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Structured and Punctured Nullstellens\"atze
Erhard Aichinger, John R. Schmitt, Henry Zhan
https://arxiv.org/abs/2506.15281 https://

Structured and Punctured Nullstellensätze
A Nullstellensatz is a theorem providing information on polynomials that vanish on a certain set: David Hilbert's Nullstellensatz (1893) is a cornerstone of algebraic geometry, and Noga Alon's Combinatorial Nullstellensatz (1999) is a powerful tool in the "Polynomial Method", a technique used in combinatorics. Alon's Theorem excludes that a polynomial vanishing on a grid contains a monomial with certain properties. This theorem has been generalized in several directions, two of which we will co…