Let G be a finite p-subgroup of GL(V), where p = char(F), and V is finite-dimensional over the field F.
Let S(V) be the symmetric algebra of V, S(V)^G the subring of G-invariants, and V* the F-dual space of V.
The following presents our solution to the above question.
Theorem A. Suppose dim(V)=3. Then S(V)^G is a polynomial ring if and only if G is generated by transvections.
Theorem B. Suppose dim(V) > 3. Then S(V)^G is a polynomial ring if and only if:
(1) S(V)^{G_U} is a polynomial ring for each subspace U of V* with dim(U)=2, where G_U = {g in G | g(u) = u, for all u in U}, and
(2) S(V)^G is Cohen-Macaulay.
Alternatively, (1) can be replaced by the equivalent condition:
(3) dim (non-smooth locus of S(V)^G) < 2.
Let R be a commutative ring with1 and R[X] be the polynomial ring over R. We determine the underlying ring R over which the polynomial ring R[X] has the property that all its prime ideals are set theoretic complete intersections.
We say that a polynomial automorphism $ϕ$ in $n$ variables is stably co-tame if the tame subgroup in $n$ variables is contained in the subgroup generated by $ϕ$ and affine automorphisms in $n+1$ variables. In this paper, we give conditions for stably co-tameness of polynomial automorphisms.
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We study Hilbert's fourteenth problem from a geometric point of view. Nagata's celebrated counterexample demonstrates that for an arbitrary group action on a variety the ring of invariant functions need not be isomorphic to the ring of functions of an affine variety. Nevertheless one can prove that such a ring of invariants is always isomorphic to the ring of functions on a quasi-affine variety. Conversely, for a given quasi-affine variety V there exists always an action of the additive group on some affine variety W such that the ring of functions of V is isomorphic to the ring of invariant f…
An automorphism $F$ of the polynomial ring in $n$ variables over a field of characteristic zero is said to be {\it co-tame} if the subgroup of the automorphism group of the polynomial ring generated by $F$ and affine automorphisms contains the tame subgroup. There exist many examples of such an $F$, and several sufficient conditions for co-tameness are already known. In 2015, Edo-Lewis gave the first example of a non-cotame automorphism, which is a tame automorphism of the polynomial ring in three variables. In this paper, we give the first example of a non-cotame automorphism which is not tam…
We previously showed that the inverse limit of standard-graded polynomial rings with perfect coefficient field is a polynomial ring, in an uncountable number of variables. In this paper, we show that the same result holds with arbitrary coefficient field. We also prove an analogous result for ultraproducts of polynomial rings.
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The lambda-ring cohomology in dimensions 0 and 1 of certain truncated polynomial filtered lambda-rings are computed. The dimension 1 case is related to deformations of lambda-rings.
An example is given of a UFD which has infinitely generated Derksen invariant. The ring is \textquotedblleft almost rigid\textquotedblright\ meaning that the Derksen invariant is equal to the Makar-Limanov invariant. Techniques to show that a ring is (almost) rigid are discussed, among which is a generalization of Mason's abc-theorem.
In this paper, we extend the well-known Hilbert's syzygy theorem to the Gorenstein homological dimensions of rings. Also, we study the Gorenstein homological dimensions of direct product of rings, which gives examples of non-Noetherian rings of finite Gorenstein dimensions and infinite classical weak dimension.
We prove an explicit uniform Chevalley theorem for direct summands of graded polynomial rings in mixed characteristic. Our strategy relies on the introduction of a new type of differential powers, which do not require the existence of a p-derivation on the direct summand.
In this paper, we extend the well-known Hilbert's syzygy theorem to the Gorenstein homological dimensions of rings. Also, we study the Gorenstein homological dimensions of direct products of rings. Our results generate examples of non-Noetherian rings of finite Gorenstein dimensions and infinite classical weak dimension.
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In this paper, we first consider the relationship between a polynomial ring $B$ over a Noetherian domain $R$ and the ring of invariants $A$ of a ${\mathbb G}_a$-action on $B$, when $A$ occurs as a retract of $B$. Next, we study retracts of a polynomial ring in general and address the questions of D. L. Costa raised in \cite{C}. Finally, we examine the behaviour of ideals and certain properties of rings under retractions.
Given a certain factorization property of a ring $R$, we can ask if this property extends to the polynomial ring over $R$ or vice versa. For example, it is well known that $R$ is a unique factorization domain if and only if $R[X]$ is a unique factorization domain. If $R$ is not a domain, this is no longer true. In this paper we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization.
The ring of integer-valued polynomials on an arbitrary integral domain is well-studied. In this paper we initiate and provide motivation for the study of integer-valued polynomials on commutative rings and modules. Several examples are computed, including the integer-valued polynomials over the ring $R[T_1,\ldots, T_n]/(T_1(T_1-r_1), \ldots, T_n(T_n-r_n))$ for any commutative ring $R$ and any elements $r_1, \ldots, r_n$ of $R$, as well as the integer-valued polynomials over the Nagata idealization $R(+)M$ of $M$ over $R$, where $M$ is an $R$-module such that every non-zerodivisor on $M$ is a n…
The principal minors of the Toeplitz matrix $\left( x_{i-j+1}\right)_{1\le i,j,\le n}$, where $x_0=1, x_k=0$ if $k\le -1$, directly determine an involution of the polynomial ring $R[x_1, ... ,x_n]$ over any commutative ring $R$.
