Andrew Snowden

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math.NT2025

Polynomial Bounds for Birch's Theorem

Amichai Lampert, Andrew Snowden, Tamar Ziegler

Let $K$ be a number field and $f_{1}, \dots, f_{s} \in K [x_{1}, \dots, x_{n}]$ forms of odd degrees. In 1957, Birch proved that if $n$ is sufficiently large then the forms always have a nontr…

math.NT2024

Two improvements in Birch's theorem on forms

Amichai Lampert, Andrew Snowden

Let $K$ be a Birch field, that is, a field for which every diagonal form of odd degree in sufficiently many variables admits a non-zero solution; for example, $K$ could be the fiel…

math.NT2024

Two improvements in Brauer's theorem on forms

Arthur Bik, Jan Draisma, Andrew Snowden

Let $k$ be a Brauer field, that is, a field over which every diagonal form in sufficiently many variables has a nonzero solution; for instance, $k$ could be an imaginary quadratic…