The real root of x*x*x - 156*x + 817 has a very large term round about the 800th position. Is there any explanation for this? It is of the order of 10**7.

I think that seeing occasional terms that are O(N^2) is to be expected, so this is somewhat unusual. If there's an analogy with the Brillhart/Stark situation, which I think <fade, fade> was for X^3 = aX+b with a,b in {6,8,10}, then one of (X+-c)^2 for small c (< 10 or 20) should have a much larger term, and it should appear earlier. Rich rcs@cs.arizona.edu