Improved bounds in Birch's theorem for forms in many variables
We improve the best known result on the number of variables needed for the smooth Hasse principle for homogeneous forms of degree \(d\geq5\).
math.NT↗
arXiv subjects
Lena Wurzinger
Publications and source records attributed to Lena Wurzinger.
The divisor function for matrices
We introduce a matrix divisor function $τ_n(T,M)$, counting factorisations $AB=M$ for $n\times n$ integer matrices $A,B$ of height at most $T$. For a fixed non-singular $M$, or for the zero matrix $M=O_n$, we prove an asymptotic formula for $τ_n(T,M)$, as $T\to \infty$, using lattice point counting. We also prove an essentially sharp uniform upper bound for $τ_n(T,M)$, for an arbitrary non-singular matrix $M$.
math.NT↗