Greetings! Welcome to the homepage about my mathematical activities.

I work primarily at the interface of number theory, combinatorics and algebra. My research investigates instances of successes and failures of certain local-global principles in number theory. This also involves using p-adic methods to address a wide class of number-theoretic problems. My research has entered an especially exciting phase, and I am excited to share the following selection of my recent works.

My most recently submitted works are about the dynamical Mordell-Lang conjecture. In particular, in one of the manuscript, we confirm the conjecture for what is known as relatively Etale systems over arbitrary endomorphisms of products of smooth projective curves. In particular, we establish the conjecture for arbitrary endomorphisms of products of smooth projective curves. You can access the preprint of this work by clicking HERE. Likewise, another recently submitted work establishes the conjecture for higher-rank radially ramified skew products in arbitrary dimensions and arbitrary fiber rank. This same work also gives an application in the form of a uniform Skolem-Mahler-Lech theorem for a fixed nonlinear orbit and a fixed polynomial-coefficient linear recurrence. You can access the preprint associated with this work by clicking HERE.

Another recently submitted work of mine is about reverse-sparsity principles associated with polynomial factorization. In this work, we prove existence of infinitely many polynomials with exactly \(m\) non-zero coefficients – each irreducible factor of which has at least superpolynomially many (in \(m\)) nonzero coefficients. This general interest result places polynomial factorization in a radically different category as compared to other algebraic operations such as composition, taking powers of polynomials etc. for which reverse-sparsity principles were established by Schinzel, by Erdős, by Fuchs, Zannier and several others. This same result also concerns a weaker constant \(K(m)\) defined by Schinzel who showed with Choudhry that \(K(m) > \text{max} \big\{2m, 0.014m^{1.22} \}\) using a result of Verdenius. Our result implies that \(K(m)\) is eventually larger than any polynomial in \(m\). You can see a preprint of this work by clicking HERE.

I also recently submitted another work on a rigidity principle regarding local-global principle for \(n^{th}\) powers. The main result of this article, colloquially speaking, is that whenever the local-global principle for \(n^{th}\) powers fails, it does not fail too badly – in a precisely defined quantitative sense. The arithmetic results in this article, surprisingly, are obtained through coding-theoretic characterizations and covering arguments in finite-chain rings.

My MR author ID is 1438156. My ORCID and zbMATH author ID can be accessed respectively by clicking here and here. The best way to reach me is through email at : (myfirstname)(mylastname)2024@gmail.com