Searcharxiv⌕ Search

arXiv subjects

Neena Gupta

Publications and source records attributed to Neena Gupta.

18 recordsLinked to original sources
arXiv2024

On the family of affine threefolds $a(x)y=F(x,z,t)$

In recent decades, linear affine threefolds have enabled researchers to solve some of the challenging problems on affine spaces. Koras-Russell threefolds, especially the Russell Cubic over $\mathbb{C}$ and Asanuma threefolds over a field of positive characteristic, are striking examples of such linear threefolds.In this paper, we apply tools from $K$-theory and theory of $\mathbb{G}_a$-actions to linear threefolds of the form $G:=a(X)Y-F(X,Z,T)\in k[X,Y,Z,T]$, over an arbitrary field $k$. We give some equivalent conditions for $G$ to be a hyperplane (i.e., $k[X,Y,Z,T]/(G)=k^{[3]}$) in the following cases: (i) $k$ is a field of characteristic zero (ii) $k$ is an arbitrary field and $a(X)$ has only multiple roots. We also establish the Abhyankar-Sathaye Conjecture affirmatively in these cases.

math.AG↗
arXiv2024

On embedding of linear hypersurfaces

Linear hypersurfaces over a field $k$ have been playing a central role in the study of some of the challenging problems on affine spaces. Breakthroughs on such problems have occurred by examining two difficult questions on linear polynomials of the form $H:=α(X_1,\dots,X_m)Y - F(X_1,\dots, X_m,Z,T)\in D:=k[X_1,\ldots,X_m, Y,Z,T]$: (i) Whether $H$ defines a closed embedding of $\mathbb{A}^{m+2}$ into $\mathbb{A}^{m+3}$, i.e., whether the affine variety $\mathbb{V}\subseteq \mathbb{A}^{m+3}_k$ defined by $H$ is isomorphic to $\mathbb{A}^{m+2}_k$. (ii) If $H$ defines a closed embedding $\mathbb{A}^{m+2}\hookrightarrow \mathbb{A}^{m+3}$ then whether $H$ is a coordinate in $D$. Question (i) connects to the Characterization Problem of identifying affine spaces among affine varieties; Question (ii) is a special case of the formidable Embedding Problem for affine spaces. In their earlier work the first two authors had addressed these questions when $α$ is a monomial of the form $α(X_1,\ldots,X_m) = X_1^{r_1}\dots X_m^{r_m}$; $r_i>1, 1 \leqslant i \leqslant m$ and $F$ is of a certain type. In this paper, using $K$-theory and $\mathbb{G}_a$-actions, we address these questions for a wider family of linear varieties. In particular, we obtain certain families of higher dimensional hyperplanes $H$ satisfying the Abhyankar Sathaye conjecture on the Embedding problem. For instance, we show that when the characteristic of $k$ is zero, $F \in k[Z,T]$ and $H$ defines a hyperplane, then $H$ is a coordinate in $D$ along with $X_1, X_2, \dots, X_m$. Our results in arbitrary characteristic yield counterexamples to the Zariski Cancellation Problem in positive characteristic.

math.AG↗
arXiv2024

On rigidity of Pham-Brieskorn surfaces

It is well known that, over an algebraically closed field $k$ of characteristic zero, for any three integers $a,b,c\geq 2$, any Pham-Brieskorn surface $B_{(a,b,c)}:= k[X,Y,Z]/(X^a + Y^b + Z^c)$ is rigid when at most one of $a,b,c$ is 2 and stably rigid when $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\leq 1$. In this paper we consider Pham-Brieskorn domains over an arbitrary field $k$ of characteristic $p\geq 0$ and give sufficient conditions on $(a,b,c)$ for which any Pham-Brieskorn domain $B_{(a,b,c)}$ is rigid. This gives an alternative approach to showing that there does not exist any non-trivial exponential map on $k[X,Y,Z,T]/(X^mY+T^{p^rq} + Z^{p^e})= k[x,y,z,t]$, for $m,q>1$, $p\nmid mq$ and $e>r\geq 1$, fixing $y$, a crucial result used in the paper "On the cancellation problem for the affine space $\mathbb{A}^3$ in characteristic $p$" by first author, to show that the Zariski Cancellation Problem (ZCP) does not hold for the affine $3$-space. We also provide a sufficient condition for $B_{(a,b,c)}$ to be stably rigid. Along the way we prove that for integers $a,b,c\geq 2$ with $gcd(a,b,c) = 1$ and for $F(Y)\in k[Y]$, the ring $k[X,Y,Z]/(X^aY^b + Z^c+ F(Y))$ is a rigid domain.

math.AG↗
arXiv2023

Retracts of Laurent polynomial rings

Let $R$ be an integral domain and $B=R[x_1,\ldots,x_n]$ be the polynomial ring. In this paper, we consider retracts of $B[1/M]$ for a monomial $M$. We show that (1) if $M=\prod_{i=1}^nx_i$, then every retract is a Laurent polynomial ring over $R$, (2) if $R$ is a perfect field and $n=3$, then every retract is isomorphic to $R[y_1^{\pm1},\ldots,y_s^{\pm1},z_1,\ldots,z_t]$ for some $s,t\geq 0$.

math.AC↗
arXiv2023

On the triviality of an $\mathbb{A}^2$-fibration over a DVR

In this paper we show that any $\mathbb{A}^2$-fibration over a discrete valuation ring which is also an $\mathbb{A}^2$-form is necessarily a polynomial ring. Further we show that separable $\mathbb{A}^2$-forms over PIDs are trivial.