Given a one-dimensional Cohen-Macaulay local ring we compare several sets of invariants (micro-invariants, Apery invariants and invariants of the tangent cone) and give explicit formulas relating them. We show that, in fact, they coincide if and only if the tangent cone of the ring is Cohen-Macaulay. Some explicit computations are also given, particularly in the case of semigroup rings.
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.
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The properties of the intersection algebra of two principal monomial ideals in a polynomial ring are investigated in detail. Results are obtained regarding the Hilbert series and the canonical ideal of the intersection algebra using methods from the theory of diophantine linear equations with integer coefficients.
We know that for a finite field $F$, every function on $F$ can be given by a polynomial with coefficients in $F$. What about the converse? i.e. if $R$ is a ring (not necessarily commutative or with unity) such that every function on $R$ can be given by a polynomial with coefficients in $R$, can we say $R$ is a finite field ? We show that the answer is yes, and that in fact it is enough to only require that all bijections be given by polynomials. If we allow our rings to have unity, we show that the property that all characteristic functions can be given by polynomials actually characterizes fi…
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We review the definition of D-rings introduced by H. Gunji & D. L. MacQuillan. We provide an alternative characterization for such rings that allows us to give an elementary proof of that a ring of algebraic integers is a D-ring. Moreover, we give a characterization for D-rings that are also unique factorization domains to determine divisibility of polynomials using polynomial evaluations.
In this paper, we prove several results on the finiteness of local cohomology of polynomial and formal power series rings. In particular, we give a partial affirmative answer for a question of L. Núñez-Betancourt in [J. Algebra 399 (2014), 770--781].
We prove constructively that for any finite-dimensional commu- tative ring R, every stably free module over R[X;X^{1}] of rank > dim R is free, i.e., R[X;X^{-1}] is (dimR)-Hermite.
Motivated by Carmichael numbers, we say that a finite ring $R$ is a Carmichael ring if $a^{|R|}=a$ for any $a \in R$. We then call an ideal $I$ of a ring $R$ as a Carmichael ideal if $R/I$ is a Carmichael ring, and a Carmichael element of $R$ means it generates a Carmichael ideal. In this paper, we determine the structure of Carmichael rings and prove a generalization of Korselt's criterion for Carmichael ideals in Dedekind domains. We also study Carmichael elements of polynomial rings over finite fields (called Carmichael polynomials) by generalizing various classical results. For example, we…
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We investigate the transfer of w-stability and Clifford w-regularity from a domain D to the polynomial ring D[X]. We show that these two properties pass from D to D[X] when D is either integrally closed or it is Mori and w-divisorial.
Using the functional interpretation from proof theory, we analyze nonconstructive proofs of several central theorems about polynomial and differential polynomial rings. We extract effective bounds, some of which are new to the literature, from the resulting proofs. In the process we discuss the constructive content of Noetherian rings and the Nullstellensatz in both the classical and differential settings. Sufficient background is given to understand the proof-theoretic and differential-algebraic framework of the main results.
Using polynomial evaluation, we give some useful criteria to answer questions about divisibility of polynomials. This allows us to develop interesting results concerning the prime elements in the domain of coefficients. In particular, it is possible to prove that under certain conditions, the domain of coefficients must have infinitely many prime elements. We give alternative characterizations for D-rings and present various examples.
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.
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…
Using the description of the Frobenius limit of modules over the ring of invariants under an action of a finite group on a polynomial ring over a field of characteristic $p>0$ developed by Symonds and the author, we give a characterization of the ring of invariants with a positive dual $F$-signature. Combining this result and Kemper's result on depths of the ring of invariants under an action of a permutation group, we give an example of an $F$-rational, but non-$F$-regular ring of invariants under the action of a finite group.
In this article it is determined which integral reflection representations of the symmetric groups and the primitive complex reflection groups of degree $2$ have rings of invariants which are isomorphic to polynomial rings.
This is an expository paper in which it is proved that, for every infinite field ${\mathbf{F}}$, the polynomial ring ${\mathbf{F}}[t_1,\ldots, t_n]$ has Krull dimension $n$. The proof uses only "high school algebra" and the rudiments of undergraduate "abstract algebra."
Let $R=k[x_1,..., x_n]$ be a polynomial ring over a field $k$ of characteristic $p>0,$ and let $I=(f_1,...,f_s)$ be an ideal of $R.$ We prove that every associated prime $P$ of $H^i_I(R)$ satisfies $\text{dim}R/P\geqslant n-\sum\text{deg}f_i.$ In characteristic 0 the question is open.
