{"paper":{"title":"Heuristic Relative Entropy Principles with Complex Measures: Large-Degree Asymptotics of a Family of Multi-Variate Normal Random Polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Michael K.-H. Kiessling","submitted_at":"2016-08-31T16:38:08Z","abstract_excerpt":"We study expected values of the polynomials $P_N^{}(z)=\\prod_{1\\leq n\\leq N}(X_n^2+z^2)$ whose $2N$ zeros $\\{\\pm i X_k\\}^{}_{k=1,...,N}$ are generated by $N$ identically distributed multi-variate mean-zero normal random variables $\\{X_k\\}^{N}_{k=1}$ with co-variance ${\\rm{Cov}}_N^{}(X_k,X_l)=(1+\\frac{\\sigma^2-1}{N})\\delta_{k,l}+\\frac{\\sigma^2-1}{N}(1-\\delta_{k,l})$. In principle these can be evaluated in closed form for arbitrary $N$, yet commonly available computer algebra handles only $N$ up to a dozen (due to memory constraints). A list of the first three expected polynomials shows that the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.08931","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}