math.AC↗
arXiv2023

On the triviality of a family of linear hyperplanes

Let $k$ be a field, $m$ a positive integer, $\mathbb{V}$ an affine subvariety of $\mathbb{A}^{m+3}$ defined by a linear relation of the form $x_{1}^{r_{1}}\cdots x_{m}^{r_{m}}y=F(x_{1}, \ldots , x_{m},z,t)$, $A$ the coordinate ring of $\mathbb{V}$ and $G= X_1^{r_1}\cdots X_m^{r_m}Y-F(X_1, \dots, X_m,Z,T)$. In \cite{com}, the second author had studied the case $m=1$ and had obtained several necessary and sufficient conditions for $\mathbb{V}$ to be isomorphic to the affine 3-space and $G$ to be a coordinate in $k[X_1, Y,Z,T]$. In this paper, we study the general higher-dimensional variety $\mathbb{V}$ for each $m \geqslant 1$ and obtain analogous conditions for $\mathbb{V}$ to be isomorphic to $\mathbb{A}^{m+2}$ and $G$ to be a coordinate in $k[X_1, \dots, X_m, Y,Z,T]$, under a certain hypothesis on $F$. Our main theorem immediately yields a family of higher-dimensional linear hyperplanes for which the Abhyankar-Sathaye Conjecture holds. We also describe the isomorphism classes and automorphisms of integral domains of the type $A$ under certain conditions. These results show that for each $d \geqslant 3$, there is a family of infinitely many pairwise non-isomorphic rings which are counterexamples to the Zariski Cancellation Problem for dimension $d$ in positive characteristic.

math.AC↗
arXiv2023

On Generalised Danielewski and Asanuma varieties

In this paper we extend a result of Dubouloz on the Cancellation Problem in higher dimensions ($\geqslant 2$) over the field of complex numbers to fields of arbitrary characteristic. We then apply the generalised result to describe the Makar-Limanov and Derksen invariant of generalised Asanuma varieties under certain hypotheses. We also establish a necessary and sufficient condition for certain generalised Asanuma varieties to be isomorphic to polynomial rings.

math.AC↗
arXiv2020

On finite generation of Noetherian algebras over two-dimensional regular local rings

Let $R$ be a complete regular local ring with an algebraically closed residue field and let $A$ be a Noetherian $R$-subalgebra of the polynomial ring $R[X]$. It has been shown in \cite{DO2} that if $\dim R=1$, then $A$ is necessarily finitely generated over $R$. In this paper, we give necessary and sufficient conditions for $A$ to be finitely generated over $R$ when $\dim R=2$ and present an example of a Noetherian normal non-finitely generated $R$-subalgebra of $R[X]$ over $R= {\mathbb C}[[u, v]]$.

math.AC↗
arXiv2019

Some results on retracts of polynomial rings

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.

math.AC↗
arXiv2019

On Double Danielewski Surfaces and the Cancellation Problem

We study a two-dimensional family of affine surfaces which are counter-examples to the Cancellation Problem. We describe the Makar-Limanov invariant of these surfaces, determine their isomorphism classes and characterize the automorphisms of these surfaces.

math.AC↗
arXiv2019

On Separable $\A^2$ and $\A^3$-forms

In this paper, we will prove that any $\A^3$-form over a field $k$ of characteristic zero is trivial provided it has a locally nilpotent derivation satisfying certain properties. We will also show that the result of T. Kambayashi on the triviality of separable $\A^2$-forms over a field $k$ extends to $\A^2$-forms over any one-dimensional Noetherian domain containing $\bQ$.

math.AC↗
arXiv2017

Nice derivations over principal ideal domains

In this paper we investigate to what extent the results of Z. Wang and D. Daigle on nice derivations of the polynomial ring in three variables over a field k of characteristic zero extend to the polynomial ring over a PID R, containing the field of rational numbers. One of our results shows that the kernel of a nice derivation on the polynomial ring in four variables over k of rank at most three is a polynomial ring over k.

math.AC↗
arXiv2015

Some K-theoretic properties of the kernel of a locally nilpotent derivation on k[X_1, \dots, X_4]

Let k be an algebraically closed field of characteristic zero, D a locally nilpotent derivation on the polynomial ring k[X_1, X_2,X_3,X_4] and A the kernel of D. A question of M. Miyanishi asks whether projective modules over A are necessarily free. Implicit is a subquestion: whether the Grothendieck group K_0(A) is trivial. In this paper we shall demonstrate an explicit k[X_1]-linear fixed point free locally nilpotent derivation D of k[X_1,X_2, X_3, X_4] whose kernel A has an isolated singularity and whose Grothendieck group K_0(A) is not finitely generated; in particular, there exists an infinite family of pairwise non-isomorphic projective modules over the kernel A. We shall also show that, although Miyanishi's original question does not have an affirmative answer in general, suitably modified versions of the question do have affirmative answers when D annihilates a variable. For instance, we shall establish that in this case the groups G_0(A) and G_1(A) are indeed trivial. Further, we shall see that if the above kernel A is a regular ring, then A is actually a polynomial ring over k; in particular, by the Quillen-Suslin theorem, Miyanishi's question has an affirmative answer. Our construction involves rings defined by the relation u^mv=F(z,t), where F(Z,T) is an irreducible polynomial in k[Z,T]. We shall show that a necessary and sufficient condition for such a ring to be the kernel of a k[X_1]-linear locally nilpotent derivation D of a polynomial ring k[X_1,...,X_4] is that F defines a polynomial curve.

math.AC↗