Undecidability of Polynomial Inequalities in Subset Densities and Additive Energies
Yaqiao Li
TL;DR
The paper extends undecidability results from graph homomorphism densities to additive combinatorics by formulating polynomial inequalities in subset densities and additive energies. It develops a framework of linear-form systems and quantum constructs to encode polynomials into density-like quantities, and uses Bollobás-type bounds to connect these quantities to graph-like extremal configurations, yielding undecidability for general density–energy inequalities. A Fourier-analytic upper bound on additive energy in terms of density is derived to enable a second undecidability result that restricts attention to densities and energies, via a reduction from Matiyasevich’s theorem. Together, the results show that determining universal nonnegativity of such polynomial inequalities is inherently undecidable in full generality within additive combinatorics.
Abstract
Many results in extremal graph theory can be formulated as certain polynomial inequalities in graph homomorphism densities. Answering fundamental questions raised by Lov{á}sz, Szegedy and Razborov, Hatami and Norine proved that determining the validity of an arbitrary such polynomial inequality in graph homomorphism densities is undecidable. We observe that many results in additive combinatorics can also be formulated as polynomial inequalities in subset's density and its variants. Based on techniques introduced in Hatami and Norine, together with algebraic and graph construction and Fourier analysis, we prove similarly two theorems of undecidability, thus showing that establishing such polynomial inequalities in additive combinatorics are inherently difficult in their full generality.