Joint IAS/Princeton University Number Theory Seminar

In the last two decades there has been much study of what happens when an algebraic curve in $n$-space is intersected with two multiplicative relations
$$x_1^{a_1} \cdots x_n^{a_n} = x_1^{b_1} \cdots x_n^{b_n} = 1 \tag{\(\times\)}$$
for $(a_1, \ldots ,a_n),(b_1,\ldots, b_n)$ linearly independent in ${\bf Z}^n$. Usually the intersection with the union of all $(\times)$ is at most finite, at least in zero characteristic. This often becomes false in positive characteristic, and I gave in 2014 a substitute conjecture and proved it for $n = 3$. I will describe all this together with more recent work with Dale Brownawell where we do the same for additive relations $(+)$; now an extra Frobenius structure has to be added, and there are no longer any direct analogues in zero characteristic.