Research

Research interests in analytic number theory and Diophantine equations.

My research lies in analytic number theory and concerns the existence and distribution of integral solutions to systems of polynomial equations. Given homogeneous forms \(F_1,\ldots,F_R\) of degrees \(d_1,\ldots,d_R\), a typical problem is to understand the counting function

\[ N(P)=\#\left\{\mathbf{x}\in\mathbb{Z}^n: \lVert\mathbf{x}\rVert_\infty\leq P, F_1(\mathbf{x})=\cdots=F_R(\mathbf{x})=0\right\}. \]

In favorable settings, the Hardy–Littlewood circle method yields an asymptotic formula of the shape

\[ N(P)=\mathfrak{S}\mathfrak{J}P^{\,n-(d_1+\cdots+d_R)} +o\!\left(P^{\,n-(d_1+\cdots+d_R)}\right), \]

where the singular series \(\mathfrak{S}\) and singular integral \(\mathfrak{J}\) encode the local arithmetic and real geometry of the system. My work studies quantitative forms of the Hasse principle for homogeneous and weighted-homogeneous equations. This involves pointwise and mean-value estimates for exponential sums, together with techniques from Fourier analysis and the geometry of numbers.

I have also studied Diophantine systems arising from magic squares, multimagic squares, and magic squares of powers. These problems connect classical questions in recreational mathematics with the analytic theory of systems of equations. Further directions that interest me include arithmetic statistics, sieve methods, and additive combinatorics.