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Varadharaj R. Srinivasan

Publications and source records attributed to Varadharaj R. Srinivasan.

8 recordsLinked to original sources
arXiv2026

Liouvillian first integrals of differential equations: A Galoisian approach

Using differential Galois theory---specifically, Magid's theory of the complete Picard-Vessiot closure---we show that a system of first-order differential equations in $n$ variables admits a Liouvillian first integral if and only if it admits a first integral in a Liouvillian Picard-Vessiot extension, if and only if it admits a first integral in its Picard-Vessiot ring. Consequently, a system with a Liouvillian first integral admits one in a differential field obtained from the field of rational functions by first taking a finite algebraic extension, then adjoining an exponential of an integral, and then an integral. This yields a new proof of Singer's theorem on Liouvillian first integrals of autonomous planar systems (Trans.\ Amer.\ Math.\ Soc.\ 333, 1992) and of its recent extension to autonomous systems in $n$ variables by Aziz et al. (arXiv:2512.15522). Our results hold over any differential field of characteristic zero with an algebraically closed field of constants; in particular, they apply to non-autonomous systems. Finally, we exhibit a system that admits a first integral in a non-Liouvillian Picard-Vessiot extension but none in its Picard-Vessiot ring and thus establishing that the Liouvillian hypothesis cannot be dropped.

math.AC↗
arXiv2026

Elementary first integrals of integral differential systems

For a field $F$ of characteristic zero with a derivation $δ$, we provide a necessary and sufficient condition for a system of differential equations \begin{equation*} δy_1=f_0, \quadδy_2=f_1,\quad\dots\quad,δy_n=f_{n-1}, \end{equation*} where $f_0\in F$, $f_1\in F[y_1]$, $\dots$, $f_{n-1}\in F[y_1,\dots,y_{n-1}]$, to have elementary first integrals.

math.RA↗
arXiv2026

Iterative Derivations on Central Simple Algebras

We prove that an iterative derivation $δ_F$ on a field $F$ can be extended to an iterative derivation $δ_A$ on a central simple $F-$algebra $A$ if the characteristic of $F$ does not divide the exponent of $A$ in the Brauer group of $F.$ For a central simple $F-$algebra with an iterative derivation, we show the existence of a unique (up to isomorphism) Picard-Vessiot splitting field and from the nature its Galois group, we also describe the structure of the central simple algebra in terms of its $δ_A-$right ideals.

math.RA↗
arXiv2024

Differential Galois Groups of Differential Central Simple Algebras and their Projective Representations

Let $F$ be a $δ-$field (differential field) of characteristic zero with an algebraically closed field of constants $F^δ$, $A$ be a $δ-F-$central simple algebra, $K$ be a Picard-Vessiot extension for the $δ-F-$module $A$ and $\mathscr G(K|F)$ be the $δ-$Galois group of $K$ over $F.$ We prove that a $δ-$field extension $L$ of $F,$ having $F^δ$ as its field of constants, splits the $δ-F-$central simple algebra $A$ if and only if the $δ-$field $K$ embeds in $L.$ We then extend the theory of $δ-F-$matrix algebras over a $δ-$field $F,$ put forward by Magid & Juan (2008), to arbitrary $δ-F-$central simple algebras. In particular, we establish a natural bijective correspondence between the isomorphism classes of $δ-F-$central simple algebras of dimension $n^2$ over $F$ that are split by the $δ-$field $K$ and the classes of inequivalent representations of the algebraic group $\mathscr G(K|F)$ in $\mathrm{PGL}_n(F^δ).$ We show that $\mathscr G(K|F)$ is a reductive or a solvable algebraic group if and only if $A$ has certain kinds of $δ-$right ideals.

math.RA↗
arXiv2023

Liouville's Theorem on integration in finite terms for $\mathrm D_\infty,$ $ \mathrm{SL}_2$ and Weierstrass field extensions

Let $k$ be a differential field of characteristic zero and the field of constants $C$ of $k$ be an algebraically closed field. Let $E$ be a differential field extension of $k$ having $C$ as its field of constants and that $E=E_m\supseteq E_{m-1}\supseteq\cdots\supseteq E_1\supseteq E_0=k,$ where $E_i$ is either an elementary extension of $E_{i-1}$ or $E_i=E_{i-1}(t_i, t'_i)$ and $t_i$ is weierstrassian (in the sense of Kolchin ([Page 803, Kolchin1953]) over $E_{i-1}$ or $E_i$ is a Picard-Vessiot extension of $E_{i-1}$ having a differential Galois group isomorphic to either the special linear group $\mathrm{SL}_2(C)$ or the infinite dihedral subgroup $\mathrm{D}_\infty$ of $\mathrm{SL}_2(C).$ In this article, we prove that Liouville's theorem on integration in finite terms ([Theorem, Rosenlicht1968]) holds for $E$. That is, if $η\in E$ and $η'\in k$ then there is a positive integer $n$ and for $i=1,2,\dots,n,$ there are elements $c_i\in C,$ $u_i\in k\setminus \{0\}$ and $v\in k$ such that $$η'=\sum^n_{i=1}c_i\frac{u'_i}{u_i}+v'.$$

math.CA↗
arXiv2023

A Classification of First Order Differential Equations

Let $k$ be a differential field of characteristic zero with an algebraically closed field of constants. In this article, we provide a classification of first order differential equations over $k$ and study the algebraic dependence of solutions of a given first order differential equation. Our results generalize parts of the work of Noordman et al. (MR4378074) and complements the work of Freitag et al. (MR4506775).

math.AG↗
arXiv2021

Integration in Finite Terms: Dilogarithmic Integrals

We extend the theorem of Liouville on integration in finite terms to include dilogarithmic integrals. The results provide a necessary and sufficient condition for an element of the base field to have an antiderivative in a field extension generated by transcendental elementary functions and dilogarithmic integrals. We also study algebraic independence of certain dilogarithmic integrals.

math.GM↗