3 From zero testing to gap zero testing

(2) Show that we can convert an instance of that problem to an instance with a gap: that is, given a set of polynomials, create a new set of polynomials where at most a small fraction can simultaneously be made zero. Or, in other words, if there is no common zero, then most of the polynomials will be non-zero at any given point. This gap comes at the price of worse arity. The new polynomials we create each depend on all the variables (while each polynomial in the original set only depended on three). This will have to be dealt with so that we can effectively check the values of individual polynomials at a point.