, Volume 29, Issue 5, pp 511–522 | Cite as

Punctured combinatorial Nullstellensätze

  • Simeon Ball
  • Oriol Serra


In this article we extend Alon’s Nullstellensatz to functions which have multiple zeros at the common zeros of some polynomials g 1,g 2, …, g n , 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 g 1(X 1), …,g n (X n ) then every residue of f modulo the ideal generated by g 1, …, g n , 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)

12E99 51E21 


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Copyright information

© János Bolyai Mathematical Society and Springer Verlag 2009

Authors and Affiliations

  1. 1.Departament de Matemàtica Aplicada IVUniversitat Politècnica de CatalunyaBarcelonaSpain

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