On uniform number theoretic estimates for fibers of polynomial maps over finite rings of the form Z/p^kZ

Let f=(f_1,...,f_m) be an m-tuple of polynomials with integer coefficients in n variables. We study the number of solutions #{x:f(x)=y mod p^k} where y is an m-tuple of integers, and show that the geometry and singularities of the fibers of the map f:C^n->C^m determine the asymptotic behavior of this quantity as p, k and y vary.

In particular, we show that f:C^n->C^m is flat with fibers of rational singularities, a property abbreviated (FRS), if and only if #{x:f(x)=y mod p^k}/p^{k(n-m)} is uniformly bounded in p, k and y. We then consider a natural family of singularity properties, which are variants of the (FRS) property, and provide each member of this family a number theoretic characterization using the asymptotics of #{x:f(x)=y mod p^k}/p^{k(n-m)}.

In order to prove our results, we use tools from the theory of motivic integration to effectively study the collection {#{x:f(x)=y mod p^k}/p^{k(n-m)}} in a uniform way.

Based on a joint work with Raf Cluckers and Itay Glazer.

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062