Smyth's Conjecture on the Mahler Measure of non-reciprocal trinomials of height $1$

Abstract

We prove a conjecture of Smyth on the Mahler measure of non-reciprocal trinomials of height $1$, determining exactly when their Mahler measures are greater than or less than $M(x+y+1)=\rho =1.381356\dots$, conjectured to be the smallest limit point of $M(P)$ for non-reciprocal polynomials. Conjecturally, our result gives the full spectrum of $M(P)<\rho$ for all non-reciprocal polynomials, not just for trinomials.

arXiv commentcomments welcome