Applied Hilbert's Nullstellensatz for Combinatorial Problems
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Bibliographic record
Abstract
Various feasibility problems in Combinatorial Optimization can be stated using systems of polynomial equations. Determining the existence of a \\textit{stable set} of a given size, finding the \\textit{chromatic number} of a graph or more generally, determining the feasibility of an \\textit{Integer Programming problem} are classical examples of this. In this thesis we study a powerful tool from Algebraic Geometry, called \\textit{Hilbert's Nullstellensatz}. It characterizes the \\textit{infeasibility} of a system of polynomial equations by the \\textit{feasibility} of a possibly very large system of \\textit{linear equations}. The solutions to this linear system provide \\textit{certificates} for the infeasibility of the polynomial system, called \\textit{Nullstellensatz Certificates}. \n \nIn this thesis we focus on the study of Nullstellensatz Certificates for the existence of \\textit{proper colorings} of graphs. We use basic ideas from \\textit{duality theory} to determine various properties of the Nullstellensatz Certificates. We give new proofs to several known results in the current literature and present some new results that shed some light on the relationship between the sparsity of a graph and the \\textit{size} of the Nullstellensatz Certificates for \\textit{$k$-colorability}.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it