On Fixed Divisors of Forms in Many Variables, I

Abstract

Let $D_{d,r}$ be the maximal fixed divisor of a primitive form of degree $d$ in $r$ variables over $\mathsf{Z}$. A formula is given for $D_{d,2}$ and estimates for $D_{d,r}$ for $r>2$. As a consequence, a question of Nagell raised in 1919 is completely answered.