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.
Mathematics Subject Classification (2000)
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