts. As an application, we will produce a method for computing generators for the first syzygy module of a subset of an $R$-subalgebra of $R[x_1,\ldots,x_n]$ where each coordinate of each syzygy must be an element of the subalgebra.
Let $R$ be a commutative ring and let $I$ be an ideal of $R$. In this paper, we introduce the cozero-divisor graph $\acuteΓ_I(R)$ of $R$ and obtain some related results.
Over an arbitrary commutative ring, correspondences among three sets, the set of trace ideals, the set of stable ideals, and the set of birational extensions of the base ring, are studied. The correspondences are well-behaved, if the base ring is a Gorenstein ring of dimension one. It is shown that with one extremal exception, the surjectivity of one of the correspondences characterizes the Gorenstein property of the base ring, provided it is a Cohen-Macaulay local ring of dimension one. Over a commutative Noetherian ring, a characterization of modules in which every submodule is a trace modul…
We develop the theory of Gröbner bases for ideals in a polynomial ring with countably infinite variables over a field. As an application we reconstruct some of the one-one correspondences among various sets of partitions by using division algorithm.
Let R be a Stanley-Reisner ring (that is, a reduced monomial ring) with coefficients in a domain k, and K its associated simplicial complex. Also let D_k(R) be the ring of k-linear differential operators on R. We give two different descriptions of the two-sided ideal structure of D_k(R) as being in bijection with certain well-known subcomplexes of K; one based on explicit computation in the Weyl algebra, valid in any characteristic, and one valid in characteristic p based on the Frobenius splitting of R. A result of Traves [Tra99] on the D_k(R)-module structure of R is also given a new proof a…
The theory of border bases for zero-dimensional ideals has attracted several researchers in symbolic computation due to their numerical stability and mathematical elegance. As shown in (Francis & Dukkipati, J. Symb. Comp., 2014), one can extend the concept of border bases over Noetherian rings whenever the corresponding residue class ring is finitely generated and free. In this paper we address the following problem: Can the concept of border basis over Noetherian rings exists for ideals when the corresponding residue class rings are finitely generated but need not necessarily be free modules?…
We study the behaviour of the finiteness of the set of associated primes of local cohomology modules, more generally of Lyubeznik functors, under various ring extensions. At first, we review the results for flat and faithfully flat extensions and we present new applications of them. Then, we focus how the finiteness property of the set of associated primes of local cohomology modules and Lyubeznik functors is transferred from extended ring to the base ring of pure and cyclically pure ring extensions. We show that finiteness property can be transferred from a ring to its pure local subring and …
Let $R$ be a commutative chain ring. We use a variation of Gröbner bases to study the lattice of ideals of $R[x]$. Let $I$ be a proper ideal of $R[x]$. We are interested in the following two questions: When is $R[x]/I$ Frobenius? When is $R[x]/I$ Frobenius and local? We develop algorithms for answering both questions. When the nilpotency of $\text{rad}\,R$ is small, the algorithms provide explicit answers to the questions.
Let $G$ be a finite graph and $K[G]$ the edge ring of $G$. Based on the technique of Gröbner bases and initial ideals, it will be proved that, given integers $f$ and $d$ with $7 \leq f \leq d$, there exists a finite graph $G$ on $[d]={1,...,d}$ with $\depth K[G] = f$ and with $\Krull-dim K[G] = d$.
We present basic properties of Gröbner bases of submodules of a free module of finite rank over a polynomial ring $R$ with coefficients in a graded truncated discrete valuations ring $A$. As an application, we give a criterion for a finitely generated $R$-module to be flat over $A$. Its non-graded version is also given.
We introduce the notion of monomial group action and study some of its consequences for Groebner basis theory. As an application we prove a conjecture of V. Batyrev and O. Popov describing the Cox rings of Del Pezzo surfaces (of degree at least 3) as quotients of a polynomial ring by an ideal generated by quadrics.
