Punctured combinatorial Nullstellensätze
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In this article we present a punctured version of Alon's Nullstellensatz which states that if $f$ vanishes at nearly all, but not all, of the common zeros of some polynomials $g_1(X_1),\ldots,g_n(X_n)$ then every $I$-residue of $f$, where the ideal $I=\langle g_1,\ldots,g_n\rangle$, has a large degree. Furthermore, we extend Alon's Nullstellensatz to functions which have multiple zeros at the common zeros of $g_1,g_2,\ldots,g_n$ and prove a punctured version of this generalised version. Some applications of these punctured Nullstellens\"atze to projective and affine geometries over an arbitrary field are considered which, in the case that the field is finite, will lead to some bounds related to linear codes containing the all one vector.