Punctured combinatorial Nullstellensätze

In this article we extend Alon’s Nullstellensatz to functions which have multiple zeros at the common zeros of some polynomials g1, g2, . . . , gn, that are the product of linear factors. We then prove a punctured version which states, for simple zeros, that if f vanishes at nearly all, but not all, of the common zeros of g1(X1), . . . , gn(Xn) then every residue of f modulo the ideal generated by g1, . . . , gn, has a large degree. This punctured Nullstellensatz is used to prove a blocking theorem for projective and affine geometries over an arbitrary field. This theorem has as corollaries a theorem of Alon and Füredi which gives a lower bound on the number of hyperplanes needed to cover all but one of the points of a hypercube and theorems of Bruen, Jamison and Brouwer and Schrijver which provides lower bounds on the number of points needed to block the hyperplanes of an affine space over a finite field.