Structured and Punctured Nullstellens\"atze

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 consider in detail: Terence Tao and Van H. Vu (2006), Uwe Schauz (2008) and Micha\l{} Laso\'n (2010) exclude more monomials, and recently, Bogdan Nica (2023) improved the result for grids with additional symmetries in their side edges. Simeon Ball and Oriol Serra (2009) incorporated the multiplicity of zeros and gave Nullstellens\"atze for punctured grids, which are sets of the form $X \setminus Y$ with both $X,Y$ grids. We generalize some of these results; in particular, we provide a common generalization to the results of Schauz and Nica. To this end, we establish that during multivariate polynomial division, certain monomials are unaffected. This also allows us to generalize Pete L. Clark's proof of the nonzero counting theorem by Alon and F\"uredi to punctured grids.