Zhengheng Bao | Searcharxiv

arXiv2026

A formula for the rank over $\mathbb{Q}(t)$ of the elliptic curve $y^2=x^3+At^6+Bt^3+C$

In this paper, we give an explicit formula for the rank and generators (up to finite index) over $\mathbb{Q}(t)$ of all non-trivial elliptic curves of the form $y^2=x^3+At^6+Bt^3+C$, which is a larger class of elliptic surfaces than the one in Kloosterman's paper (2026) and Desjardins and Naskrecki's paper (2024), namely $y^2=x^3+At^6+C$. Our proof provides a new method to find this formula using generators of the geometric Mordell-Weil group while this geometric Mordell-Weil group (which is a $\mathbb{Z}[ω]$-module) no longer decomposes into rank-one submodules by the methods in Kloosterman's paper (2026) and Desjardins and Naskrecki's paper (2024). Moreover, our proof uses only a few elements in the Galois group of the coordinates of the generators and only light computations by hand.

math.NT↗

arXiv2021

A non-commutative Nullstellensatz

Let $K$ be a field and $D$ be a finite-dimensional central division algebra over $K$. We prove a variant of the Nullstellensatz for $2$-sided ideals in the ring of polynomial maps $D^n \to D$. In the case where $D = K$ is commutative, our main result reduces to the $K$-Nullstellensatz of Laksov and Adkins-Gianni-Tognoli. In the case, where $K = \mathbb R$ is the field of real numbers and $D$ is the algebra of Hamilton quaternions, it reduces to the quaternionic Nullstellensatz recently proved by Alon and Paran.

math.RA↗