Sources and novelty boundary
Checked September 22, 2026.
What Karp proves and what he conjectures
Primary source: D. Karp, arXiv:1203.1482, https://arxiv.org/abs/1203.1482.
The introduction labels the coefficient and stability assertions as Conjectures 1 and 2. Section 3 proves coefficient positivity for the shifted unit-parameter specialization. Section 4 labels the higher-order claims as Conjectures 4, 5, and 6. The manuscript here follows Karp's explicit multinomial normalization of P_n^r; the prose generating-function line in the source omits the n! appearing in its derivation. This positive scalar convention has no bearing on the counterexamples.
The new manuscript derives its arbitrary-shift two-row-minor identity and its higher-order Cauchy--Binet identity explicitly. It uses the standard rising-factorial Vandermonde identity, finite differences, Cauchy--Binet, and the classical Hurwitz criterion. Those tools are not claimed as new. The proofs are not transcriptions of Karp's special-case proof, which uses elementary symmetric functions and a coefficient sign-change argument.
Other primary sources checked
- Journal record: https://www.mathnet.ru/eng/znsl5267, DOI 10.1007/s10958-013-1438-y.
- Original Hurwitz article: https://gdz.sub.uni-goettingen.de/download/pdf/PPN235181684_0046/LOG_0026.pdf, DOI 10.1007/BF01446812.
- Author-hosted bibliography and later-work search results were consulted for orientation; they were not used as proof of an unresolved current status.
Searches included the paper title and targeted combinations of Karp, conjecture numbers, coefficient positivity, rising factorial, and Hurwitz stability. They did not identify a definitive later resolution of these exact claims in the materials inspected. This is not an exhaustive citation search and does not establish historical priority or that each claim remained open until this manuscript. The mathematical proofs and disproofs stand independently of a priority determination.
Earlier paper
The separate P56 manuscript is available at: https://github.com/Hwzw/algebra-number-theory-drafts/tree/main/papers/P56-karp-rising-factorial-zeros
Its positive-endpoint Conjecture 3 theorem is not contradicted by the present r=3 counterexample or by the PF_2-only stability input.