Smallest Known Measures of Two-Variable Polynomials

The following list records all known polynomials in two variables having Mahler's measure greater than 1 and less than 1.37. Most polynomials come from one of the following five families of six-term polynomials, or hexanomials:

P(a,b) = x^max(a-b,0)((1-x^a) + (1-x^b)y + x^b-a(1-x^a)y²)/(1-x),
Q(a,b) = x^max(a-b,0)((1+x^a) + (1+x^b)y + x^b-a(1+x^a)y²),
R(a,b) = x^max(a-b,0)((1+x^a) + (1-x^b)y - x^b-a(1+x^a)y²),
S(a,b,e) = 1 + (x^a+e)(x^b+e)y + x^a+by²,
T(f(x)) = f(x) + f^*(x)y.

A few sporadic polynomials are labeled using the rows of their coefficient matrix, for example, the third polynomial in the list denotes the polynomial 1+x+(1-x²+x⁴)y + (x³+x⁴)y².

  1.)  1.2554338662666087457   P(2,3)
  2.)  1.2857348642919862749   P(1,3) or P(2,1)
  3.)  1.3090983806523284595   [++000, +0-0+, 000++]
  4.)  1.3156927029866410935   P(3,5)
  5.)  1.3247179572447460260   T(1+x-x3)
  6.)  1.3253724973075860349   P(3,4)
  7.)  1.3320511054374193142   P(2,5)
  8.)  1.3323961294587154121   S(1,3,+)
  9.)  1.3381374319388410775   P(3,2)
 10.)  1.3399999217381835332   P(4,7)
 11.)  1.3405068829308471079   P(3,1)
 12.)  1.3497161046696958653   T(1+x2-x7); [+++0--]
 13.)  1.3500148321630142650   P(3,7)
 14.)  1.3503169790598690950   S(1,4,-)
 15.)  1.3511458956697046903   P(4,5)
 16.)  1.3524680625188602961   P(5,9)
 17.)  1.3536976494626355711   Q(1,6)
 18.)  1.3567481051456008311   P(4,3)
 19.)  1.3567859884526454967   P(5,8)
 20.)  1.3581296324044179208   [++00000, +0---0+, 00000++]
 21.)  1.3585455903960511404   P(4,1)
 22.)  1.3592080686995589268   P(4,9)
 23.)  1.3598117752819405021   P(6,11)
 24.)  1.3598158989877492950   S(1,6,+)
 25.)  1.3599141493821189216   T(1+x+x8); [+0-+0-+]
 26.)  1.3602208408592842371   P(5,7)
 27.)  1.3627242816569882815   P(5,6)
 28.)  1.3636514981864992177   S(3,5,+)
 29.)  1.3641995455827723418   T(1-x2+x5)
 30.)  1.3644358117806362770   [+000, 00++, ++00, 000+]
 31.)  1.3645459857899151366   P(7,13)
 32.)  1.3646557293930641449   P(5,11)
 33.)  1.3650623157174417179   S(2,7,-)
 34.)  1.3654687370557201592   P(5,4)
 35.)  1.3659850533667936783   [++000, ++0-0, 00000, 0-0++, 000++]
 36.)  1.3661459663116649518   P(5,3)
 37.)  1.3665709746056369455   P(5,2)    
 38.)  1.3668078899273126149   P(5,1)
 39.)  1.3668830708592258921   R(1,5)
 40.)  1.3669909125179202255   P(7,12) 
 41.)  1.3677988580117157740   P(8,15)
 42.)  1.3678546316653002345   T(1+x4+x11); [+-0+0-+0-+]
 43.)  1.3681962517212729703   P(6,13)
 44.)  1.3682140096679950123   P(1,9)
 45.)  1.3683434385467330804   [++00000, ++0-0++, 00000++]
 46.)  1.3687474425069274154   P(6,7)  
 47.)  1.3689491694959833864   P(7,11)
 48.)  1.3697823199880122791   S(1,9,+)

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