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  "path": "/abs/2603.24890v1",
  "publishedAt": "2026-03-27T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "P. Horak",
    "I. Semaev"
  ],
  "textContent": "**Authors:** P. Horak, I. Semaev\n\nLet n denote the number of variables and m the number of equations in a sparse polynomial system over the binary field. We study the inconsistency probability of randomly generated sparse polynomial systems over the binary field, where each equation depends on at most k variables and the number of variables grows. Associating the system with a hypergraph, we show that the inconsistency probability depends strongly on structural properties of this hypergraph, not only on n,m, and k. Using inclusion--exclusion, we derive general bounds and obtain tight asymptotics for complete k-uniform hypergraphs. In the 2-sparse case, we provide explicit formulas for paths and stars, characterize extremal trees and forests, and conjecture a formula for cycles. These results have implications for SAT solving and cryptanalysis.",
  "title": "Inconsistency Probability of Sparse Equations over F2"
}