Block determinants of the two-locus sampling recursion: degree, positivity, squarefreeness, coprimality, and Galois groups
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…
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