Multiplicative orders on varieties

In this talk we discuss intersections of varieties over the closure of Fp and small multiplicative groups, in the spirit of certain conjectures put forward by Poonen in xed characteristic. Our approach is based on transference to number elds using a quantitative Nullstellensatz and estimates on additive relations in multiplicative groups of low rank. We also establish stronger results `on average' for most p which may be seen as a renement of the well-known Erdos-Murty theorem related to the Artin conjecture. 1

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