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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…

Oh look, my band has a new single out today .. *nudge* *nudge* .. look at it, internet, look at it!
Slight change of pace with this one too, what started in our writing sessions as a prototypical melodeath track went somewhere very different by the time it reached the studio. Different influences were indulged, vocal style swapped, structure tweaked .. Cliché to say it but I'm proud of this one for being a true example of just throwing out the template, not rushing and letting th…