engleski

Root separation for reducible monic quartics

We study root separation for reducible monic integer polynomials of degree four. Let H(P) be the height and sep(P) the minimal distance between two distinct roots of a separable integer polynomial P(x), and sep(P) = H(P)^{; ; -e(P)}; ; . We show that lim sup e(P) = 2, where limsup is taken over all reducible monic integer polynomials P(x) of degree 4. One part of our proof relies on the construction of a family of polynomials with very close roots. Coefficients in these polynomials are constructed using Fibonacci sequence. We also examine the case of families of polynomials which have polynomial and not exponential growth of coefficients.

root separation ; integer polynomials

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