{"id":"palomar/PALOMAR-2026-10-29-000018","url":"https://github.com/cenozoic-works/rhone-theory/tree/babec8449a724c270450c7c0e2cd3985ccc2a989","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-29-000018-v1.json","repo":"https://github.com/cenozoic-works/rhone-theory","tags":["math.AC","math.NT","msc:12F10","msc:11R32","msc:12E05","msc:92D10"],"score":0,"title":"Block determinants of the two-locus sampling recursion: degree, positivity, squarefreeness, coprimality, and Galois groups","trust":"high","domain":"math.AC","source":"palomar","authors":["X. Lee","N. Volkov"],"reviews":0,"subject":"math.AC","summary":"This entry formalizes algebraic properties of the block determinants E(i, j) of the two-locus sampling recursion, viewed as polynomials in the recombination rate ρ. E(i, j) is the determinant of an explicit (min(i, j) + 1)-square tridiagonal matrix whose entries depend on i, j, the mutation rates θ_A and θ_B, and ρ. Its degree in ρ is min(i, j) under a stated nondegeneracy condition, and it is positive for i, j ≥ 2 with nonnegative parameters. For nonnegative real mutation rates, every root in ρ of E(5, 5) and of E(7, 7) is real and negative. At explicit rational parameters with negative…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-29","reactions":{},"formalized":true}