Some Problems InvolvingRazborov-Smolensky

Several recent results in circuit complexity theory have used a representation of Boolean functions by polynomials over nite elds. Our current inability to extend these results to superrcially similar situations may be related to properties of these polynomials which do not extend to polynomials over general nite rings or nite abelian groups. Here we pose a number of conjectures on the behavior of such polyno-mials over rings and groups, and present some partial results toward proving them.