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Values of random polynomials at integer points

This material is based upon work while both authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California during the Spring 2015 semester, supported by the National Science Foundation Grant DMS 0932078 000.
JSA: Partially supported by NSF CAREER grant DMS 1559860, NSF grant DMS 1069153, and grants DMS 1107452,1107263,1107367 "RNMS: GEometric structures And Representation varieties (the GEAR Network)".
GAM: Supported by NSF grant DMS 1265695.
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