In this paper, we extend the characterization of $\mathbb{Z}[x]/\ < f \ >$, where $f \in \mathbb{Z}[x]$ to be a free $\mathbb{Z}$-module to multivariate polynomial rings over any commutative Noetherian ring, $A$. The characterization allows us to extend the Gröbner basis method of computing a $\Bbbk$-vector space basis of residue class polynomial rings over a field $\Bbbk$ (Macaulay-Buchberger Basis Theorem) to rings, i.e. $A[x_1,\ldots,x_n]/\mathfrak{a}$, where $\mathfrak{a} \subseteq A[x_1,\ldots,x_n]$ is an ideal. We give some insights into the characterization for two special cases, when $…
In this paper, we investigate the structure of skew power series rings of the form $S = R[[x;σ,δ]]$, where $R$ is a complete filtered ring and $(σ,δ)$ is a skew derivation respecting the filtration. Our main focus is on the case in which $σδ= δσ$, and we aim to use techniques in non-commutative valuation theory to address the long-standing open question: if $P$ is an invariant prime ideal of $R$, is $PS$ a prime ideal of $S$? When $R$ has characteristic $p$, our results reduce this to a finite-index problem. We also give preliminary results in the "Iwasawa algebra" case $δ= σ- \mathrm{id}_R$ i…
Let $K$ be a Gorenstein noetherian ring of finite Krull dimension, and consider the category of cohomologically noetherian commutative differential graded rings $A$ over $K$, such that $H^0(A)$ is essentially of finite type over $K$, and $A$ has finite flat dimension over $K$. We extend Grothendieck's twisted inverse image pseudofunctor to this category by generalizing the theory of rigid dualizing complexes to this setup. We prove functoriality results with respect to cohomologically finite and cohomologically essentially smooth maps, and prove a perfect base change result for $f^{!}$ in this…
We obtain two equivalent conditions for m polynomials in n variables to form a p-basis of a ring of constants of some polynomial K-derivation, where K is a UFD of characteristic p>0. One of these conditions involves jacobians, and the second - some properties of factors. In the case of m=n this extends the known theorem of Nousiainen, and we obtain a new formulation of the jacobian conjecture in positive characteristic.
Consider the diagonal action of $SL_n(K)$ on the affine space $X=V^{\oplus m}\oplus (V^*)^{\oplus q}$ where $V=K^n, K$ an algebraically closed field of arbitrary characteristic and $m,q>n$. We construct a "standard monomial" basis for the ring of invariants $K[X]^{SL_n(K)}$. As a consequence, we deduce that $K[X]^{SL_n(K)}$ is Cohen-Macaulay. We also present the first and second fundamental theorems for $SL_n(K)$-actions.
We introduce a general constructive method to find a p-basis (and the Ulm invariants) of a finite Abelian p-group M. This algorithm is based on Groebner bases theory. We apply this method to determine the additive structure of indecomposable modules over the following Dedeking-like rings: ZCp, where Cp is the cyclic group of order a prime p, and the p-pullback Z --> Zp <-- Z of Z+Z.
In this paper, we examine the behavior of ideal-adic separatedness and completeness under certain ring extensions using trace map. Then we prove that adic completeness of a base ring is hereditary to its ring extension under reasonable conditions. We aim to give many results on ascent and descent of certain ring theoretic properties under completion. As an application, we give conceptual details to the proof of the almost purity theorem for Witt-perfect rings by Davis and Kedlaya. Witt-perfect rings have the advantage that one does not need to assume that the rings are complete and separated.
In this paper we briefly discuss \Rings --- an efficient lightweight library for commutative algebra. Polynomial arithmetic, GCDs, polynomial factorization and Gröbner bases are implemented with the use of modern asymptotically fast algorithms. \Rings can be easily interacted or embedded in applications in high-energy physics and other research areas via a simple API with fully typed hierarchy of algebraic structures and algorithms for commutative algebra. The use of the Scala language brings a quite novel powerful, strongly typed functional programming model allowing to write short, expressiv…
