arXiv subjects
Publications and source records attributed to Sumit Chandra Mishra.
In this article, we characterize the isotropy groups of certain special monomial and Jouanolou-type derivations of polynomial rings over fields of characteristic zero. Under suitable conditions, we determine the structure of these isotropy groups.
Let $k$ be a field of characteristic zero. Let $m$ and $α$ be positive integers. For $n\geq 2$, let $R_n=k[x_1,x_2,\dots,x_n]$ with the $k$-derivation $d_n$ given by $d_n=(1-x_1x_2^α)\partial_{x_1}+x_1^m\partial_{x_2}+x_2\partial_{x_3}+\dots+x_{n-1}\partial_{x_n}$. We prove that for integers $m\geq 2$ and $α\geq 1$, $d_n$ is a simple derivation on $R_n$ and $d_n(R_n)$ contains no units. This generalizes a result of D. A. Jordan. We also show that the isotropy group of $d_n$ is conjugate to a subgroup of translations.
Let $k$ be a field of characteristic zero, and let $i$ and $n$ be positive integers with $i\geq 2$ and $n>i$. Consider a non-invertible $k$-derivation $d_i$ of the polynomial ring $k[x_1,\ldots,x_i]$. Let $d_n$ be an extension of $d_i$ to a derivation of $k[x_1,\ldots, x_n]$ such that $d_n(x_j)\in k[x_{j-1}]\setminus k$ for each $j$ with $i+1 \leq j\leq n$. In this article, we undertake a systematic study of the isotropy groups associated with such non-invertible derivations. We establish sufficient conditions on $d_i$ under which the isotropy group of the non-invertible simple derivation $d_n$ is conjugate to a subgroup of translations.
We study residually transcendental extensions of a valuation $v$ on a field $E$ to function fields of hyperelliptic curves over $E$. We show that $v$ has at most finitely many extensions to the function field of a hyperelliptic curve over $E$, for which the residue field extension is transcendental but not ruled, assuming that the residue characteristic of $v$ is either zero or greater than the degree of the hyperelliptic curve.
Let $K$ be a complete discretely valued field with the residue field $κ$. Assume that cohomological dimension of $κ$ is less than or equal to $1$ (for example, $κ$ is an algebraically closed field or a finite field). Let $F$ be the function field of a curve over $K$. Let $n$ be a squarefree positive integer not divisible by char$(κ)$. Then for any two degree $n$ abelian extensions, we prove that the local-global principle holds for the associated multinorm torus with respect to discrete valuations. Let $\mathscr{X}$ be a regular proper model of $F$ such that the reduced special fibre $X$ is a union of regular curves with normal crossings. Suppose that $κ$ is algebraically closed with $char(κ)\neq 2$. If the graph associated to $\mathscr{X}$ is a tree (e.g. $F = K(t)$) then we show that the same local-global principle holds for the multinorm torus associated to finitely many abelian extensions where one of the extensions is quadratic and others are of degree not divisible by $4$.
Let $k$ be a field with $u$-invariant $\leq2$. Assume further that $k$ is not quadratically closed, $\mathsf{char}(k)\neq 2$ and $|k|\geq 5$. It is known that the covering number of both $\text{SL}_2(k)$ and $\text{PSL}_2(k)$ is three, while their extended covering number is four. In this article, we completely describe the product of two conjugacy classes in $\text{SL}_2(k)$ and $\text{PSL}_2(k)$. Further, we also describe the product of three conjugacy classes (at least two of which are distinct) in $\text{SL}_2(k)$ and $\text{PSL}_2(k)$.
It is shown that a valuation of residue characteristic different from $2$ and $3$ on a field $E$ has at most one extension to the function field of an elliptic curve over $E$, for which the residue field extension is transcendental but not ruled. The cases where such an extension is present are characterised.
Given integers $k,l\geq 2$, where either $l$ is odd or $k$ is even, let $n(k,l)$ denote the largest integer $n$ such that each element of $A_n$ is a product of $k$ many $l$-cycles. In 2008, M. Herzog, G. Kaplan and A. Lev conjectured that $\lfloor \frac{2kl}{3} \rfloor \leq n(k,l)\leq \lfloor \frac{2kl}{3}\rfloor+1$. It is known that the conjecture holds when $k=2,3,4$. Moreover, it is also true when $3\mid l$. In this article, we determine the exact value of $n(k,l)$ when $3\nmid l$ and $k\geq 5$. As an immediate consequence, we get that $n(k,l)<\lfloor \frac{2kl}{3}\rfloor$ when $k\geq 5$, which shows that the above conjecture is not true in general. In fact, the difference between the exact value of $n(k,l)$ and the conjectured value grows linearly in terms of $k$. Our results also generalize the case of $k=2,3,4$.
Given integers $k,l\geq 2$, where either $l$ is odd or $k$ is even, let $n(k,l)$ denote the largest integer $n$ such that each element of $A_n$ is a product of $k$ many $l$-cycles. In 2008, M. Herzog, G. Kaplan and A. Lev proved that if $k,l$ both are odd, $3\mid l$ and $l>3$, then $n(k,l)=\frac{2}{3}kl$. They further conjectured that if $k$ is even and $3\mid l$, then $n(k,l)=\frac{2}{3}kl+1$. In this article, we prove this conjecture. We also prove that $n(k,3)=2k+1$ if $k$ is odd.
In this article, we provide counterexamples to a conjecture of M. Pellegrini and P. Shumyatsky which states that each coset of the centralizer of an involution in a finite non-abelian simple group $G$ contains an odd order element, unless $G=\text{PSL}(n,2)$ for $n\geq 4$. More precisely, we show that the conjecture does not hold for the alternating group $A_{8n}$ for all $n\geq 2$.
Let K be a complete discretely valued field with residue field k and F be a function field of a curve over K. Let L/F be a Galois extension of degree n. If n is coprime to char(k), then under some assumptions on k(e.g. k is algebraically closed or a finite field) and on the geometry of the curve, we show that there is a local-global principle with respect to discrete valuations for the norms from the extension L/F.