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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.24890 |
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| _version_ | 1866912982911418368 |
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| author | Horak, P. Semaev, I. |
| author_facet | Horak, P. Semaev, I. |
| contents | Let 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24890 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Inconsistency Probability of Sparse Equations over F2 Horak, P. Semaev, I. Probability Computational Complexity Let 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 |
| topic | Probability Computational Complexity |
| url | https://arxiv.org/abs/2603.24890 |