## Systems of bihomogeneous forms of small bidegree

### Volume 213 / 2024

Acta Arithmetica 213 (2024), 325-368
MSC: Primary 11D45; Secondary 11D72, 11P55
DOI: 10.4064/aa230525-23-10
Published online: 27 March 2024

#### Abstract

We use the circle method to count the number of integer solutions to systems of bihomogeneous equations of bidegree $(1,1)$ and $(2,1)$ of bounded height in lopsided boxes. Previously, adjusting Birch’s techniques to the bihomogeneous setting, Schindler showed an asymptotic formula provided the number of variables grows at least quadratically with the number of equations considered. Based on recent methods by Rydin Myerson we weaken this assumption and show that the number of variables only needs to satisfy a linear bound in terms of the number of equations.